KEK Report 89-22 January 1990 H
Proceedings of the Workshop Topology, Field Theory and Superstrings"
KEK, Tsukuba, Japan November 6-10, 1989
edited by Makoto KOBAYASHI and Shin'ichi NOJIRI
NATIONAL LABORATORY FOR HIGH ENERGY PHYSICS © National Laboratory Tor High Energy Physics, 1990 KEK Reports are available from: Technical Information & Library National Laboratory for High Energy Physics 1-1 Oho, Tsukuba-shi Ibaraki-ken, 305 JAPAN Phone: 0298-64-1171 Telex: 3652-534 (Domestic) (0)3652-534 (International) Fax: 0298-64-4604 Cable: KEKOHO FOREWORD
The workshop "Topology, Field Theory and Superstrings" was held at KEK on November 6-10. This workshop was devoted to various recent developments in field theories and string theories, e.g. the deformations of conformal field theories, exactly solvable models, link theory, quantum gravity, topological field theories, superconductivity, etc. More than 50 talks, including several introduc tory lectures, were given. Many of 130 participants were young students. We hope that they generate the new currents in physics toward 1990's.
This workshop was supported by the Grant-in Aid for scientific research on Priority Areas from the Ministry of Education, Science and Culture. We wish to thank the speakers and participants for their efforts which made this workshop a successful one.
Makoto Kobayashi
Shin'ichi Nojiri CONTENTS
Conservation Laws in a Perturbed k=l SU(2) Wess-Zumino-Witten Model
K-I. Kobayashi, T. Uematsu i
On the Integrals of Motion away from criticality
J. Kodaira, Y. Sasai and H. Sato < g
Toda Field Theory
R. Sasaki 95
Algebraic Aspects of the Deformation of Conforroal Field Theory
M. Fukuma 27
On q-analog Racah-Wigner Algebra, Yang-Baxter Relations
and Witten's Quadratic Relation
M. Nomura 35
Link Polynomials, Linking number and Exactly Solvable Model
T. Deguchi 45
Exact Solution of the Fully Anisotropic Kondo Problem
Y. Quano 77
Comments on Super Schwarzian Derivatives
S. Matsuda 87
Lie Algebra Cohomology and N=2 Superconformal Field Theory
S. Hosono and A. Tsuchiya 109
Parafermions as Z^ orbifold and level 3 SU(3) Kac-Moody Algebra
A. Fujitsu 123 c=3d Algebra
S. Odake ^33
N = 2 Superconfoimal Symmetry based on N = 1 Snpersymmetric Non-Compact Group Current Algebra
S- Nojiri 143
Spontaneous Deformation of Maximally Symmetric Calabi-Yau Manifolds and A Realistic Four-Generation Model
T. Matsuoka I53
Torus-Orbifold Equivalence in Compactified String Theories
M. Sakamoto }gg
Scattering Amplitude of String on Orbifold
S. Nima ^79
Operator Formalism on Higher Genus Riemann Surfaces
S. Ojima 291
Renormalization Invariant Effective Action of the cr-Model String Theory
Y.X. Cheng 207
Symmetry of Cutoff Theory and Symmetry of String Theory
K. Sakai 219
Canonical Quantization of Witten's String Field Theory in Mid-point Time For malism
M. Maeno 229
Non-Polynomial Closed String Field Theory
K. Suehiro 241
— ii — Wormhole Solutions in the Skyime Model
A. Iwazaki 253
(2+l)-Dimensional Gravity with the Cosmological Constant
Y. Fujiwara and J. Soda 261
Linearized Analysis of (2+l)-dimensional Einstein-Maxwell Theory
J • Soda 273
BRS Current and Related Anomalies in Two-dimensional Gravity and String Theories
H. Suzuki 281
BRST Quantization of Chern-Simons Gauge Theory
H. Imai 283
Super Wess-Zumino-Witten Models from Super Chern-Simons Theories
N. Sakai and Y. Tanii 293
Topological Quantum Field Theories In Higher Dimensions
I. Oda 305
Action Principle for Chiral Bosons
K. Harada 323
Path-Integral Quantization of a Particle Coupled with Chern-Simons Gauge Field
C. Itoi 33!
Effective Actions of ID and 2D Heisenberg Anliferromagnels in CPl Represen tation
H. Mukaida 341
Nonlinear Sigma Model and Quantum Antiferromagnets
H. Yamamoto 351
— 111- "String Amplitudes": What Can We Do about the Divergent Integrals?
K. Amano oyy
Necessity of Finite Size Terms in WZW model
-coherent-state path integral approach-
S. Iso, C. Itoi and H. Mukaida ooq Conservation Laws in a Perturbed k=l 1. Introduction and Motivation SU(2) Wess-Zumino-Witten Model* In the last several years, there has been much interest in two-dimensional conformai field theories(CFT's) [1,2] in connection with the problems of string compactifications and also statistical models of critical phenomena. KEN-ICHIRO KOBAYASHI Conformally invariant field theories in 2-dimensions possess infinite dimen Lyman Laboratory of Physics, Harvard University sional conformai algebras known as Virasoro algebras which enable us to deter Cambridge, MA 02138, USA mine n-point correlation functions. This situation is quite in contrast to the conforrnal field theories in higher dimensions. The essential ingredient for the solvability here, in the 2-dimensional case, is the existence of the infinite number and of the conservation laws, which is more general criteria for the solvable models.
Two-dimensional conformai field theories describing the critical points of sta tistical systems correspond to the renormalization-group fixed-points in a larger TSUNEO UEMATSU set of 2-dimensional field theories which are generally scale non-invariant. Now it has become increasingly interesting to study 2-dimensional field theories away Department of Physics, College of Liberal Arts and Sciences from critical points or off-critical behavior of conformai field theories [3,4,S,22]. Kyoto University, Kyoto 006, Japan It has also been suggested that integrable lattice models carry an infinite dimen sional algebraic structure characteristic to CFT even away fxcm critical points ABSTRACT [61.
Since an infinite number of conserved quantities are crucial for solving the We study conservation laws in the SU(2) Wess-Zumino-WiUen (WZW) model theories exactly, as mentioned above, there has recently been numerous works on with the level k = 1 perturbed by a certain relevant operator. The perturbed the conservation laws in the conformai field theories away from criticality which system is a special case of the sine-Gordon theory to the lowest order in the are achieved by applying some relevant perturbations on the original theories perturbation theory, and it turns out that there exist extra conserved currents [7,15,16,17]. In fact, it has been known that there exist higher integrals of motion due to the SU(2) symmetry in the original WZW model. in perturbed conformai field theories such as minimal models of CFT and W- algebras [8,9].
On the other hand, a well-known type of solvable models apart from CFT's • Talk presented by T. Uemitaii at KEK Worbhop on Topology, Field Theory and is soL'ton theory. In ref.[10], Sasaki and Yamanaka investigated the higher inte Snperatrlng Theory, Nov.6lh - 10th, 1989. »• Fellow of Nishina Memorial Foundation grals of motion in the quantum sine-Gordon system. Their prescription to get
- 1 - -2- conserved quantities is writing down mutually commuting 'Polynomial'functions 2 S = ±Jd zdVdV (2) of the Virasoro generators. Recently Eguchi and Yang [11] have studied the deformation of the Virasoro minimum models perturbed by the (1,3) operator where v{z, z) = \{4(z) + fc)]. which leads to the sine-Gordon theory and clarified the connection between the This system is invariant under the transformation: f —»
and $ as: field theory [23].
In this talk, based on our work [18], we discuss the conservation law in the J+(z)=: e,V5«*> : J+{z) =: e^*" : SU(2) Wess-Zumino-Witten (WZW) model [12,13,14] with the level k = 1 and J-(j)=:e-^*(-"': J-(z) =: e-'^W: (3)
therefore with the central charge c = L perturbed by a certain relevant operator. 3 3 J [z) = -Ld4,{z) J {z) = -j=dfc) We will show that for a special value of the /? in the sine-Gordon theory, we get extra integrals of motion in addition to those obtained by Sasaki and Yamanaka where : : means the usual normal-ordered product. We note that they are suitably
[10], 2 single-valued under ^(^) —• 0(0) + 2irr at r = \/y/2. The operators J*, J satisfy the following operator algebras: 2. SU(2) WZW Model 1 2J3(u)) + J (*)'-(»)~ 77-^72 2 + Now let us consider the SU(2) WZW model with the level * for which the (1 — tu) (z — w) central charge is given by c = jjj^?.
The operator product expansion (OPE) reads
(z —iu)* (z-w)' z — w where we have used the following formula:
•/'(») , dwJ'(w)
WH~^ + ^ (l) ofl (z - w) : e""**') :: e''*W : = (z - iu) : e'«*W+>0«»). (5)
(z — w)2 z — w Here we normalize the holomorphic scalar field 0 as:
where T[z) and J"(z) (a = 1,2,3) denote the energy-momentum tensor and the <0(sMui)>=-ln(z-iii) (6) SU(2) currents, respectively.
The k = 1 i.e. c = 1 case is realized by a free boson - 3- -4 - 3. Perturbation and Sine-Gordon Equation and we obtain dA = \(AQ)-i + XdB (12) We now add to the action S of this theory a relevant perturbation term as t t follows: where S - X^ / $(vi,w)d~w$(v>,w)d2 . (7) B = E 7Ti»l'(-1)*~lfl'*'", x X 2 d, (8) 8tA = \6,X , (13) < >»= TX £ /•••/ -j < *M •••*(»«) >° <<« ••• y* withX = X' + B, since its integral gives a conserved charge (integral of motion): where Zx = £» „ J-~f%< *(»i) • • • *(y„) >o d'lft • • • dhM. Let A' be diA(z, z) and in the lowest order of perturbation theory we get •^idzA{z,z) = 0 (14) 8,A(z,i) = \d, f d'w[Aiz)i[w,ui)] (9) For a perturbation of the form We now introduce our notation for the operator product expansion: $ = eV*eil>* + e-#*e-Wi (15) k A{z)i{w, w)= £ (* ~ w) (A>»)k{w, w) (10) the lowest-order perturbative calculation leads to the sine-Gordon equation [11]. This can be seen with the suppression of the anti-holomorphic part as follows: where —IAt denotes the sum of conformal dimensions of A and In terms of this expression (10), equation (9) becomes B(d4>) = X6 J (PwlBtHeW* + «"**)] (16) r -'" = -tA/3(ew - e_,'M) dsA = Xd,{ d'w J^{z - w)-V*)-t("'."')] (11) Note that 8S* = 7sin(3« (17) k l d1(z-w)- = ~^dJ-S(:-w) where 7 = 2X0. -5- -6- Now we choose the value 3 = 1/V7: for these conserved currents in terms of the field 4>: j\++){z) =: e'J^«-> : The $ has the conformal weight ({, j) and it is a conformal field which is invariant jf(z) =-. {{dW - \{H)" - fwxa'fllM •• (20) under the Z* transformation:^ , 0 — - , — 4>. This theory is a perturbed k=l SU(2) WZW model and also a special case of a quantum sine-Gordon theory [11,11]. In the k=l case, the primary fields are only 1 and e" '*', and hence the ./{"'(j) =: «-a^«" : operator product can be written in terms of the descendants of those primary fields [20]. For example, the expression for j\ (r) is obtained from 4. Conserved Currents . eiy/24(i) .. eiV2*{v) ._. (z _ w)2 . giV^W+^/WM . _ (21) As discussed in refs.[10,ll], this system has an infinite number of conserved quantities. But, in our particular case with the perturbation (18) we obtain much Now we will show these quantities are in fact higher integrals of motion in more conserved quantities than in the general case. For conformal weight 4 we our perturbation theory. First of all, let us show that : is a have obtained 5 conserved currents(charge densities): conserved current under perturbation. Since y<++1 = (J+J% = ((•/+•/+)), :ea'/i*i'):Hw,tti) +) 3 + j\ = -2[(y+J )2 + [J'J+h] = -i((J J')h = -J-jy : e'ts** ... -^#(») . ! ; \/WM '^«») -^*C») 2 e + 2 ; e e : B) + + + 3 (r-w) (i-w) 4 = -2((j rh+{j-j h - ww,] = -mj nh - 2((/v)),] (22) we have j<-> = -2[( r j')2 + {J*J-h] = -<(irj')), j[-) = (j-j-), = ((J-j-)h Ui++)*)-, = |8(e'**.-**) (23) (19) where we have introduced the symmetrized product ((.45))^ = |[(>li?)jt+(B>l)i]. and therefore j\ is a conserved current. Possible another proof for conser- Namely, we have 5 local integrals of motion: ((J+ J+))i, ((J+J3))2, vatiun of j\ ' is to show that 0((J+J+))j is proportional to a total derivative {{J+J')h ~ 2((JVa))!, {{J-J3)), and{{J-J-)),. Among these 5 conserved bv taking into account, the conformal weight of ((J+^+))2 and the dimension of currents, J\°'(2) turns out to be identified with the weight 4 conserved current the coupling constant A following the Zamolodchikov's argument [7,8,9]. Conser found in the general case [10,11], To see this, we write down another expressions vation of the remaining currents can now be easily seen by noting the following - 7 - expressions What is remarkable here is that the conserved quantities (19) form a SU(2) multiplet. This is supposed to be due to a remnant of SU(2) symmetry existing 7<++) = (J+J+h , A+) = (J-(J+J+h)-x in the unperturbed system. j™ = (j-( j-( /+j+),)_ )_, = (J+(J+(J-J-M-i)-i <24> < 1 In refs.[10,ll], the energy-momentum tensor contains a Feigin-Fuchs linear + ++) 2 4"' = (J (J-J-)2).! , j\ = (J-J-h term d (jt in order to connect the sine-Gordon theory with the minimal theories for which c < 1. Their energy- momentum tensor which we denote by T is related The above relations can be proved by using the following 'Leibniz' rule whi'.h toourT = -\ :{d f = T + ^dJ3 = -^:(d^:+i^-d^ (28) {(ABUC), = (A{BC)i)-i - (BiAC).,), (25) where the coefficient of the linear term has been determined so that the e77"4' The conservation can be demonstrated by explicit calculation using the bo should be the screening operator. son representation. But more general arguement is provided by the following theorem. The weight 4 conserved quantity in the sine-Gordon theory is calculated to be Tk. If X is a conserved current or eouivalently if(X$)~\ is a total derivative, 2 J 4 then (J~X)-\ is also another conserved current. The proof goes as follows. By 7i = (ff)o=:-|(a « +jW) : (29) using the Leibniz rule (25) for I = -1 we have up to total derivatives. Since ((J-X).,*)-! = (J-fX*).,)., - (,Y(.T*)-i)-i (26) WHS1*) = -{d1*? + diWKO1*)} (30) Noting that in general (AdB).i is a total derivative, the first terra of (26) becomes fi coincides with the j\ ': J3 = 0 component of our weight 4 multiplet up to a total derivative. And also we note an overall factor and total derivatives. The conserved charges obtained from T4 and J^ are the same up to an overall constant factor. The conserved quantities (•/-$)_! =: e"'^*e7T* : (27) obtained in refs.[10,ll) are all expressed by T and J3 in our notation. Although we only investigated the case of weight 4, we expect that for higher conformal Since (Jf *)_i is a total derivative, (Xe ^T*)_i are also total derivatives. There weights these quantities form SU(2) multiplets as well. Thus it is conjectured fore the second term turns out to be a total derivative. Hence eq.(26) is also a that there exist an infinite number of conserved quantities which contain the total derivative, which implies that (J~X)-\ is a conserved current. From this conserved charges of ref.[10] as a subset. argument, starting from J\ , we can show the conservation of J^ ' and j[ '. In a similar manner, j\~' and /,{""' are seen to be conserved currents. - 9- -in- 5. Concluding Remarks ls] D. A. Kastor, E. J. Martinec and S. H. Shenker, Nucl. Phys.B316(1989)590; E. J. Martinec, Phys. Lett. 217B(1989)431; C. Vafa and N. Warner, It would be interesting to extend the present analysis to general simply-laced Phys. Lett. 218B(1989)51. algebras as Eguchi and Yang studied in the case of sine-Gordon theory. Secondly, [6] H. Itoyama and H. Thacker, Phys. Rev. Lett. 58(1987) 1395;Nucl. Phys. although we only studied the k = 1 case in this paper, it should be explored for B320[FS](1989)541. higher levels. In particular, the k = 1 case is described by three fermkms [21], and [7] A. B. Zamolodchikov, JETP Lett.46(1987)160. the explicit computation might be carried out. More generally, arbitrary level k could be treated by boson representations. Although our analysis is still at [8] A. B. Zamolodchikov, Kiev preprint ITF-87-65P(1987). preliminary stage, it would be intriguing to study the possible connection of the [9] A. B. Zamolodchikov, 'Integrable Field Theory from Conformal Field The extra conserved charges with spectrum and S-matrbc structures of sine-Gordon ory', in the Taniguchi Conference on Integrable Models(1988). theory. In this connection, from the argument in refs.[24,22], other special values [10] R. Sasaki and I. Yamanaka, in Advanced Studies in Pure Mathematics 16 of j3 axe v'V" for n = 3,5, • • • (n = 4 is the present case), for which we expect (1988)271. extra conserved charges. Finally, the present analysis is totally based on the lowest-order perturbation theory. Therefore it is extremely important to see how [11] T. Eguchi and S. K. Yang, Phys. Lett. 224B(1989)373; University of Tokyo the present result will be affected by possible higher-order corrections. preprint, UT-554(1989). [12] E. Witten, Commun. Math. Phys. 92(1984)455. This work was partially supported by the Grant-in-Aid for Scientific Re [13] A. M. Polyakov and P. B. Wiegmann, Phys. Lett. 131B(1983)223 ; 141B search from the Ministry of Education, Science and Culture (#63540216 and (1984)223. #591160202066). [14] V. G. Knizhnik and A. B. Zamolodchikov, Nucl. Phys. B247(1984)83. [15] J. L. Cardy, Phys. Rev. Lett. 60(1988)2709. [16] J. Kodaira, Y. Sasaki and H. Sato, Hiroshima University preprints(HUPD- References 8902 and 8912,1989) [1] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Fhys. 241 [17] P. Griffin, Fermilab preprint, FERMILAB-PUB-89/153-T(1989). (1984)333. [18] K. Kobayashi and T. Uematsu, Fermilab preprint, FERMILAB-PUB- [2] D. Friedan, Z. Qiu and S. Shenker, Fhys. Rev. Lett. 52(1984)1575. 89/174-T(1989). [3] A. B. Zamolodchikov, JETP Lett.43(1986)730 ;Sov. J. Nucl. Phys. 46 [19] P. Ginsparg, Nucl. Phys. B295(FS211(1988)153. (1987)1090. [20] R. Dijgkraaf, E. Verlinde and H. Verlinde.Commun. Math. Phys. 115 [4] A. W. W. Ludwig and J. L. Cardy, Nucl. Phys. B285(FS19](1987)687. (1988)649. -il -12- [21] A. B. ZamolodchiJcov and V. A. Fateev, Sov. J. Nucl. Phys. 43(1986)657. [22] T. Eguchi, "Deformation of Conformal Field Theory ", Talk given at this Workshop. [23] R. Sasaki, Toda Field Theory ", Talk given at this Workshop. H. W. Braden, E. Corrigan, P. E. Dorey and R. Sasaki, Phys. Lett. 227B (1989)411; Preprints RRK89-26(UDCPT 89/35) and RRK-89-29 (UDCPT 89/53), 1989. [24] V. A. Fateev and A. B. Zamolodchikov, "Conformal Field Theory and Purely Elastic S-Matrices ", Landau Institute preprint (1989). -13- 1. Introduction In general almost all IM which exist at the fixed point are destroyed away from On the Integrals of Motion away from criticality criticality. However it was pointed out by Zamolodchikov that an infinite set of nontrivial conserved currents may survive for perturbed CFT by a particular field. Therefore one can get a non-conformal field theory which is integrable. This fact has inspired many efforts to study the problem of deformations of CFT.' One of the recent results is the paper by Eguchi and Yang. They pointed out an interesting relation between the deformation of CFT by the (1,3) operator and the sine-Gordon systems in the lowest order of perturbation theory. J. KODAIRA, Y. SASM AND H. SATO In the deformation of CFT with the central charge c = 1 — 6/p(p+l) where p = Departmen' if Physics, Hiroshima University 3,5,..., the existence of higher order terms in IM was noticed by Zamolodchikov "' Higashisenda-machi, Naka-ku, Hiroshima 730, Japan using power counting arguments. However the exact value of them has not yet been obtained.' Furthermore it is quite non-trivial whether higher order terms prevent the existence of IM for the higher spin currents. In order to prove the existence of the infinite number of IM and related problems for such models, it is necessary to reveal the structure of higher order corrections. ABSTRACT 2. Origin of higher order terms We discuss higher order corrections in the integrals of motion for the unitary minimal model with the central charge c = 1 - 6/p(p + 1) perturbed by the (1,3) In this section we shall give a framework to study IM and clarify how the higher operator. After clarifying their origin, we give an explicit expression in the case of order corrections appear in IM. Let us consider the action following to tef.[2], p=5 for the spin 4 current. *-*>-i?/W*' (21) where HT is the action of the unitary minimal CFT with the central charge c = 1 - .(Ayi (P = 3,4,..,) and *. (* for short) is the scalar primary field with its conformal dimension being (1 — -^-, 1 — -£y). For this system it was suggested that one can construct some local fields which satisfy the conservation laws, -15- -16- d,T, = 9.Q,.,, (2.2) 8, hits the step function H[\: — y]~ — a-), s denotes the spin and r, is a differential polynomial of the stress tensor T e.g. d-M-- - y.-l1 - -5) = (-" - »,•)*(!= - y,l2 -"2). T, = r, T =: T1 : etc.. In general the operator Q has the following structure,"1 4 since there is no ^-dependence in the correlation functions . Inserting the OPE of T, with $, 1 Q, = AG! + A'r a./f;. (2.3) ^(--W^=E(7r^rA^W. The second term of eq.(2.3) appeal a only when p is an odd number and HJ = 0 by the power counting arguments. The index i runs over the independent operators into eq.(2.4), we get //] which will appear with spin a. The explicit form of the first term was discussed in ref.[4]. In order to find conservation laws, we follow the perturbative method. The correlation function will be estimated by expanding it into the perturbative series as follows; i , 2 2 x{z-y)S(\z~y\-a)d yd\...d y„, (2.5) where ( )g means the correlation with the cut ofT functions, ( )H = (r,(r,z).Y(x,x)) ( ) H{\y - JTjl2 - a2)H[\y - y,\2 - a2).... In the above calculations, we as sumed x ^ z {\x — z\ > o). When integrating with respect to y, only the single = \ £ / ^T~ where .Y is a product of arbitrary fields and Z = (l)v In the right hand side, the correlation is evaluated in the unperturbed theory using the operator product WXh-zZJ %W^W (2.6) expansion (OPE) of CFT. Now it is important to note that the r.h.s. of eq.(2.4) n! l J is not well defined due to the ultraviolet divergences. So we adopt the cut off x dr (XAm»(.-)«(»I)... •(*,)), d\ ... d y„. 1 regularization.'" ' We insert a sufficient number of step functions into the r.h.s. of Since we consider the situation z / x, the existence of the conservation law means, eq.(2.4) to regularize them, H(\z - ^ - a*)H{\,A -yf- a')H(\y, -„/-»')... , Therefore we transform the r.h.s. of eq.(2.6) such that only one derivative d. to sit outside the correlation function. When moving d. into (if m > 3) or outer (if where a is a short-distance cut off parameter. When operating d, to the both sides, t 2 and v« are separated by a. t Only the angular integrations remain due to the ^-function. -17- -18- m = 1) the correlation function, 0, hits also the step functions. This procedure of the higher order terms at order A"'+1"2. There is no higher order terms when p leaves some contributions to the resulting expressions, is an even number. Fallowing the above formalism let us consider the spin 2 current. From eq.(2.6) one can write, S.-( >„ = («.. Jtf + D-'-SiWI---»,•!' -»')< >*• (2-7) 1 This is equivalent to taking into account the equal-time commutator terms tn the operator formalism. And these contributions turn out to be the higher order cor (2 8) rections to Q .. For just the same reason stated before eq.(2.6), the second term + (BfiMtiJ... Taking account of the conformal dimension of $, it is easy to recognize that 8, (TA'l^Ml-AJfl, <**),,. only the 2±! point contractions in the correlators of eq.(2.6) can produce a single pole after integrating w.r.t. y ,... , y^ , For example, in the case of p=3, x by the power counting argument and an explicit check of this result is given in 1 ref.[9]. Next we consider the spin 4 current Tt =: T :. We can write down eq.(2.6) {•(.)•(»)> = jj^jr as follows; And in the case of p=S, $ has the conformal dimension A = A = 2/3 and the single pole in : — y. will appear from the following fusion /' *(*)*(y,)*(y,) In order to see this singularity, let us integrate the three point function, - d, {X(Llx + 2AX_,)*(.-)*(9l).. )u (2-9) / c + ^j ("_,*-,•(*)*(»,) • • •),]*», • • • **. • •h I'll In the following section we show how to calculate the contribution from the second r'(i - A)r(2A-i) 1 ••irC 3 term of eq.(2.7) and determine the higher order corrections to IM in the case of r (A)r(2-2A) |s — j/jI'tSA-i) • p = 5. where C is the coefficient function for $3."' Putting A = 2/3 we can see the single pole structure in the anti-holomorphic part. Hence we understand the appearance -19- -20- 3. The spin 4 current with p=5 + £ds J (dM'Wy,Wy,)T[x))a In principle we may perform the calculation in the same way as in the p=3 x {= - Sl)«(|r - y,f - a=)ffj<*V»j, (3.2a) 1 1 case. ' In the case of p > 5, however, the situation becomes complicated consider A ( 9 h = £ / -^f- > (O^zWy,) • • • *to.>T(*)>ff A, • • • d*yn ably compared to the p=3 case. The extra (higher order) contribution is produced only after the partial integration of the (p + l)/2 point function as explained in - ^dtJ{Dii(z)9(y1Wy,)Tlx))a sect.2. Therefore the information on the OPE of Am$ and $ is not enough to X (z - y,W|r - y,|! - J)H d\d\ , (3.2b) determine the higher order corrections. The best thing we can do is to calculate 2 the value of the correlation function. We can identify an operator uniquely which where Dz = d\ arid DA = £_,. The second terms of eq.(3.2) contribute to reproduces the same result consulting the structure of the operator algebra at criti the higher order terms in IM. Substituting the amplitudes and performing the cality. Since X in eqs.(2.8) and (2.9) can be an arbitrary field or products of fields, integrals1'1 we obtain we choose X to be an appropriate field to calculate the correlation functions. For the spin 4 current, it is sufficient to calculate the correlation function with 3 I, = \8S {d] Now choosing ,Y to be T(i), let us consider the spin 4 current with p=5. From The second term {x — z)"* can be interpreted as the correlation function eq.(2.9) we must consider the following integrals: -c(T[2)T{x)) due to the operator algebra. Therefore we can conclude A { 1 h" £ [ -^-8U8, /,-/,- Ad, (fl»«(z, z)X)k + A'-^jfd + 10*)fl, (T(r, z)X)x . (3.3) A '3 = £ / ^ <#•(-)*(»,) • • • *(Vn)X)„ d\ ...d*ynl (3.1c) After tediua calculations for I% and It in the same way as /j, we finally obtain the h = AEIr{-^(8,i. i>{'WyO-Myn)x} d\...d'y . (3.id) 1 B n expression for Q}; When the derivative 8, is moved in the r.h.s. of eq.(3.1), we get the expressions 3 for /. Q7{z, z) = M^L^tyz, z) + a,Li^(z,!)) + A oT(r, z), (3.4) where / / A a d 1 ="i = E/ ^f" ' («i»(--)*(».) • • • •(».m«))jr \ • • • *V» -21- -22- terms (m > 2) of OPE have already at least one derivative outside the correlation function. So the simple pole term is relevant to the dangerous one. A calculation for the higher spin current would be interesting related to the above problems. ACKNOWLEDGEMENTS 4. Conclusion and Discussion We thank K.-J. Hamada and S.-K. Yang for useful suggestions and valuable discussions. In this paper we have shown how the higher order terms in IM are generated and how to calculate them for the minimal CFT perturbed by (1,3) operator. A careful REFERENCES treatment of the ultraviolet divergences was shown to be necessary to determine the higher order terms. The explicit expressions were obtained in the case of p=3 1. A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nud. Phys. B241 and 5 for the lower spin currents. The higher order corrections to the higher spin (1984) 333. currents and also the higher p(= 2n + 1) case can be estimated just in the same 2. A.B. Zamolodchikov, Kiev preprint ITF-87-65 P, Kiev(1987). way apart from the following technical complexity; 3. A.B. Zamolodchikov, JETP Lett. 46 (1987) 160. (i) we need the (p+ T)/2 point functions in the case of general p. Moreover the 4. A.B. Zamolodchikov, Integrable Field Theory from Conformal Field Theory, (p — l)/2-integrals correspond to the calculations of (p — l)/2-loops diagrams. in : the Taniguchi Conference on Integrable Models (1988). (ii) we must evaluate correlation functions including various ,Y's (denoted by 5. A. Leclair, Princeton preprint PUPT-1124(1989); A'.) which are chosen appropriately to discriminate different terms in C3 _ for the J J B.A. Kupershmidt and P. Mathieu, Phys. Lett. 227B(1989)245; higher spin currents. For example, for the spin 6 current, the higher order terms P. Griffin Fermilab preprint FERMILAB-PUB-89/153-T (1989); are dT2 and 33T, and we choose -Y, to be T3 and d'T. Then we obtain the K. Kobayashi and T. Uematsu, preprint FERMILAB-PUB-89/174-T(1989); simultaneous equations for the coefficients of the higher order terms. P. Christe, preprint UCSBTH-89-29 (1989), and references therein; However the method explained in this paper seems to be the only one to obtain H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Preprint RRK 89-26, the higher order corrections as far as we follow the perturbalive formulation of the UDCPT 89/35 (1989); and references therein. problems. 6. T. Eguchi and S.-K. Yang, Phys. Lett. 224B (1989) 373. For the higher spin currents (» > 10), there exist operators which are not total 7. R. Sasaki and I. Yamanaka, in Advanced Studies in Pure Mathematics 16 derivatives in the right hand side of cq.(2.2) by the power-counting arguments. (1988) 271. Those dangerous terms which destroy the conservation laws appear only in the 8. J. Kodaira, Y. Sasai and H. Sato, Mod. Phys. Lett. A (1989), in press. higher order corrections.''' Therefore one should examine whether those dangerous terms vanish or not. 9. J. Kodaira. Y. Sasai and H. Sato, Prog. Theor. Phys.83(1989), in press. As we have shown in sect.2, the higher order terms appear when we move 10. J.L. Card)', 'Conformal Invariance and Statistical Mechanics' Les Houches the operator 0. into or outer the correlation functions. However the higher pole Summer school on Field, Strings and Statistical Mechanics, (1988). -23- -24- HTTP Hiroshima University Toda Field Theory RYU SASAKI Research Institute for Theoretical Physics Hiroshima University, Takehara, Hiroshima 725, Japan The subject of this talk is covered by the following references: [1] H. W. Braden, E. Corrigan, P. E. Dorey and R. Sasaki, 'Extended Toda Field Theory and Exact S-Matrices', Phys. Lett. B 111 (1989) 411-415. [2] H. W. Braden, E. Corrigan, P. E. Dorey and R. Sasaki, 'Aspects of Perturbed Conformal Field Theory, AfBne Toda Field Theory and Exact S-Matiices', Proc. of the XVIII International Conference on Differential Geometric Meth ods in Theoretical PhyBics: Physics and Geometry, Lake Tahoe , USA 2-8 July 1989. preprint RRK 89-26 (1989). [3) H. W. Braden, E. Corrigan, P. E. Dorey and R. Sasaki 'Affine Toda Field Theory and Exact S-Matrices', NucJ. Phys. B (1990) in press, preprint RRK 89-29 (1989). [4] II. W. Braden, E. Corrigan, P. E. Dorey and R. Sasaki, in preparation, -25- Algebraic Aspects of the Deformation of The (Euclidian) quantum field theory is characterized by the following constituents: 4>(z,z) = (tj>(z,z)) : fundamental fields (1) Conformal Field Theory* t 2 5[0] = j d zC{ Masafumi Fukuma* The vacuam expectation value of a functional A = A[ Department of Physics, University of Tokyo the path integration Bunkyo-ku, Tokyo 113, Japan The (improved) energy momentum tensor is defined through the action S[$], and if translation^ invariance is required to be preserved after quantization the following identity holds: Abstract We point oat that the Tod a lattice hierarchy known in soli ton theory is rel 9,(ru,j)-..) + a,(0(=,5)...) = o, (4) evant for the description of the deformation of rational conform al Held theories BI ,e, , . W]9(»,£)-.«-»M the system perturbed from Q coset conformal model. SO. (6) In this theory, eqs. (4) (5) mean that rfi.i) (resp. T{z,z)) is holomorphic (resp. anti- holomorphic) in the bracket { )0 : (r)0 = {T[z))a (resp. (f )o = (f^Do). Furthermore, "Talk presented at the Workshop on Field Theory, Topology and Suptrslring, Nov. 9-13, 1989 the field T{z, z) has the following operator product expansion: KEK. Tsnknba, Japan 1 E-mail address: ftikumaOtkyvax.hepnet (r,I,rW..0oH(^ + ^ + ^ + Ie,,..,D, (7, where c is the central charge. -27- -28- Deformation of conformal field theory is defined by adding a term to the CFT action In the following, we will consider field theoretical models, so that the number of S„[# degrees of freedom will be infinite, labeled by the spatial coordinate o\ Furthermore, in 2D field theory, /(r) is conserved if it is expressed as the integral of the time component S,M = S [fl-j/d»z *(*,*)• (8) 0 of a conserved current: Henceforce, we assume, for simplicity, that the deformation field $(2,z) is a primary I = jda JA",T), (12) field with conformal weight [h, h = h) in the bracket ( )o: aTJT{0,T)-d„JAo,T) = a, (13) (T(»)*(ii(, w) • • -)o = ((, ' ' + v + reg.) •••)„. (9) \z — u/j' z — w where a and r are spatial and time coordinates on a cylinder, respectively. After Wick's It is easy to show that, in the perturbed system, the trace of the energy momentum rotation and mapping to the plane, this conservation law is represented as tensor, 0(z, z), is proportional to the deformation field $(s, z) up to order g2 [1]; a,j{z, z) + a,J(z, s) = o (j s J„ J s J,). (u) be checked whether there exist a sequence of pairs (Jnt Jn) with a property fl,(J„(z,S)), + e,(/.( ,z)), = 0. (15) 2D-CFT is known to be integrabte and plays an important role in the classification i of 2D critical phenomena. Our problem is whether the perturbed system (away from Before starting to find out the conserved currents (Jn, J„), it is useful to notice that criticality) is also integrable. CFT has an infinite number of conserved currents. In fact, through the holomorphy of First of all, we must clarify the concept of lntegrablllty [5]. In the classical Hamil- the energy momentum tensor T(z), the pair (.//(z), 0), J/(z) = f{z)T{z), is a conserved tonian mechanics, to solve a physical system means to decide an orbit for time evo current for any holomorphic function /(z): lution (specified by a Hamitonian H{p,q)) on the phase space r3„ (with coordinates (Pii fl')r * — li *' • in)- We can solve this problem easily if there exist n independent M->l{'))<> = 0- (16) {unctions /j(p, 0iU.t>,*))o = u. (17) and the Hamiltonian depends on p, q only through them; H = ff (/). A system is called completely integtable when there exist such (mutually commuting) conserved quantities In this limit, commutativity is expressed formally as {/,), and [ip, I) which are obtained from (p, q) by a coordinate transformation on V7„ [fdzJ„(z),fdzJ (z)] = Q. (18) are known as the action-angle variables. m -29- -30- Now our concern is to know whether such pairs (Jn, Jn) exist for various deforma system, by showing the following scheme tions caused by primary fields $(s,z). In general, of course, perturbed systems do not Toda lattice hierarchy D KP hierarchy allow the existence of such integrals of motion except {Jt = T, J\ = 0), which comes Q teduction $ reduction from the translatianal invariance. Zamolodchikov, however, noticed [1] that, in the case Q Toda field theory D g KdV-type theory (20) of coset conforma! model Gk © Gi/Gk+i (where Gk is an affine Lie algebra with level k), perturbed conformal field theory D unperturbed chiral conformal field theory. there exist several conserved currents in the perturbed system which is deformed by Furthermore, by studying the structure of conserved currents, this unperturbed system $ad}0inf • Here $adSoini' is the primary field which corresponds to the Virasoro character can also be described by the Toda field theory associated with simpk Lie algebra Q. xirfjomc appearing in GKO decomposition of characters of trivial representations of &and&: Let us explain the meaning of the above diagram. Far more detailed explanation, see refa. [3][4]. R Toda lattice hierarchy (TLH) is the system from which Toda field theories associated Eguchi and Yang then pointed out that, if coset conformal model is represented by with various affine Lie algebras can be obtained by so-called reductions (G reduction free bosons, the deformation field $ = $ : «'*•* : corresponding to the 0-th simple root a0 of ff, and that the perturbed system the KdV-type theories can be obtained by similar reduction pocedure, and the reduced will be described by the Tod a field theory associated to the same affine Lie algebra 0 system has the symmentry of affine Lie algebra G {Q reduction of KPH) [7][8]. One of PI- the most prominent properties, which is relevant to the deformation theory, is that KPH can be obtained from TLH by formally neglecting the dependence on antiholomorphic It is recognized that the most suitable and powerful method to construct the con coordinate, and that Q reduction of TLH leads to the {? reduction of KPH [6]. One can served quantities and find the action-angle variables, is the inverse scattering method, further show that the holomorphic part of G Toda field theory is connected with the developed by Leningrad school. Furthermore, this method is known to be applicable to G KdV-type theory through the generalized Miura transformation, so that the KdV- the Toda field theory. Thus, it is expected that the integrable deformation of conformal type system is completely determined by the representation theory of the W algebra field theory can be systematically dealt with in the framework of the inverse scattering associated with G {W<* algebra) [3][4]. On the other hand, it is known that, for special method of the Toda field theory. Then, however, there will arise the following prob values of the central charge, representation space of the W$ algebra corresponds to the lem: what corresponds to the unperturbed canformal field theory? We point out that HUbert space of the coset conformal model constructed from G [9][10][ll]. This agrees the (quantum) KdV-type system corresponds to the chiral sector of the unperturbed with our assumption that the starting (unperturbed) CFT is G coset conformal model, thus the commutatibity of this diagram (20) is assured. This scheme is shown by using a semiclassical argument based on the classical in- -31- -32- verse scattering method of the Toda field theory, that is, commutatibity of conserved 1090; Intern. J. Mod. Phys. A 3 (1988) 743; A 4 (1989) 4235. quantities means the one with respect to a Poisson bracket {, }. Quantization of our [2] T. Eguchi and S.-K. Yang, Phys. Lett. B 224 (1989) 373. scheme (20) is now investigated and is commented at the end of the ref. [4]. [3] M. Fukuraa, T. Takebe, Univ. of Tokyo preprint UT-547 (1989). Finally, we mension the conserved currents (J {z, 5), J {z,z)), n n [4] M. Fukuma, Univ. of Tokyo preprint UT-548 (1989), to appear. a y„( ,5) + 3,J„(.',i) = o, (21) i J [5] L. Faddeev, in: Recent Developments in Field Theory and Statistical Mechanics, in the perturbed system. eds. J.-B. Zuber and R. Stora (Elsevier Science Publishers, 1984) p. 562. There exist an infinite number of the conserved currents with the following prop [6] K. Ueno and K. Takasaki, Adv. Stud. Pure. Math. 4 (1984) 1. erties: (1) Holomorphic part of them, Jni can be written completely in terms of the [7) M. Sato, lectures given at the Univ. of Tokyo (1982). W algebraic currents while the antiholomorphic part Jn cannot. (2) All of the /„ are proportional to jj = 0, which can be identified with the deformation field $ by (10), [8] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, in: Proceedings of RIMS Sympo and further (3) the J\ = 0 is nothing but the vertex operator corresponding to the sium, eds. M. Jimbo and T. Miwa (World ScienRtic, 1983) p. 39. D-th simple root of Q. These facts (l)-(3) also assure that (A) the unperturbed system is described by the Q coset conformal model which corresponds to Q Toda system, and [9] A. B. Zamolodchikov, Theor. Math. Phys. 65 (1986) 1205. that (B) the integrable deformation is caused by the vertex operator associated with [10] V. A. Fateev and S. L. Lykyanov, Intern. J. Mod. Phys. A 3 (1988) 507. the 0-th simple root of Q, [11] V. A. Fateev and A. B. Zamolodchikov; Nucl. Phys. B 280 [FS18] (1987) 644; F. A. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, Nucl. Phys. B 304 Acknowledgements (1988) 348, 371; I would like to thank Prof. T. Eguchi for valuable discussions and encouragement S. Mizoguchi, Phys. Lett. B 222 (1989) 226; Univ. of Tokyo preprint UT-538 throughout this work. I also acknowledge Profs. A. A. Belavin, H. Ooguri, K. Ueno, (1989). A. B. Zamolodchikov and Drs. A. Kato, H. Kunitomo, S. Mlzoguchi, E. Murayama, S. Odake, K. Ogawa, K. Takasald, S. -K. Yang for useful discussions and comments. In particular, 1 thank Dr. T. Takebe for stimulating discussions at an early stage of this work. References [1] A. B. Zamolodchikov, JETP Lett. 43 (1986) 730; Sov. J. Nucl. Phys. 46 (1988) -33- -34- I. Introduction In recent years, the 8-j symbol (I.e. the Racah coefficient) On q-Analog Racah-Wigner Algebra, Yang-Baxter Relations has proven [1-3] to be a solution to Yang-Baxter (Y-B) relations and Wltten's Quadratic Relations for IRF (interaction round a face) model [4,51. The 6-j symbol generally means recoupling coefficients of quantum groups [6-81. Masao Nomura Here, we first give a summary of q-analog W-R algebra of SUq(2) in connection with Y-B relations: W-R algebra means the tensor operator algebra based on n-J symbols, Wlgner-Eckart Institute of Physics, College of Arts and Sciences, theorem, etc. [9] Next, we discuss two sets of quadratic rela University of Tokyo, Komaba. Tokyo, 153 tions, one given by Woronowicz [10] and the other by Witten til). The quadratic relations are suited for formulation of q-analog Wigner-Eckart theorem [121. and their realization by boson operators [13] are easily obtained. Specification of the boson operators Is simpler than that of the standard one [14.15]. We abbreviate discussions on Y-B relation for vertex model, Abstract: on W-R algebra Involving rapidity parameters [16] and boson Racah-Wlgner (R-W) algebra of quantum group SUq(2) is summarized operator realization of the quadratic relations. in connection with Yang-Baxter (Y-B) relations. Two sets of II. Links of n-J symbols to Y-B relations and to representations quadratic relations In operators, one given by Woronowlcz and the of braid groups other by Wltten, are linked to relations for 51)^(2) generators and are used to formulate a q-analog Wlgner-Eckart theorem. We regard the 6-J symbol as a more fundamental quantity than the 3-J symbol (i.e. the Clebsch-Gordan coefficient) [17], because of the following re ons. (a) The 6-J symbol is Invariant under Interchange of q by l/q, while the 3-J symbol is not. (h) The 6-J symbol is a solution to the IRF model, (c) An asymptotic limit of the 6-J symbol gives the 3-J symbol and another limit leads to so-called R matrix element: The 6-j -35- ~3fi- symbol acts as a fusing and braiding Matrix. Here, Wu-Kadanoff- matrix satisfying the Y-B relation for the vertex model. Wegner transformation Is realized In a very natural way [2J. The operator R is linked to the 6-J symbol such that [2] II-l. The 3-J symbol as asymptotic limits of the S-J symbol , ,2S*a+b*c+d j a s+b s*e 1 [2S+e+f*1]! 1) The 3-J Is expressed by B-J symbols as [2] ^»'" Id S.cS.fJ [«••!,•«,] I , adb-e.e-c -(x(b)-fx(c)-x(e)-x(f) }/4 (bc-ef)/2 , ,,2S*Za*2d £(c+d+f)/6f aa a b e e I r =(R 'f-e.b-f q q 11m (-1) q • ( •/ [2S+1J »-•» ls+S+dd S+c 5+f-S+f>J , 1/2 -l/2,b+c-e-f x (q -q ) . (7) , _ -b-c d {c+d-2fl/8 ( b.f-c,d-f I e,-c*d) ±W [2e*l] ( 1)a + qi at qa- (1) We can express R in various ways [2]: is. b e \ (2) ' \ f- d-f a-1 a m 'f-o 0 d-f c-da-&'^ q- hh l'- 2' mjnij where q— means q or q.~ according as q>l or q Ji*J2-J -(x(J,)*x(Jz)-x(J))/2 la],(^/2.q.a/2)/(ql/2.q-l/2h (3) -S(-l) q J We have defined the q-analog 3-J symbol by (2). The 3-J and the x (JiJ2mi'm2' ' Jn1)qU2JlII12'nl ' Jm)q (8) B-J symbols obey the same kinds of symmetries as those of SU12). {-mitmi' + lmi-i-m!')(m2<-m2'))/4 .„ n^-mj' II-2. R matrix elements as asymptotic limits of the B-J symbol = q (qi'z -q"1''1 ) Let R be the operator defined by [21 ! 2 1 / [ Ji+Oi) I [Jx-mil! [J2-m2]! [J2*ni2] "\ ^ (8) * lmx-mil! VU^l! [Jjtnil ltJ2*m2l I [J2-m2l! J ' JlJ2 R m 4 "jV (J1J2)Jni>q F(JiJ2J' l/q<(JlJ2)J I < > If Ji"J2. the right-hand side of (6) gives the element of the Jl*J2-J "jV _1 JiJ2 Hi'.nig' JiJo mi'.m,' in which m-i uin mi nin -tx(Ji)*x(J2)-x(J))/2 (B1 F(J1J2J)- 1 • Jl J J (10) where x(a):a(n»l). As is known, R expresses the scattering "0" " 'mppfi ' 1" 2 ' where q and 2J^+1 here correspond to t and N in [18], respec symbol, expressed as triple products of the 6-J symbol, implies tively. From (9) It Is shown that the Y-B relation of the IRF model [2]. This discussion in SU(2) Is nontrivial one [1]. J]J2 Jz*Jz -n(n-l)/4 n -nj /2 n nJ_./2 n R =q £ q U-q"1) q (J_) xq (J„) /[n1! n>0 II-4. Kinds of expressions of 3-j and 8-J symbols (11) The 6-J symbol Is expressed as [8] in which J_ and J+, respectively, are generators of SUq(2). : a b el 'A (abe)A (acf)A(cde)A (dbf) II-3. A symmetry of 9-j symbol as Y-B relation for IRF model Id c ffJJ xZ i-1)z [z«-l]!{[z-a-b-e]l[z-a-c-f].[z-b-d-f]! We define q-analog 9-J symbol by [2] z l -a b e! •' (z-d-c-e]! [a+b+c+d-z]! [a+d+e+f-z] I [b+c+e*f-z] 1 }~ . 2z -{x(z)-fx(h)*x(d)*x(e)|/2 c d t) =£(-!) q [2z*l] (IB) i h k gJ q We have an alternative expression [2], /a c hi f b d k 1 r e f gi ll2) fab e { H • 3 c+d+e >k g I *H =(-l) (bae)(cde)/{(caf)(bdf)> Ik g zJlc z f-Hz a b J Id c fJ The 9-J Is not invariant under Interchange of q and 1/q. (-l)a~z[a+z]![c»f-zlI[b-c+d+z] I z [a-z]t[b-e+z]1[bte+ztl]l[-ctf+z]![-btctd-z]! ' The 9-J symbol obeys the symmetry relation such as [2] a q-analog of the expression exploited in [19]. We have defined a b e ^ f a e b\ A -x'(A)/2 > V(a+b-c]1fa-b+clIfa+b+c+ll1\l/x 2 (18) c d f f =(-1) q { c f d f , (13) nhr \ '* . aDC' ^ [-a*b*c]i ) h k gJ q lh s kJ 1/q These produce four kinds of expressions of the 3-J symbol. Applying (1) to (IB) gives q-analog of Van der Waerden form. The 9 q-analogs of Racah's first form, of Wlgner form and of Majumdar A=I ai =a+b+c+dte+f+g*h+k, (14) t form result from application of (1) to (17). We present here the 9 last two forms: The Wlgner form is expressed as x'(A)»I1x(a1)=.x(a)tx(b)+ — *x(k). (15) -39- -40- 1/2 III. Two sets cf quadratic relations, one by Woronowlcz and the ( [J»m11[J-nll[2J»l] > other by Witten. and their links to defining relations for SUq(2) * VtJi+mjl! [J2*m2]! [Jj^-mj,]! [J2-m2]!,) z-tj?tm2 z(J1+J2-J*l)/2 We try to rewrite the standard SU (2) relations for xE(-l) q q generators In a set of quadratic relations. For details, see < [J2*J -nii-z)![J1-ni1*2]! [13]. One solution we get Is X (IS) [z]![J1-.J2-m+z]![j*m-z|![J-J1*J2-zl! " Jz/2 Jz/Z Tl=-q- 3jHT\ , T.1=q" J./fW , As the Majumdar form, we have Jz TQ=q" (JtJ.-J.J»)/[2] , (22) m m (JlJ2 l 2 I Jni)q These obey 1 4 1/2 1/2 -1 1/2 IT-ITO-I"1TOT-I=T-I • <23' :Jl*m1|l[J2«m2]UJ»m]![J-m]l[2J*l]\ [J -m ]I[J -m ]! I 1 1 2 2 which are reduced to what Woronowicz [10] obtained in theories of matrix pseudogroups: We Identify (23) with the result given in z z(m-J-l)/2 [2Jz-z]l[Ji-J2*Jtz]l «I|-1) q p. 163 of ref. 10, If we put v =q1/2, Ao'U+Q)1'8!!. Aj^tlt-qJTo [z] I[J-J2+m1+zlI[J2*m2-z]I[J1*J2-J-z]! and A =(l+q)1/zT_ . (20) 2 1 Another set of solutions Is for operators T , Kinds of recursion relations for 3-J and 6-J symbols come m out [131 In connection with boson operator realization of J 2 2 (J 2)/2 Tl=-q-< *- " Jt//T2T , T.1=,- ^ j.//TiT , quadratic relations discussed below. For example, we have tW 1)/a ,8J 1)/a T0.(,- «* jti. - ,- "- j.J,}/[.] . (24) (mi m2)/4 1 q " ,Jl Ja 1)/4 1 -a * * {[Jl-1*l][JB-«a*l|} « theories without recourse to the standard relations of SU_(2). (Jl Jz 1,/4 1 1/Z 1 Z x (j1*l/2,J2*l/2,m1»l/2,m2-l/2 I Jm)q -q~ * * QT_1T1-q- T1T_1.f0 . q TQT1-q- '' T1T0»T1 , 1/2 1/2 1/2 x ([J1-m1*ll[J2*m2*l!) (J1tl/2,J2tl/2,m1-l/2,mz*l/2 I Jm)q. (21) q T.1T0-q" T0T.1=T.1 . * (25) -42- A question arises If (25) Is deducible In the same way as References (24) was In ref. 10. We get a positive answer for It. To obtain [1] H. Nomura, J. Phys. Soc. Jpn. 57, 3653 (1988). (24) Woronowicz adopted a special matrix representation of his [21 M. Nomura, J. Math. Phys. 30, 2397 (1989). operators a and a" (p.132 of ref. 10). We obtain (25) If we [3] L. Alvarez-Gaume. C. Gomez and S. Sierra, Phys. Lett. 8220(1989)142. adopt an alternative matrix representation so as to replace [4] C.N. Yang, Phys. Rev. Lett. 19(1997)1312. 2, 2 uz fitaW and fx (a*) =• u of ref.10 by t {a)-i/' and 1 [5] R.J. Baxter, Exactly Solved Models in Statistical Mechanics f^(B*)= i/1'z, respectively: y =q In this case. (Academic, London, 1982). We can express the operator Tm acting on I Jm> as [13] [6] V.G. Drinfeld, Sov. Math. Dokl. 32(1985)254. [7] M. Jlmbo, Lett. Math. Phys. 10(1985)63. m Tm. I Jm>—q" (ljiii'm;j m*m')q J [2J][2J*2]/[2J I J m*m'> , <2B) [8] A.N. Klrlllov and N. Yu. Reshetikhin: Representations of t.ie Algebra Uq(2), q-Orthogonal Polynomials and Invariants a form of Wlgner-Eckart theorem. In [12), the present author of Links. Preprint, 1988. formulated a q-analog Wlgner-Eckart theorem. [9] L.C. Bledenharn and J.D. Louck, Encyclopedia of Mathematics and Its Applications, edited by G.-C. Rota (Addlson-Wesley, Comultiplicatlon for (23) was investigated in [10], It is Reading, Massachusetts) Vols.B and 9. modified so as to be compatible with the standard SUq(2) as [10] S.L. Woronowlcz, Publ. RIMS, Kyoto Univ. 23(1987)117. [11] E. Witten, 'Gauge Theories, Vertex Models, and Quantum (U (n 0,)J (1, Groups.' preprint, IASSNS-HEP-89/32, 1989 (May). MV=Tn(l)8I(2)*q- * ' * ®Tm(2) . (27) [12] M. Nomura, To appear In J. Phys. Soc. Jpn. 59, No.2 (1990). In [13] we obtain the following comultiplicatlon for (25): An alternative description of the quantum group SU_(2) and the q-analog Racan-Wigner algebra. A (T ).T (l)l3I(2)*q"Jz(1,®T (2) (28.1) [13] M. Nomura, Quadratic relations for q-deformation of SU(2) ±1 ±1 ±1 and q-analog boson operator realization, Preprint, 1989. and [141 L.C. Bledenharn, J. Phys. A. Math. Gen. 22(1989)L873: 'A q-boson realization of the quantum group SU„(2) and the theory of q-tensor operators', 'On q-tensor operators for 2Jz(l, l/2 1/Z quantum groups', Preprints, 1989. fl (T0).f0(l)8I(2)*q" 8T0(2)>(q -q" )D , (28.2) [15] A.J. Macfarlane, J. Phys. A. Math. Gen. 22(1989)4581. where D stands for the operator expressed by Chang-Pu Sun and Hong-Chen Fu, J. Phys. A. Math. Gen. 22 (1989)L983. M. Chalchian and P. Kullsh, Preprint,19B9. _1/2 J tl, 1/2 Jztl) D=q " Z T_1(l)8T1(2)*q " T1(l)8T_1(2) . (28.3) [16] M. Nomura, J. Phys. Soc. Jpn. 58, 2694 (1989). [17] V. Pasquler, Comm. Math. Phys. 118(1988)355. In the q-»l limit the operator D tends to 2((J 'J )-J J )• It 1 z lz 2z [181 Y. Akutsu, T. Deguchl and M. Wadatl, J. Phys. Soc. Jpn. 56 z '•> possible to express q In terms of Tm (or Tm). (1987)3039. [191 M. Nomura, J. Phys. Soc. Jpn. 58. 2877 (1989). -43- -44- UT-Komaba 89-27 1 Introduction December 1989 The Yang-Baxter relation is a sufficient condition for the solvability of Link Polynomials, Linking number and Exactly models in statistical mechanics and field theoties such as 1-dimensional Solvable Models* quantum spin chains, 2-dimensional lattice systems, many body sys tems in (l+l)-dimensions, etc.. [1,2,3,4,5,6,7] Fot vations models this Tetsuo Deguchi relation is written in terms of the operators A";(u) as [1,3,13,14], *,(«)*,+,(« + v)Xi(v) = Xi+1(v)Xi(u + i.)Jf.-+i(«), Institute of Physics, College of Arts and Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan XiWWv) = *;(«)*.(«). I>"/I > 2. (1) Abstract The Yang-Baxter relation in this form has an advantage that we can easily see connection of solvable models to the braid group. Through a general method we construct link polynomials from ex Recently, the Yang-Baxter relation has been found to be a key to actly solvable models in statistical mechanics. Various examples are explicitly shown. From the crossing symmetry we derive link poly several fields in mathematical physics. Various link polynomials [8,9, nomials with the graphical calculation. By use of transformations we 10,11,12] and their extensions are obtained from exactly solvable mod obtain different Link polynomials from a solvable model. els through a general method. [13,14,15,17,18,19,20,21,23,24,25,26,27] The purpose of this paper is to show a general theory for construc tion of link polynomials from exactly solvable models in statistical mechanics. The outline of this paper is given in the following. In §2, vertex models, IRF models and faclorized S-matrices are introduced. In §3, the braid group and the mothod for construction of the representations are explained. In $4, link polynomials are constructed. The crossing 'Based on Talk given at the Conference "Topology, Field Theory and Super Strings" symmetry is used for the graphical calculation of the link polynomials. in KBK, Tsuktiba, November 6 ~ 10, 1989. This report !s alio based on Talk al the In §5, some examples are shown. In §6, transformations of solvable Workshop Phyiici, Braidi and Linti in 1989 Banff NATO Advanced Stndy Institute, The models are explained. In §7 we give concluding remarks. Banff Centra, August 13 ~ 26, 1989. -45- -46- 2 Exactly solvable models Let us considei IRF models. The Boltzmann weight of an IRF model w{a,b,c,d;u) is defined on a configuration {a,b,c,d} round a 2.1 Solvable models in statistical mechanics face (Fig.l). IRF models have constraints on the configurations. The symbol Let us explain solvable models in two-dimensional statistical mechan b ~ a denotes that the "spin" 6 is admissible to the "spin" a under the ics. [3,24] There ace two types of solvable models, veitex models constraint of the model. If the conditions b ~ a,a ~ d, b ~ c and c ~ d and IRF models. (Fig.l) Let us first consider veitex models. The aie all satisfied, then the configuration {a,b,c,d} in Fig.l is called to be allowed. The Boltzmann weights for not-allowed configurations are set to be 0. For IRF models the Yang-Baxter relation is written as 5Z «"(*• d> c.a! v)vi(d, e, /, cj u + v)w(c, /, g, a; v) = (3) "'X' <<$>• c = '£w(d,t,c,b;v)w(b,c,g,a;u + v)w(c,e,f,g;v,) (4) a c Fig. 1 (a) vertex configuration (scattering process) {i,j, k,t}. (b) IB.F configuration {a, b, c, d}. The IRF configuration a, b, c, d in Fig.l corresponds to the vertex configuration in Fig.l by i = a—d,j = b—a, k = b-c and t = e—d. We Boltzmann weight (statistical weight) w(i,j,k,l;u) of a veitex model refer to this correspondence as Wu-Kadanoff-Wegner transformation defined is fot a configuration {i,j, k,t) round a vertex. Here the pa [34,18]. In general, we can transform any (unrestricted) IRF model rameter u is called spectral parameter which controls the anisotropy into a vertex model by taking the Wu-Kadanoff-Wegner transforma (and strength) of the interactions for the model. tion and taking a limit [18] which brings "the base point" w0 of the The Yang-Baxter relation is a sufficient condition for the commu- IRF spin states into infinity:wo -* oo. tativity of the tiansfer matrices of the model. In this sense it gives the solvability of the model. There are various methods to calculate phys 2.2 Factorized S-matrices ical quantities (free energy, one-point function, etc.) for the solvable models, such as Bethe ansatz method, corner transfer method, inver Let us introduce factorized S-matrices. We write the amplitude of sion method, etc.. [3,28] Models whose Boltzmann weights (or matrix the scattring process: »' —• it, j -+ I as SjJ(u) (Fig.l), where u is the elements) satisfy the Yang-Baxter relation are called to be solvable. rapidity difference. In general, the "charge" variables i,j,k and I of For vertex models the Yang-Baxter relation is given by SjJ(u) take vector values (weight vectors). The factorized S-matrices represent the elastic scattering of particles where only the exchanges £ u/(6, c, q, r; u)w(a, k, p, c;u + v)w(i, j,a,b;v) of momenta and the phase shifts occur. The rapidity difference of the abc = £ w(a, b, p,q;v)w(i,c,a,T;u + v)wU,k, b, c;u). (2) scattering particles can be depicted by the angle in the diagram. It abc -47- -48- is known that factorized S-matrices are mathematically equivalent to 4) crossing symmetry (Fig.3) corresponding solvable vertex models. [29] 3 When Sjjf(u) is non-zero only for the case i + ;' = k +1, we say that the model has "charge conservation" property. [13,14,23,24] *"-*»-o(388)'- < > The Yang-Baxter relation for the S-matrices reads as Here, we have used the notation J for the "antiparticle" of/. We E 3?00S2(« + «)sj|(«) = E s£(tOs»(« + «)s>>). (5) abc abc This relation is often referred to as the factorization equation. [1,6,5,7] X. 2.3 Basic relations = M'xoy" TVs>c The Boltzmann weights for most of solvable models satisfy the fol lowing basic relations in addition to the Yang-Baxter relation. [13,14, 3 * Fig. 2 Crossing symmetry. 18,19,23,24] In this subsection we write the relations in terms of the factorized S-matrices. assume that r(j) = l/r(j). Note that the second inversion relation 1) standard initial condition and the crossing symmetry define the crossing multipliers. The Boltzmann weights for most of IRF models satisfy the basic S}}(« = 0) = 6ufijh. (6) relations corresponding to (6)-(9). For example, the crossing symme 2) inversion relation (unitarity condition) try is E 33*(«)S&(-") = p[u)p{-u)SuSjk, (7) w{a,blc,d]u) = w(b,c,d,a;\- «) ($$$)' " - (1°) mp where p{u) is a model-dependent function. where { -49- -50- For the 2-dimensional lattice systems the symmetry describes the in- 3 Braid group variance of the system under 90 degree rotation. Note that from the standard initial condition and the crossing symmetry, the inversion 3.1 Braids and closed braids relation and the second inversion relation are derived. We shall see We introduce braids and the braid group. [30] The braid group B is the basic relations and the Yang-Baxter relation are related to the lo n denned by a set of generators, 6i, • • •, b -i which satisfy cal moves on link diagrams, known as the Reidemeistet moves in knot n theory. &;&i-fi&; = &;+ii;&;+ii bthj = bjbit\i-j\>2. (14) 2.4 Yang-Baxter operator It is known that any oriented link can be expressed by a dosed In order to see the connection of exactly solvable models to the braid braid. The equivalent braids expressing the same link are mutually group we introduce Yang-Baxter operator X;(u). [13,14,18,23,24] The transformed by a finite sequence of two types of operations, Markov operator is, in statistical mechanics, a unit constituent of the diagonal- moves I and II. The Markov trace $(•) is a linear functional on the to-diagonal transfer matrix. [3] For factorized S-matrices we define representation of the braid group which have the following properties Yang-Baxter operator by (the Markov properties): 1, +1) Xi(») = £sfi(tOJ< «-««iS««ii I.ftAB) = (BA),A,BtBn, (IS) abci S^'+a) ! A In terms of the Yang-Baxter operators, the Yang-Baxter relation for a(A) = {Tf)- ^{^< H{A),A(Bn (18) vertex models and IRF models is written in the same form. Here e{A) is the exponent sum of 6,'s in the braid A, which is equiva lent to the writhe of the link diagram. For instance, UA = &}4jJ6a&i"1, then t(A) = 4-2 + 1-1 = 2. -51- -52- 3.2 Construction of the braid operator models. The Markov trace takes the follwing form [13,14,23,24,25] The braid operator G(+);, the inverse operator G(-); and the identity = r:WnM) V / are given by (13,14] ' Tr{H{n)) mn)]t\z:::c; = f[r\ )6i]. ( ) 9 aj 24 G(±); = ulimo.X;(±")/P(±«). (I ) I = Xi(0). (20) For the models with the crossing symmetry (and the second inversion The limit u —* 00 (more precisely, an infinite limit in a certain direc relation), r(p) is nothing but the crossing multiplier of the model. We tion in the complex u-plane) requires that the Boltzmann weights be present sufficient conditions for the Markov properties explicitly. We parametrized by hyperbolic (trigonometric) functions. Hereafter we can show that the trace #(•) denned in (24) is the Markov trace by write the matrix elements of the braid operator as proving for the Markov property I the "charge conservation" property and for the Markov property II the following conditions: G%±) = Jta, s£(±«)M±u). (21) J E6G°J(±)r (6) = x(±) (independent of o). (25) Then we can express the braid operator (19) constructed from the Yang-Baxter operator as The r-factors are related to x(±) as f/r = x(-)/x(+). We can prove the extended Markov property, [18,20,23,24,25] which 1 +1) +2) (B) G(±)i = EaWG^(±)i< '<»...®eW®e^ ®/<'' ®---®/ . (22) is an extension of the Markov property with finite spectral parameter. It is sometimes convenient to write the matrix elements of the braid £•*"&(»)*(*) = J5f(«i >?)P(«) ( independent of a), (26) operator b where the function H{ii;r)) is called characteristic function. [ lib 1- VK'" 4 Construction of link polynomials where the symbol E" represents the summataion over admissible multi- indices ti : ii+i ~ /,-for i = 0, • • •, n - 1 with lt> being fixed. Then the 4.1 Construction of the Markov trace Markov trace <£(•) is written as We shall obtain link polynomials by constructing the Markov trace w AeBn (28) on the representations of the braid group derived from the solvable *-M))' ' -53- -54- where I(n) is the "identity" operator for n strings. where J(a, c) = Sac is the Kroneckei delta. We can prove the extended Markov property also lor IR.F models. [18,20,23,24,25] 4.2 Graphical calculation 4 ~c d The crossing symmetry is signilicant in algebraic and graphical aspects of the knot theory. For solvable (vertex and IRF) models with the crossing symmetry, the Yang-Baxter operator becomes the Temperley- Lieb operator at the point u = A. [19] In fact, setting Ei = Xi{\), (29) we find that the operators {£?,} satisfy the following relations [31] Fig. 3 Scattering with u = A corresponds to annihilation- creation process. EiEi±\Ei = Ei, We can regard the elements r(c)S(c, 3) and r(a)S(a, I) as the weights " E? = q±Et, for the pail-annihilation diagram and the pair-creation diagram, re- EiEj = EjEi, |i-j|>2, (30) spectively(Fig.4). b where the quantity g1'^is related to the crossing multipliers r(a) (or *(.')) by [13,14,18,19] ^TA^-y q' = J^r5^'), for S-matrix (vertex model), (31) i = ETrr.forlRF model, (32) where in (32) the summation is over all states b allowable to a. The relations (30) are the defining relations of the Temperley-Lieb algebra. Let us consider the graphical meaning of the relations (30). From the crossing symmetiy and the standard initial condition we have Fig. 4 Elements of link diagram. (Fig.3) [19,23] (1) pair-annihilation diagram: r(c)6Ctd] (^(a)/^)) • 1 3 (2) pair-creation diagram: r(a)6aii; (\i(c)/^(6)) ' . (3) braid diagram with e = -1: G%(+); G(a, b, c, d; +). = r(a)6(a,b)T(c)6(c,d), (33) (4) braid diagram with t = 1: G^(-); G(a, b, c, d; -). -55- -56- Then, the Yang-Baxter operator at u = A is depicted as the monoid It is easy to see that a(L) is invariant under the Reidemeister moves diagram, by which the Temperley-Lieb algebra is explained. This (Fig.6), and therefore a(L) is a topological invariant of the link L. interpretation is consistent with a fact that the energy at the point A Thus we have shown that the link polynomials constructed from solv is related to the pair-creation energy. able models with the crossing symmetry can be graphically formu For IRF models, the weights {V'(a)M&)}1/3 and Mc)M&)}1/2 lated. The monoid diagram and the weights for the creation and correspond to the pair-annihilation and pair-creation diagrams, re annihilation diagrams were used by L.E. Kauifman for the Bracket spectively (Fig.4). polynomial which gives a graphical calculation of the Jones polyno We can formulate link polynomials with the crossing symmetry di mial. [32] The graphical calculation is named "state model". rectly on link diagrams. Link diagram £ is a 2-dimensional projection of a link L. The writhe w(L) is the sum of signs for all crossings C; in the link diagram (Fig.5): w(L) = VcACi), (34) r We calculate statistical sum Tr(L) on the diagram L by the rules given ia Fig.4. n Fig. 5 Sign <=(C). « = 1 « = -1. in The link polynomial for the link L is calculated as •*>-<*»!$!• <"> Fig. 6 Reidemeister moves. where Ka is the trivial knot diagram (a loop) and the constant c is defined by a relation We have remarks. The graphical formulation applied to closed GiEi = cEi, (36) braids yields the Markov trace (Fig.7). For the link polynomials with or by the crossing symmetry, the formulation based on the Markov trace is equivalent to the graphical formulation. This viewpoint is consistent -57- -58- with the braid-plat correspondence [33]. 5 Various Examples The link diagrams are considered as the Feynman diagrams for the high energy processes of charged particles and the link polynomials as 5.1 iV-state vertex model the scattering amplitudes. At the lowest point in the diagram there From the JV-state vertex models a hierarchy of link polynomials are occurs a pair creation and at the highest point a pair annihilation. obtained by the general method presented in §3 and §4. [13,14] The Further, if we regard the link diagrams as distorted 2-dimensional model corresponds to the factorized S-matrices with spin s particles, lattices, the link polynomials are considered as the partition functions. where N = 2s + 1. For the case N = 3, there are 19 vertex configura To conclude this section, we put emphasis on the fact that the tions. [35] The Boltzmann weights of the iV-state vertex model can be crossing symmetry has the algebraic and graphical meanings. Alge systematically calculated by using recursion relations. [34] Therefore, braically, the symmetry leads to the Temperley-Lieb algebra (and the an algorithm for construction of the hierarchy of link polynomials has braid-monoid algebra). been established. [13,14] From the iV-state vertex model (asymmetrized by the symmetry breaking transformation) we get the braid operator which satisfies an JV-th order relation: [13,14] (G( - ft)((?,- - Cs) • • • (Gi - CN) = 0 (38) where for j = 1,2, • • •, JV d = (-iy+JMW-0-tiO-i), t = eM. (39) We call a relation for G; such as the relation (38) reduction relation Fig. 7 Equivalence of the Markov trace and the graphical of the braid operator. The crossing multiplier for the asymmetrized calculation. JV-state vertex model is [13,14] r(fe) = e-u = I"*'2, Jfe = -3,-3 + 1, •••,3, (40) Graphically, the pair-creation and pair-annihilation diagrams are in where troduced through the crossing symmetry. 3 = (N- 1)12. (41) The extended Markov property [18,24] is satisfied with the character istic function given as [23,24] „, sinh(iVA-u) , . ff(u;A)= , I J-. v(42) sinh(A - u) ' -59- -60- The constants r and f are (ar;n) = (1 - *)(1 - zt)---{\ - 2in_1) for n > 1, r = l/(l + t + --- + tw-1), (43) = 1 for n = 0, f = i"-V(l+ * + •••+ 2 3 4 6 7 a(Xa+) = i(l-i +i )a(I+) + (t -t + < )a(Zo) knowledge of the composite Yang-Baxter operator. -r8a(I_), (JV = 3) (46) 3 i 3 i 6 i 1 3 i e s 5.2 Graph state IRF model a(X3+) = t l{\-t +t-t )a(.L7+) + t(l-t + t + t-t +t )ol{L+) 3 8 a +(»/»(_ 1 + t-t + t )a(X0) -t °<*(£-), (tf = 4). (47) We can construct solvable IRF models corresponding to arbitrary graphs in any dimensions. [36,18] Let us express the constraint of the In (45), by X+, Xo and X_ we have denoted links which have the model by a graph. In the graph each point represents the spin state. configuration of 6;, 6f and ft"1, at an intersection. Similarly, Lj+, X+, When a spin c is admissible to d then the point for c is connected to the Xo and X- in (46) and X , X , X+, Xo and X_ in (47) should be 3+ 3+ point for d. For ADE type graphs the models are called ADE models. nndeistood. [37] There also exist solvable models with elliptic parametrization for We can also present a general expression for the braid matrix de extended Dynkin diagrams [36,38]. rived from the JV-state vertex model. The symbol t^N'e'> denotes the Let us construct the graph state IRF models. [18] We solve the charge submatiix acting in the sector of the total charge c (c = i+j = eigenvalue equation for the graph k + i). !>(&) = Atfa), (51) m+ |e| |e| i (*W)». = (-l) "( .»,••', (49) Qmn(z)=/ i//(5) = s\n{5 -H + uo), (52) Wnml ' (t;m-n)(ljn)(ti;n) v -61- -62- where a = (ai,a2) and n = (711,712). Constructing the Temperley-Lieb with operator 7 = «-'" for A^_v B%\ CW and Dm\ (61) f) = ^-ipi+.d+'a fox flJJ), CCi) and 2? We can consider vertex models corresponding to the graph state (The explicit forms of the crossing multipliers are given in [17]). Using IRF models under the Wu-Kadanoff-Wegner transformation and the the reduction relations and the Markov traces, we obtain the (gener base-point-infinity limit . [27] We call them vertex models in TL class. alized) skein relations: [18,19,27] From these vertex and IRF models we have multi-variable «(£+) = (l-l)tfm-1>'aar(Xo) + t",«(X-)fcii4jii.i. (65) braid matrices.[27] x 1 o(Ij+) = (l-t + f!)e-V +«°- »-a{L+) 5.3 ABCD IRF models +(t + 0t- p)e-'KK+»l*-i)). a(x0) The IRF model corresponding to affine Lie algebra i4„Li (Bm , Gm , _l/8e-*<™+»C<'-i)). o(£_), 1 { Z?(J') is called A ^ (B J,\ cli\ DW) model. [39] The crossing pa for Si,1', ci!5 and D^1 (66) rameter A and the sign factor a are denned as where A = mu/2, a = 1 for A^, (55) t = e~liw. (67) A = (2m - l)w/2, a = 1 for si!', (56) For A)l'_i model, the Alexander polynomail is obtained by the limit A = (m + l)w, A = (m - l)w, a = 1 for DW', (56) Link polynomials thus obtained are one-variable invariants for each fixed 771. It is noted that m is independent of t. We now have two where u is a parameter. The reduction relations are variables t and m. The link polynomial constructed from J4„_J model (Gi - l)(Gi + 75) = Ofoi^-i, (59) corresponds to the two-variable extension [10,11] of the Jones polyno 7 1) 1) ) (G; - 1)(G, - /?)(G; + I ) = 0forJ5L ,cL andU^ , (60) mial. The link polynomails from Bm , Cm , Dm models correspond -63- -64- to the Kauffman polynomial [12]. We thus have explicit realizations of where u is a free parameter, p. and V are arbitrary vectors, and if is a the Kauffman polynomial and the two-variable extension of the Jones vector such that (j + k) • eis an integer for any weight vectors j and k. polynomial (HOMFLY polynomial). The braid matrices constructed Using these transformations C, P and T invariances for the S-matrices by Turaev [40,41] correspond to the vertex-model analog of the present can be broken. There are symmetry breaking transformations for IRF 1 ) braid matrices constructed from A^_lt B^ ', cL", Z?m IRF models. models corresponding to those for S-matrices. [18,23,?4] From the IRF models we can construct biaid matrices and the Markov The transformed matrix elements satisfy the crossing symmetry trace for the vertex models by the Wu-Kadanoff-Wegner transforma with tion and the base-point-infinity limit. [18] For example, from A-type f(£) = r(£)e:Bp[-2ifc./IA]. (73) IRF models we derive the multi-state vertex models [42] related to Under the transformations the standard initial condition and the first SU(n). From the Markov trace [20] for the IRF model we have that inversion relation are invariant. The second inversion relation holds for [40] for the vertex model. the transformed matrix elements with the crossing multipliers modi fied as (73). 6 Transformations of solvable mod 6.2 Deformation of solvable models els We shall show that the symmetry breaking transformation changes 6.1 Transformations algebraic structure of the Yang-Baxter operator. Let us take the 6- vertex model. We set ft = 7 = 6 = 1 and consider only the trans The solvable models with charge conservation condition are invari formation <>,;,*{(u) in the following discussion. We assume that the ant under several transformations. [34,24,27] We introduce symmetry vector ~jt is parallel to the weight vectors and write it simply as p.. breaking transformations (or gauge transformations) [34] for factor- The Boltzmann weights of the 6-vertex model are given by ized S-matrices (equivalently, for the Boltzmann weights of the vertex ci/21/2- , _ -i/2 -1/3, > _ sinh(A - u) models ): ? 8 s%~$M = stf-Ufa) = 1. (75) Sjj(u) - S;*(u) = a0l«(tt)A>lW7tf,M*y.MS#(tO. (6 ) c-1/2 -1/2, ^ _ si/3 1/2 , v _ sinhu 69 ar 80|W = ezp[7r/=l(J+£)•£], (72) see that the Yang-Baxter operator of the 6-vertex model satisfies a - 65 - cubic relation: The braid matrices are given in the following, (i) p = | ('*>-*a^') (*«-<' -Si") (l 0 0 °) 0 0 -t* 0 G$(+) = 1 (84) 0 1-t 0 If we apply the symmetry breaking transformation or^jufu) with ^ = 0 0 ]/ ±1/2 to the 6-vertex model, then we have an " asymmetrized" 6-vertex \° (") -i < » < k model with the nontrivial crossing multipliers: (l 0 0 r(k) = ezp(-tt), k = ± 1, for =\, (78) 1 °\ U 0 0 0 GS(+) = (65) r(k) =exp(kX), k = ±i-, foiu = -j, (79) 0 0 0 0 0 The transformed Yang-Baxter operator -X'i(u) satisfies a quadratic 1° */ (iii) p = -\ relation: (1 0 0 o\ 0 1-t -ti 0 G°S(+) = (86) Further, we can decompose the Yang-Baxter operator A"i(u) as 0 -a 0 0 0 0 1/ *,(«) = pfflW + /(")£<)> (81) 1° The braid matrices for the cases (i) and (iii) are equivalent if we in where terchange np-spin and down-spin. They have the Markov traces. It is , . sinhfA - u) remarked that they satisfy the defining relations of the Hecke algebra "(U) = sinhA ' [9]: fW = siihfT^)' (82) GjGj+iG; = Gi+iGjG;+i, and £,• is the Tempeiley-Lieb operator. For the symmetry breaking GiGj = GjGi, for \i - j\ > 2, transformation aij,iw(u) with arbitrary ft, we find that the transformed Yang-Baxter operator X,(u) satisfies a cubic relation. The Yang- G] = (1-OGi + iJ. (87) Baxter operator satisfies a quadratic relation only when fi = ±1/2. The operator G; can be decomposed into the Temperley-Lieb operator By changing the value of the parameter fi, we get different repre as [9] sentations of the braid group from the Yang-Baxter operator. Here G, =/-!*£,. (88) after in this sub-section, we set In the case (ii) , the operator G, satisfies a cubic relation: i = eip(2A). (83) (G, - /)(G? - tl) = 0. (89) -67- -68- 6.3 6-vertex model and linking number automatically becomes the Maikov tiace and also the link polyno mial. By making use of the general method piesented we shall constiuct the Markov tiace for the braid group representations derived from the 6- veitex model. Fiom the asymmetrized 6-vertex model with ft = ±1/2, 7 Concluding Remarks we obtain the Jones polynomial by using the transformed crossing We have shown that various link polynomials aie systematically con multipliers in the tiace. The Jones polynomial has a quadratic skein structed from exactly solvable models. We remark that the Alexander relation corresponding to the quadiatic reduction relation. polynomial is derived fiom the free feimion model. [44,45] Therefore Foi the symmetric 6-veitex model with fi = 0, the crossing multi we can constiuct from solvable models all known link polynomials such plier is equal to 1 and then the Markov trace is a. trace on the braid as the Alexander polynomial, the Jones polynomial, etc.. matrix (the case (ii)) with the trivial matrix just (H = /). We thus The existence and properties of the link polynomials [13,14] con obtain a link polynomial with a cubic skein relation: structed from the .AT-state vertex model [34] can be proved also by o(X3+) = a(L3+) + ia(£+) - ta(La), (90) the construction of composite models (fusion method) in terms of the Tempeiley-Lieb algebra and the graphical formulation derived from wheie L„+ is the link which has n twist at a crossing point in the the crossing symmetry. [19] Note that the combination of the cross link diagram. This link polynomial is also obtained by soving directly ing symmetry and the Tempeiley-Lieb algebra characterize the link the defining relation of the braid group with the assumption that the polynomials. braid matrix does not satisfy the charge conservation condition. [43] We can construct composite solvable models from the graph-state The link polynomial (90) for a link with two strings is determined IRF models and vertex models in TL class. [18,19,27] From these by the linking number of the link. For a link with two strings, the link composite models we obtain the link polynomials constructed from polynomial is given by a(I) = 1 + 1", where v is the linking number the iV-state vertex models. [19,27] of the link. Therefore the link polynomial can be considered as a g- Due to the limited space we have omitted the discussion for con analogue of the linking number. Thus, from the 6-vertex model we struction of two-variable link invariants [16,17,23,24,25] which may be obtain two different link polynomials, the Jones polynomial and the regarded as two-variable extension of the link polynomials constructed link polynomial related to the linking number. [27] It is remarked that from A type composite vertex and IRF models. In the papers [16,17] there are many multi-variable link polynomials related to the linking an algorithm for calculation of the two-variable link invariants for any number. [27] links has been established, and some examples have been given. We have a comment. In the case of the symmetric 6-vertex model, Some class of braid matrices obtained fiom vertex models related the Maikov trace is given by the ordinary trace: -69- -70- analogue of universal enveloping algebra of the Lie algebra. [41,46] References For example, the matrix elements of the braid operator obtained from the JV-state vertex model are also calculated by using the knowledge [1] C.N. Yang: Phys. Rev. Lett. 19 (1967) 1312. of su(2) [47]. [2] R.J. Baxter: Ann. of Phys. 70 (1972) 323. We may also consider solvable vertex models related to Lie super- [3] R.J. Baxter: Exactly Solved Models in Statistical Mechanics (Aca algebras. They are extensively studied. [48,49,50,51,45] demic Press, 1982). Recently there are some attempts to obtain braid matrices by solv [4] L.A. Takhtadzhan and L.D. Faddeev, Russian Math. Surveys 34 ing the defining relations of the braid group. [43,52] Connection of (1979) 11. these braid matrices to solvable models is an interesting problem. [5] A. B. Zamolodchikov and A.B. Zamolodchikov, Ann. of Phys. There are several problems in physics related to the braid group. 120 (1979) 253. [53,54,55,56,57,58] Interestingly, solvable models and conformal field theories share many mathematically similar points in common. [59,54, [6] M. Karowski, H.J. Thun, T.T. Truong and P.H. Weisz: Phys. 60,61,62,63,64,65] Through the fusion rule, mathematical structures Lett. 67B (1977) 321. analogous to IRF models appear in conformal field theories. [65,66,67] [7] K. Sogo, M. Uchinami, A. Nakarrmra and M. Wadati: Piog. It may be instructive to compare the viewpoints of field thoeries and Theor. Phys. 66 (1981) 1284. statistical mechanics. [8] J.W. Alexander, Trans. Amei. Math. Soc. 30 (1928) 275. It seems that there are many interesting problems concerning ap [9] V.F.R. Jones: Bull. Amet. Math. Soc. 12 (1985) 103. plications of link polynomials to physics, chemistry and biology. We [10] P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickerish, K. MilleU and hope ; the knowledge exhibited in this paper will be helpfull for studying those applications of the link polynomials. A. Ocneanu: Bull. Amer. Math. Soc. 12 (1985) 239. [11] J.H. Przytycki and K.P. Traczyk: Kobe J. Math. 4 (1987) 115. [12] L.H. Kauffman, On Knots (Princeton University Press, 1987); Trans. Amer. Math. Soc. (to appear). Acknowledgements [13] Y. Akutsu and M. Wadati: J. Phys. Soc. Jpn. 56 (1987) 839. The author would like to express his sincere thanks to Prof. M. Wadati [14] Y. Akutsn and M. Wadati: J. Phys. Soc. Jpn. 66 (1987) 3039. and Dr. Y. Akutsu for continuous encouragements, critical reading [15] Y. Akutsu, T. Deguchi and M. Wadati: J. Phys. Soc. Jpn. 56 of the manuscrip and fruitful collaborations on which this paper is (1987) 3464. based. [16] Y. Akutsu and M. Wadali: Commun. Math. Phys. 117 (1988) 243. -71- -72- [IT] T. Deguchi, Y. Akutsu and M. Wadati: J. Phys. Soc. Jpn. 57 [30] J.S. Birman: Braids, Links and Mapping Class Groups (Prince (1988) 757. ton University Press, 1974). [18] Y. Akutsu, T. Deguchi and M. Wadati: J. Phys. Soc. Jpn. 57 [31] Temperley H.N.V. and Lieb E.H., 1971 Proc. Roy. Soc. London (1988) 1173. A322 251. [19] T. Deguchi, M. Wadati and Y. Akutsu: J. Phys. Soc. Jpn. 57 [32] L. H. Kauffman, Statistical Mechanics and The Jones polynomial, (1988) 1905. preprint 1987 (to appear in Proceedings of 1986 Santa Cruz Con [20] T. Deguchi, M. Wadati and Y. Akutsu: J. Phys. Soc. Jpn. 57 ference on the Artin Braid Group); IHES/M/88/46 Septembre (1988) 2921. preprint 1988. [21] M. Wadati and Y. Akutsu: Ptog. Theor. Phys. Suppl. 94 (1988) [33] J.S. Birman and T. Kanenobu, Proc. Amer. Math. Soc. 102 1. (1988) 687. T. Kanenobu, Math. Ann. 285 115. [22] M. Wadati, T. Deguchi and Y. Akutsu: in Nonlinear Evolu tion Equations, Integrability and Spectral Methods, ed. A. Fordy [34] K. Sogo, Y. Akutsu and T. Abe, Prog. Theor. Phys. 70 (1983) (Manchester Univetsity Press, 1989). 730,739. [23] Y. Akutsu, T. Deguchi and M. Wada'ti: in Braid Group, Knot [35] A.B. Zamolodchikov and V.A. Fateev, Sov. J. Nucl. Phys. 32 Theory and Statistical Mechanics, ed. C.N. Yang and M.L. Ge (1980) 293. (World Scientific Pub., 1989) p. 151. [36] Y. Akutsu, A. Kuniba and M. Wadati, J. Phys. Soc. Jpn. 55 [24] M. Wadati, T. Deguchi and Y. Akutsu: Phys. Reports 180 (1989) (1986) 1466. 427. [37] V. Pasquier, J. Phys. A: Math. Gen. 20 (1987): L217, L221. [25] T. Deguchi, M. Wadati and Y. Akutsu: Adv. Stud, in pure Math. [38] A. Kuniba and T. Yajima, J. Phys. A: Math. Gen. 21 (1988) 519; 19 (1989), Kinokuniya-Academic Press, p. 193. J. Stat. Phys. 50 Nos. 3/4, (1988) 829. [26] M.Wadati, Y. Akutsu and T. Deguchi,.to appear in Lecture Notes [39] M. Jimbo, T. Miwa and M. Okado, Commun. Math. Phys. 116 in Mathematics (Springer-Verlag, Berlin, Heidelberg, 1989). (1988) 353. [27] T. Deguchi : Braids, Link Polynomials and Transformations of [40] V.G. Turaev: Invent. Math. 92 1988 527. Solvable Models, preprint UT-Komaba 89-11 July 1989. [41] N. Yu. Reshetikhin, LOMI preprints E-4-87,E-17-87, Leningrad [28] H.J. de Vega, Adv. Stud, in pure Math. 19 (1989), Kinokuniya- 1988. Academic Press, p. 567. [42] I.V. Cherednik, Theor. Math. Phys. 43 (1980) 356. [29] A.B. Zamolodchikov, Commun. Math. Phys. 69 (1979) 165. O. Babelon, H. J. de Vega and C. M. Viallet, Nucl. Phys. B190 -73- -74- (1981) 542. [60] E. Verlinde, Nucl. Phys. B300 [FS22] (1988) 360. Cheiie L. Schultz, Phys. Rev. Lett. 46 (1981) 629; J.H.H. Peik [61] G. Moore and N. Seiberg, r-hys. Lett. B212 (1988) 451; IASSNS- and G.L. Schulti, Phys. Lett. 84A (1981) 407. EEP-88/39, Princeton preprint 1988. [43] Z.Q. Ma and B.H. Zhao, J. Phys. A: Math. Gen. 22 (1989) L49. [62] K.H. Rehren and B. Schroer, Einstein Causality and Artin braids, [44] T. Deguchi, J. Phys. Soc. Jpn. 58 (1989) 3441. preprint 1988. [45] T. Deguchi and Y. Akutsu, preprint UT-Komaba 89-26. [63] J. Frohlich, Statistics of fields, the Yang-Baxter equation, and [46] A.N. KiriUov and N.Yu. Reshetikhin, LOMI preprint E-9-88, the theory of knots and links, preprint 1988. Leningrad 1988. [64] H.C. Lee, M.L. Ge, M. Couture and Y.S. Wu, Int. J. Mod. Phys. [47] M. Nomura, 3. Math. Phys. 30 (10) (1989) 2397. A, 4 (1989) 2333. [48] P.P. Kulish, preprint RIMS 615. [65] M. Wadati, Y. Yamada and T. Deguchi: J. Phys. Soc. Jpn. 58 P.P. Kulish and N.Yu. Reshetikhin, Lett. Math. Phys. 18 (1989) (1989) 1153 143. [66] P.Di Francesco and J.B. Zuber, Saclay preprint S,Ph-T/89/92. [49] H. Saleur, Saclay preprint PhT/89-136. [67] P. Ginsparg, Harvard University preprint HUTP-89/A027. [50] M. Chaichian and P. Kulish, preprint HU-TFT-89-39. [51] T. Deguchi, A. Fujii and K. Ito, preprint UT-Komaba 89-25. [52] H.C. Lee, M. Couture and N.C. Schmeing, Chalk River preprint CRNL-TP-1125R December 1988. [53] Y.S. Wu, Phys. Rev. Lett. 52 (1984) 2103. [54] A. Tsuchiya and Y. Kanie, Adv. Stud, in Pure Math. 16 (1988), Kinokuniya-Academic Press, p. 297. [55] G.W. Semenoff, Phys. Rev. Lett. 61 (1988) 517. [56] A.M. Polykov, Mod. Phys. Lett. 3A (1988) 325. [57] C. Rovelli and L. Smolin, Phys. Rev. Lett. 61 (1988) 1155. [58] E. Witten, Commun. Math. Phys. 121 (1989) 351. [59] A. Kuniba, Y. Akulsu and M. Wadati, J. Phys. Soc. Jpn. 55 (1986) 3285. -75- -76- In this talk we generalize the s — d exchange model to the fully anisotropic case. We emphasize that the reduction to the one dimensional system is based on the localfdelta- function like) interaction but not on the rotational mvariance of the system. Hence we may discuss the anisotropic magnetic alloys under the same scheme. Here we shall point out that Exact Solution the fully anisotropic Kondo model is also integrable. of the Fully Anisotropic Kondo Problem* 1. Now we consider the one-dimensional system which consists of N conduction electrons and one impurity. The electrons and the impurity are spin 1/2 fermions. The impurity is Yas-hiro Quano localized at r = 0, the center of the system. The electrons have the linear dispersion relation, interacting with the impurity through the delta-function like spin-spin coupling. Essentially Department of Physics, University of Tokyo our problem is a scattering problem r,[ the electrons and the impurity. Bunkyo'ku, Tokyo, Japan We start from the following seccnd-quanlized Hamilton!an Abstract H = I'.fc{(-.>;(x)aI,Mr) + £ 2JMV>:(xK9tMx)x:sjU»i(*)}, (1) It is shown that the fully anisotropic one-dimenaioDaJ Kondo problem is complete]} where <7°, S° are unit matrices, and a', S*(i = 1,2,3) are Pauli matrices representing the spin integrable based on the equivalence with the inhomogeneotia eight vertex mode]. of the electron and the impurity, respectively. VCfV'a) is the creation (annihilation) operator of In the Kondo problem[2] one studies the cooperative phenomena of the conduction elec the electron, xl(x*) 'a tne creation (annihilation) operator of the impurity, respectively, or, a trons in the presence of an impurity in the magnetic alloys. Several models, such as the a - d stand for spin indices. exchange model, the Anderson model, are considered to understand the magnetic property These operators hold the following anticommutation relation: of the system which consists of the electrons and impurities. The a — of exchange model is an effective theory of the Anderson model in a certain limit. R(x),lW} = W(*-a). (2) Surprisingly, Andrei[3] and Wiegmann[4] have independently shown that this many-body problem is exactly solvable under the following assumptions: {x-.yJ} = ** (3) {i)A Bingle localized impurity, whose only degree of freedom is its spin, U located at the {x.,Xb} = {xZ,Xfc} = a. center of the system. The interaction with the electron is represented by the delta-function Of courBe V£(z)i V*a(x) andx« xj ace anticommute. Now we restrict ourselves tothesubspace like, spin dependent potential. ( satisfying xlx* = 1. This restriction is necessary to put the impurity spin be 1/2. (u)Aa only the a wave interacts with the Impurity, the problem reduces to the one dimen When JQ = 0 and J = J) = J eq.(l) is the Rami!Ionian of the a - d exchange model sional one. x 3l (the Kondo model with SU(2) symmetry). (iii)The kinetic energy of the electron Is approximately linear because at low temperature We notice that the number operator, N — Jdx^(x)i}t [x), commutes with the Kondo the electron states lying far from the Fermi surface ate negligible, a Hamiltonian. Hence the Hilbert space of the system is decomposed into the direct sum of the -Tali talk Ii baaed on Ref.fl] sectors where the number operator N takes an eigenvalue. -77- -78- In the N particle sector the eigenslate of the unperturbed HamUtonian Tio is given by wt{u) - u/,(u) = pe(2i))H(ti-n)8(u + TJ), (10) wi(u) + wi(u) = pH(2r,)e(u-t))9(u + n), i*o(Ai.---.M)a...aw = /nd*,«p(«'EMj)^n^.(*.')i°). w ' .=1 j=i .=i IB,(B)-I«,(U) = pH(2r,)H(u-r,)H(u+r>). where the reference state |0) is defined as In the special case J, = J2 = J3, eq.(7) gives R matrix in s - d exchange model[5]. *.(i)|0) = X.|0> = 0. (5) 3. Next we shall show that for the sector of N electrons and one impurity the Bethe ansatz In the relativisttc quantum field theory the state defined by (5) is an unphysical vacuum. is valid. Let us consider the following state We consider, however, the non-relativistic system. Hence there exists no antiparticle in our problem. As we choose the origin of the energy at Fermi surface, even negative energy states |«(*i,--•,*»))„....» = [fld*,ertiJ:^MiI[rA*MZH*i)W)i~a'l, (ID are assined to the particle states. The state (4) stands for the one what all the levels lower ' i.l j=] i-1 l where Pl (ni+aj)/2 = ui0(A) + uij(A), (a2 - n,)/2 = ui,(A) -w,[X), (9) P.jP,, = 1, Rj0Ro, = 1, FijR.oRjo = R-joR-oP,,. (15) (ao + oj)/2 = Wg(A)-wg(A), (a}-ao)/2sui,(A) + W](A), Eqa.(15) ensure the validity of the Bethe ansatz for our problem. where u^tij's are represented using Jacobi theta functions H(u),8(u) of modulas k as 4. Now we clearify the equivalence between our problem and some vertex models[7]. From Lhe results obtained in the preceding section we can establish the wave function if we know to0(i.)+uij(u) = p0(2r;)e(u-n)ll(u + n), -79- -80- one of $(Q), for example $(/). It is necessary to impose a suitable boundary condition to Hence the fully anisotropic Kondo problem is equivalent to the inhomogeneous eight vertex determine $(/) and the spectrum of the system. Let us consider the system in a periodic box model. The inhomogeneity is derived from the existence of the impurity. Further we point of length L foe convenience. If we write * = $(/) for simplicity, periodic boundary conditions out that our problem is equivalent to the inhomogeneous six vertex model when Ji = J3. lead to i T[>p!::;i!*^:.:J = e"*'*:::;:l;, 06) 5.To solve the eigen value problem (16), we construct the monodromy matrix where T(u):!::°»Afl"+' = Z ft *&'*(« + *)/Mbi))", (23) •wt&3i=(p,-,):;*;;., • • • (p,„)f# (ROfi?.(p»„)f#„ • • • r^flr^. |S,).=D ' and summation is taken over /? in the above expression (see Fig. 2). where We notice that the following identity »i = (A - n)&,. (24) The operator a? acts on the Hiibert space Ho, = ZwMaga;, (17) S=®iL>!l„ li = CJ, (25) Pi, = Z<*M<,ra*/M,), (18) and is defined as where o-f = / ® • - - 0 cr** ® •I. (26) m(u) + w (u) = />e(2n)0(u-r>)H(u + n), U co n 3 It is evident that TJj'j and the transfer matrix X(u) = llfitmfis*! ^( ) > cide when u = r}. u/(i(u) - u/3(u) = />0(2r/)H(u - n)0(u +n), Baxter[6] has pointed out the commutativily tui(u) + uij(u) = pH(2n)0(u - r;)0(u + n), (19) [T(ii),T(»)] = 0 (27) u),(u)-i»a(u) = pH(2n)H(u-n)H(u + n), *(«) = />e(0)H(u)e(ii). (2D) is equivalent to the factorization condition (15). Regardless of the inhomogenely of the system, the above generalised eight vertex model is By introducing the operator completely integrable. Recall the existence of a family of the quantum TC-matricessatisfyin g ft K{u, «)(£,(«) » CM) = (£,(«) 8 £i(a))«(u,«) (28) we can rente T[j] as follows: IB the most essential point for solvability of the homogeneous eight vertex model. The explicite form of TZ-rnatrices are given by TW'—" «55) °(*'•'- M) 'a 0 0 d\ 0 6 c 0 #(2fj) W>°', i d 0 0 a 4(2") " "'•;•.'•' ^(2„) 0(2„) • -81- -82- where and we adopt the basis which diagonalues cr1. ' a(u, u) = 9(2>;)9(u - ti)H(u - v + 2n), To calculate the eigenvalue of T(u) = Ai,j(u) + DI,I(U), we use the following commutation l(u, o) = H(2rj)9(u - u)9(u - « + 2IJ), (30) relations: c(ti, u) = 9(2t))H(u - t/)8(u - v + 2i7), B»,, (u)B4-ij(«) = B i(»)Bt {u), (38) d(u,«) = H(2n)H(u - v)H(u - u + 2n), +1 i w+ +u A»,,(u)B i,;-i(f) = a(u,v)Bi,,_j(u)A . (u)-fl_ (u, ulBv.i-jtuJAk+u-itv), Because ft(u,t/ ) depends only on u — u, the inhomogeneity 4+ i+u 1 1 Du(u)Bt+u.i(ii) = or(u,n)B»+2j(u)D»+ij-,(ti) + A+i(u,u)Bn.w(u)D4+1j.i(v), ^(iiji-rtu+ui), £(")-£(» + «.), (31) where a(u, ti) = h(u - v - 2n)//i(u - v), ft(«, o) = n(2r/)A(Tk + u - u)/7i(u - u)A(?i)< gives 7t(u, u) no change. Therefore our problem is one of the realization of the most generalized $(ui,-•-,UAr) is an eigenstate of T(u) under the following condition which cancels out Baxter's model[6]. the unwanted terms £ /•(•!„ + v,- + r,) „ A(u„ - u„ + 2n) ,. _ , , , . 6. First we introduce a family of gauge-transformed matrices[8] nMu„+v,-r,) = n_Muo_Uj_2,).(«-l.-,M) (39) and tts eigenvalue is given by putting u = 17 rk,,(u)=JC'(«)T(«WM=f y;\ y;\) . (32) T{n)#(ull...,«M) = A(n)*(uI)...,uJr), (40) where k,l g 55. Whole family of gauge transformations A/|(u), which depends on an integer k where and arbitrary complex parameters a and (, is given by the expression «,(„,. («' +a'-> H(« + 2i, «)/*(i,)\ + To obtain the above results we have used the fact that $t(ui, • • •, uu) is symmetric under the \ e{« + 2Ji7 - u) e(( + 2/n + u)/n(7,) y1 permutations of Ui,-• • ,UM, which is based on the commutation relation (38). where, A(ti) = H(u)9(u), and T| = 2-'(s + 1) + 2ln - K. Here, K is the complete elliptic integral of modulus k. 7. By taking the logarithm of (16) and (39) and using (41), we obtain Next we construct the following state *yi = 1*1, + ' logtfA + n) + £ #.,(11.), (42) *(«i, ••,"M) = £*I(UI,- ••,««), (34) isz where Ua's are determined by solving the set of the following equations, where tf^iK) + vi(un + A - i)) = 2T./„ + J] v^(«. - ua)- (43) *i(ui,•••>«*) » B,+1.,.,(u,) • • • B,+Mj-„(u»r)n'-", (35) a*" and AC = (]V + l)/2. Here the family of the local vacuum states fl' is denned as Here /,, J„ S Z and i#>,(c<) = log(/i(a + n)//i(o - n)), iVj(a) = log(/i(a + 2rj)/n(c< - 2n)). n' = Ugg.-guj, (36) From (42) and (43), the energy of the system is given by where B = -7-E'j + "iog#(A + 'rt + Jv5>,(ti.) («) w,= /H(s + (2i + 2/+l), + „l)\ | +J ) (45) ' \ 9(s + (2i + 21 + \)n + »,)/' fE'i+fj;j.+" »i!«»+i)-T^'"("" -' ' -83- -84- where n = N/L. The first teem of the r.h.s. of eq.(45) is the energy of the spinless free fennions. ia's are the quantum numbers of the (so called) spin momenta[5]. The last two terms give the energy shift due to the existence of the impurity. 8. Thermodynamic properties are discussed in Ref.[9]. Reference [l]Y.Quano, Tokyo Univ.Preprinl UT-551(1989) [2]J.Kondo, Prog.Theor.Phys.32(1964)37 [3]N.Andrei, Phys.Rev.Letl.45(1980)379 [4]P.B.Wiegman, JETP Lett.31(1980)364 [5]N.Andrei, K.Furuya and J.H.Lowenstein, Rev.Mod.Phys.65(1983}331 [6JR.J.Baxter, "Exactly Solved models in Statistical mechanics", Accademic Press, Lon don, 1982 [7]Y.Quana, Talk al the Workshop on Superstring, RIMS(Kyoto Univ.), November, 1989, unpublished [8]L.A.Takhtajan and L.D.Faddeev, Russ.Math.Surv.34:5(1979)ll [9]Y.Quano, Tokyo Univ.Preprint UT-552(1989) -85- KUCP-0021/89 1. Introduction December 1989 In (super)conformal algebras (super)conformal anomaly appears as central terms of the algebraic relations. The central charge sitting in front of the terms characterizes the corresponding (super)conformal theory. For example, the mini mal models of the N=0 conformal algebra are specified by the value of the central charge c given as c=l . 6 ,,, m = 3,4,5,-- (1) Comments on Super Schwarzian Derivatives* m(m + l)' v ' Thus, physics of conformal field theories can be discussed by use of the infinites imal form of conformal anomaly. On the other hand, if we are concerned with the mathematical structure such as moduli space of Riemann surfaces on which SATOSHI MATSUDA conformal field theories are defined, we are bound to consider the finite form of the anomaly associated with the finite conformal transformations Department o{ Physics College of Liberal Arts and Sciences *-£ = *(.-) (2) Kyoto University, Kyoto BOB, Japan The corresponding anomaly is given by the so-called Schwarzian derivative [1] ABSTRACT <*•'!-T-Kir) (3) We propose and discuss a powerful method to obtain systematically the This expression corresponding to the N=0 superconformal algebra (SCA) is generic forms of super Schwarzian derivatives for various cases of higher N su- well known. However, it is quite nontrivial to obtain the finite forms of juper perconformal algebras. Schwarzian derivatives corresponding to finite superconformal transformations of niynerN [2,3,4,5]. In the present talk we are primarily concerned with the SCA's of higher N. We shall propose and discuss a powerful systematic method to obtain the generic finite forms of super Schwarzian dierivatives corresponding to those finite Invited talk given at the KEK Workshop on Topology, Field Theory and Superatring Theory, November 6th - 10th, 1389, KEK, Tiuknba, Japan (to be published in the Proceedings). superconformal transformations for various cases of higher N. -87- In order to substantiate our points, we first discuss the general mothod of to satisfy the composition law obtaining the expressions of (super) Schwarzian derivatives from the operator product expansions (OPE). It is pointed out that the method works beautifully S(z,=)=(§),S[z,S) + S(zJ) (6) for lower N, but that it becomes increasingly tedious and almost impossible to rely on it for higher N, especially for N=4, unless one knows the answer or the S(z,z) is called the anomalous term, and its solution satisfying the Cayley's identity (6) is given by the Schwarzian derivative (3). The correctly normalized expression close to the answer. Therefore we propose a practical method to guess S(z,z) is the correct derivative form of a super Schwarzian derivative up to a normalization factor of a simple scalar function. The correct factor of the scalar function can 5(,fl.^..}.^.!(5)'} (7, be identified through the composition law. Our strategy is that we shall only rely on the general method mentioned above, based on the OPE, just to doubly The general method to obtain the correctly normalized form of 5(2, z) is the check the correct normalization factor, thus achieving the rigorous proof of the following: First write down the operator product expansion (OPE) for L(z) correct final expression. LUiW't) = 4- + ^T3^ + 9"L{*2) + (regular term,), ( , = n-z,) (8) For the simplicity of our presentation we illustrate our strategy and practical 2l method in detail for the case of the N=0 SCA. Then we argue that our approach and substitute the expression (5) on both sides of Eq.(8) to obtain the relation can be generically extended to higher N. We consider each case of N=l, U(l)- for Z(i). Require the OPE similar to Eq.(8) to hold for the transformed L[z), extended N=2, and SU(2)-extended N=4 SCA's separately. then arriving at 2. The General Method for (Super) Schwarzian Derivatives LHS of (8) dh \2 / dz•> \ 2 - i I + pe Br ern 2.1 N=0 SCA ( IT) KIT) (*i) (*a) ( *«' ' ") A primary field i(n) = i(n) + 212 *(„) + izl 5"(2,) + |«J, f[tt) + ••• (10) while the energy-momentum tensor (the Virasoro operator) L(z), being the de scended field of the unit operator, as to obtain the relations 2 £(--) = (§) £(2) + 5(.-,J) (5) 2j = 2, + 2,J 2, + -ZU Z, +•• : Consequently, for the consequtive transformations z —• r —• z, S(i, z) is required 212 = 215 215 (l+1^=15+^7= 12 + ---) -89- -90- Making use of these in Eq.(9), we get Super energy-momentum tensors J'{Z): k 2 f(z) = - 2T*'(i) + -nfJGU) + -j=G(zye - sVfl L(Z) + iV'* 9a>e d,T (z) LHS of (8) = ^-+-rl{2(;'2) L(-:,)+l(£r-0jl) }+(<*>» 'ingvlar term*) (12a) - ±={ie)i*id,G{z) + •±*i9B)a,0(zWe + i(w)Jfl,»r'(x) Similarly for the right hand side of Eq.(8), we get (Ha) RffS of (8) OPE: = 4- + f-r + —) lip-)1 £(*"»> +s ^ '*>)} + ("s"'0"-*"•"») J J(Zi)J [Z2) - onJ'(Zi) - - = -5- + -j-j2(zJ)3 £(zj) + 25(*i, =2) 1 + (/e*» singular terms) (126) -z^~T""~zT,— "a7"~~^r"'x'"i'~~zl. Thus, equating the two equations (12a) and (12b), we finally obtain the correctly 1 fluo-'ffu 1 1 + ^—BnDiJ (Z,) - -——2—[D2J'{Z,))Bl2 + -€ —=5 J [Z,) 2~l i normalized expression (7) of S(z, 2) already given above. 12 * ^12 ^12 Now let us note that the global conformal transformations which are free of + 3Z'12 V Z\, 4* Zft the anomaly are given by the projective transformations, i.e., (146) Finite superconformal transformaion: S{z,z) = 0 «=> i(j) = ii±i (ad-lc=l) (13) cz + a j'{Z) = M{'{Z,Z) J'{Z) + S'{Z,Z) i (1S) The point here is that the anomaly free condition can be regarded as the differ M''(Z, Z) = \{ (16) 2.2 SU(2)-EXTENDBD N=4 SCA It is quite formidable to follow the general method literally in order to obtain the super Schwarzian derivative for the case of the N=4 SCA. Let me convince These formulas were obtained by Matsuda and Uematsu in the recent papers you by presenting the lengthy complicated formulas which will be involved in [5,6]. You will agree with me that it is quite a formidable task to calculate that attempt. the expression of the super anomaly S*[Z,Z) according to the general method -91- -92- presented above without actually knowing the answer or knowing the expression 3.1 N=0 SCA close to the answer. One has to be more practical and surely to be a little wiser The global conformal transformations are given by the projective transfor to get the following final answer [5]: mations i &{Z%Z) = -kD where which gives K(Z,Z) = ±(Djb)(D%) (18) K{z,z)~z' = -—5_y or -l= = -±= = a + d (20) Suprisingly enough, the answer is remarkably simple. In the next section we shall discuss the pracLxal method for the super anoma where K(z, z) is the local scaling factor foi- the transformation z —» i = z(z). lies. We have the relation dz = K{z,z) dz (21) 3. The Practical Method for (Super) Schwarzian Derivatives for the infinitesimal distances dz and dz in complex space. Therefore the lowest The practical method on which we shall elaborate is strongly motivated by the order differential equation can be written as observation (13) mentioned before. Suppose we know the global (supei, informal transformations which are free of anomalies. Then ask; what is the lowest order 0'(y=)=O (22a) differential equation such that the global transformations be the solution to it? We point out that the differential form thus obtained turn out to be the correct Actually, notice that anomaly expression up to a normalization of a scalar function. The correct factor of the scalar function can be identified by the composition law. The overall c number normalization can be easily determined by the infinitesimal form in the OPE. The normalization factor thus identified can be doubly checked by the The requirement of the composition law as well as the infinitesimal form of S(z, z) above general method of section 2, since we are already aware of the expression in the OPE determines the correct normalization factor. Therefore we get close to the answer. This last step of calculation at the same time completes the rigorous proof of the differential form of the (super) Schwarzian derivative including the normalization factor. Let us explain our point in the case of N—0 SCA. One may wonder if the differentia] form is unique. The answer is no. For -93- -94- example, we could have obtained the differential equation in the form is expressed as / 5(z,5) = ^a'(--^)^aM-') = o (29) a(v Ffl(Jr))=0 (226) Thus we have the solutions However, the nonuniqueness is only superficial, as is obviously exemplified by the 3 1 equality S £(r) = 0 <=> e(z) = £_, +e0z + e\z (30) The above 'hiee parameters z0, z2, e can be identified in terms of the generic parameters e_i, eo, e\ as They all lead to the unique Schwarzian derivative form up to the normalization factors which have to be correctly identified by the composition law and the OPE E_i =-i2, ea = -e=l-a, £i = ZQ (31) anyway. We only remind you at this point that the first form of Eq.(25) is more The corresponding generators for the global transformations are given by d, zd, generic than the second, as you will see in the cases of higher N SCA. z2d. The quantized forms are the Virasoro operators La, L±\: The algebraic structure of the projective transformation d — £-i, zd — La, z*d -» L (32) The algebraic structure of the global transformations (19) can be more trans +l parently studied by the following parametrization for zy —* z\ —* z\ It is well known that these operators form the SL(2,C)/Z2 ~Sp(2)~SU(l,l) algebra with no anomaly. -— = zi = 20H , zu = z\-zj (26a) 3.2 N=l SCA where z\{zi) is connected to the unit transformation as The N=l SCA is studied over the superspace Z = (z,9). Under the general £i —» z\ at za, z: —• 0, or —» 1 (264) transformations (z, 9) — (z(z, 9),9{z,B)) the covariant derivative D transforms We have three parameters (z0, zj, a — 1 + e) to describe the infinitesimal trans D = (DB)b + (Dz-6DS)di, D = -^ + 99„ b = ^ + 6di (33) formations. The conformal vector field e(z) to describe the general conformal It is required that D transforms homogeneously under the superconform&l trans transformations is defined by formations: «r = 5-.- = «(*) (27) D = (D9)D (34a) Thus we have Thus we have the superconformal condition: K(z,z)=l + dl(z) (28) Dz = 9(D9) (344) For the infinitesimal transformations the vanishing of the Schwarzian derivative The super projective (global superconformal) transformations Z\ = (si.^i) —» -95- -96- Z\ = (zi.^i) satisfying Eqs.(34a) and (34b) can be found as an extention of the overall c number normalization. Thus we get [2] Eq.(26) for the N=0 case: -— = z\ = r0 H = , — = »i = S0 H =— (J5) 4 V D9 D8 DB ) where the superdistances Zu and 0i2 are defined by The algebraic structure of ike N=l superprojective transformation Z»<=zi- z-i -hh, hi=h-fa with DiZ =Bu (36) n The infinitesimal superconformal transformations iZ = (6z = i — z,6B = 9 — 9) can be described by the superconformal vector field v(Z) [2] as The transformations are parametrized by three bosonic (roi*!i« = 1 + e) and two fermionic (Sa.^j) variables. Defining the infinitesimal superdistance v{Z) = Sz + 969 = c(z) + 9$(z) dZ = dz + Hi (37) se = ^Dv, Sz = v-9se = v-hDv Thus we have we have DB = 1 + i&t> (44) iZ = K(Z,Z)dZ (38) The vanishing condition of the super Schwarzian derivative takes the form where the local scaling function K(Z, Z) is given by K{Z, Z) = d,z + 9dj = (DB)' (39) S(Z, Z) = iVKDB. (-^j=) = iDt?., = 0 (45) Substituting the transformation (35) into Eq.(39), we find Therefore the solutions are given by 3 2 , f fl *(z) = 0 f e[z) = e-\ + eo* + «i* K{ZuZi) = {D19l) = jrzl or -J= = -±- = -^(-^n+a-^uJo) ' \fl'/3(z) = 0 \/J(*) = /».I+J9^ (40) From this the lowest order differential equation can be written as The generic parameters in Eq.(46) correspond to the above bosonic and fermionic parametrizations as Bla,(^)=o («) s_l = -zj, to = -t = V - a, «! = z0, 0_^ = -20j, 3}. = 280 (47) The composition law for S[Z, Z) determines the scalar function to be multiplied to the derivative form, and the.innnitesimal form o(S(Z, Z) in the OPE specifies and the corresponding generators are given by the Virasoro operators La,L±t, -97- -98- and the supercharge operators G±i as loop of the super Riemann surface satisfies those conditions, thereby causing the interchange B <-» B. From here on we only consider the case of U(l) holonomy 7 5, -L_i, z5, + i«a« - La, z 0, + zBd, - Lt with no global twist where it is always possible to choose the trasition functions to * (48) satisfy the second conditions, which we shall generically call the chiral conditions. dt-9d, — G_i, zBs-Bzdz — <7i Thus we have The bosonic generators are known to form the orthosympletic group OSp(l| 2)cs SO(l)x Sp(2). f/(l) holonomy : D = (D0)B, D = {D~8)5 with Di = ~BDB, Dz = BD~S and chiral conditions DB = DB = 0 3.3 U(1)-EXTENDBD N=2 SCA (53) The superspace for the N=2 SCA is described by the two Grassmann vari The global superconformal transformations with U(l) holonomy are given by ables: Z = (1,8,8). The corresponding covariant derivatives are defined as D + S8 & = +1 Bd 5 49 -Te \ " W 2 " M = < < ) z\ £\i z\ £\i (54) 1 . -Voe^MiJ + Voe-^ffutfq -a - j- = z\ = *o -i r= + •=— Under the superconfol.nal transformations D and D should transform homoge neously where the superdistances are given by D = (DB)D + (DB)D, D = (Di)b + (bi)b (50) which require that the following conditions be satisfied: Zu = ri-*l--i(Mi + Mj). «13=«I-9J, »u = «i-»j (55a) Dz = BD8 + BDB, Dz = BDB + BD9 (51) with the relations Since D' = D' = 0, we find from Eq.(51) DiZ„ = \ii7, OiZi2 = i*,j (556) (08")(D5) = O and (flfl)(08) = O (52a) and the infinitesimal form Therefore we must have dZ = di + hfldB + Bd§) (55c) either DB = DB = 0 or DB = D~B = 0 {52b) In Eq.(54) we have four bosonic (JO,*3,S = ar-1,^) and four fermionic (flo,0o,Si, The first conditons in Eq.(52b) correspond to general 0(2) holonomy with twist 9j) parameters to describe the transformations z\ —» i] —• z\. The local scaling where at least one transition function in the consecutive patches around a closed function K(Z, Z) satisfying dZ = K{Z, Z) dZ is defined to be and calculated from -99- -100- Eq.(54) as global superconformal transformations K(Z ,Z ) = (Ui«i)(5i*"i) = 0,Ji + i(»ifl„ S~i +81M1) = -^r'A (56) 1 1 S(Z,Z) = 0 => [D,D]d,V = 0 (62) From this we find by putting K\ = K[Z\,Zi) V(£) can be expanded in terms of the generic parameter functions e(z), fi[z), 0(z) and i(z) as V[Z) = <=(*) + B0[z) + Sft*) + ij«W (63) and obtain the lowest order differential equation as Then Eq.(62) implies that (UiA-SiOi)(^=0 (58) Si = 0 ri(r)=«o /?(*) = /?_•+,8^ [D,D\ d,V = 0 (64) 3 This derivative form, after being normalized by the composition law and the fl 0 = O 0{') = L'. + P>.z ! OPE, correctly gives the super Schwarzian derivative for the N=2 SCA [3]: a3£ = o , c(i) = e_i + coi + €ii The generic parameters correspond to the four bosonic and four fermionic 2 V . if' . - (59) parameters in Eq.(54) as v cj f d.DB L S.J38 (d,B)(B,8)] ' 2 L" Dfl 5l (Dfl)(5l)J 6_i = -zj, to = -E = 1 - or, ei = *0, *o = 2iV, (65) TAe algebraic tincture of N=S global svpcrconformal transformations The superconformal vector field V(Z) describes the infinitesimal supercon- The corresponding global generators in derivative form can be found straight formal transformations [3] forwardly from Eqs.(60),(63) and (64), and their quantized expressions are given by the Virasoro operators Io,£±i, the supercharge operators G±L,G±L, and SB = DV, 6B = DV, 6i = V - l-BDV - ^BDV (60) the U(l) charge operator % which form the closed subalgebras of the TJ(1)- extended N=2 SCA. The set of the bosonic operators form the orthosympletic The chiral conditions are satisfied due to the relations D1 = D1 = 0. K(Z, Z) gToup OSp(2 | 2) = SO(2)xSp(2). can be expressed as K(Z, Z) = (D6)(D'B) S 1 + {D,D}V = 1 + 9,V (61) Thus we obtain the following condition on V[Z) describing the infinitesimal -101- -102- 3.4 SU(2)-EXTENDED N=4 SCA Let us define the superdistance Zn and the infinitesimal superdistance dZ: In the zhiral formuhtian [6] of the SU(2)-extended N=4 SCA four Grassmann Zu = n-zi-\(e\h* + 9\Jl), dZ = dz + ^9.d6' + B°d9a) (71) a variables 9 ,9a{a — 1,2) are introduced such that they should transform as a doublet (9°) and an anti-doublet (Ba) of the SU(2) Kac-Moody algebra. The With the K[Z,Z) introduced above we have the local scaling relation dZ = corresponding covariant derivatives are denned by K(Z, Z)dZ. The new feaure here is that in our chiral formulation we also need to introduce the chiral superdistances Zu±: °'=w+y-°» °°=wAB°d' (66) Zn± = ^12 ± !»t'i»u. (72o) Now the cAiW superconformal conditions where 9'n = 8f - SJ, Sa. = §u - J». (724) (67) h i M = o, b'z - \{b'h)'e = o Putting Zi at the origin in the superspace, we also have are imposed in order to preserve the SU(2)|oc,|xSU(2)gi0b,i structure of the N=4 (72c) SU(2)-extended SCA under the superconformal transformations. Under the chi- ral conditions (67) (9") and (0„) do not mix with each other. Consequently the Note that the chiral superdistances satisfy covariant derivatives Da, D" transform homogeneously as J?.**-** {D>f>-=1 (73) h i and Da = {Dj )bb< 5" = [D%)b (68) From Eq.(67) we obtain With a judicious consideration we can write down the global superconformal transformations in terms of the chiral superdistances as b (Dj'){D Sc)=6\lHZj) (69) iiS! ft j. ,., yg(«^»»)' »i._* _f , Mh*- ). where l K(Z, Z) = d, s + -9a8j' + \i'd,$, (70a) Also we have the following expressions for K(Z,Z) from Eq.(69) £i_ " " Zn- Z\%- K{Z, Z) = koJ^b'Si) = y/dct{D9)det(D~6) (706) where we have six bosonic (ro,i!,e = a—1,0) and eight fermionic (^.flooi^J, ^Ja) — 103 — -104- parameters. By substituting Eq.(74) into Eq.(70) K(Z, Z) can be calculated as the superconformal vector field E(Z) < SB" = B'-8" = D'E, 6§a = Ba - Ba = D.E (79) Sz = z~z = E- ~iaSf - U°SBa Since K{Z, Z) turns out to be the product of the chiral distances with ± suffices, The chiral conditions (67) require E(Z) to satisfy it does not help to naively take the inverse square root of it as we have done for the D„DiE = 0, D"DhE = 0 (80) cases of lower N, in order to obtain the lowest order differential equation. Instead, we should take the logarithm of K[Z, Z). Then the variables with ± suffices can The solution to these is given by be separated into additive terms, and as a result the lowest order differential equation can be obtained in a transparent manner due to the properties (73) of E(Z) =<.(*) + 8,F(') ~ &(*)«" + 5» z K(Z, Z) is calculated to be Dlab\]nK(ZuZ,) = S\ ft£l=, b\D,.\aK{.ZuZ,)=6\ Sp± (76) K(Z, Z) = i(iJ.*')(5°«j) 2 1 + i{Z)„ D'}E = l + d,E (82) therefore we obtain Thus for the infinitesimal global superconformal transformations the vanishing D (ai)\DblnK(Z,Z) = t) (77) a of the super Schwarzian derivative (78) requires the following relation Again the requirement of the composition law together with the OPE determines S'(Z, Z) = Q => -k Do-'Dd.E = 0 (83) the normalization for the derivative form of the super anomaly for the N=4 SU(2)-extended SCA as [5] which implies that Do-'Dd.E =iflV9S3e + tia'd'P + d^c'B - hjSB^Ba^P) + hs > <=*• • The algebraic structure of SU(£)-exUnded N=f global aupcrconformal iranj/or- fl'fl. = 0 *°* j AM = P-L. + Pi.' matians 6°e = 0 U(-) = «-i+ — 105— -106- Eq.(74) as REFERENCES ?a. = 2ifl, e_! =-zj, £Q = -£ = 1-O, (\ = 20 [1] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B241 (85) (1984)333. Pi; = -<>'„ P\ = ii, /9_j. = -«».. /8ia = «o. [2] D. Friedan, in Proc. Santa Barbara Workshop on Unified String Theories, The corresponding global generators in derivative operator form can be found eds. M. B. Green and D. Gross (World Scientinc,1986) p.162. from Eqs.(79), (81) and (84). The global superconformal generators in quantized form are given by the Virasoro operators La, L±i, the SU2) charge operators [3] J. D. Conn, Nucl. Phys. B284( 1987)349. TJ, and the supercharge operators G*.%, G±L,- The bosonic generators form the [4] K. Schoutens, Nucl. Phys. B295JFS21] (1988)634. special superunitary group SSU(1,1| 2) =SU(l,l)xSU(2). [5] S. Matsuda and T. Uematsu, Super Schwanian Derivatives in N=*4 SV(i)- Exiendtd Svverconformal Algebras, Kyoto University preprint, KUCP- 4. Conclusions 0018/89, September 1989 (to be published). To conclude, we present the summary of super Schwarzian derivatives: [6] S. Matsuda and T. Uematsu, Phys. Lett. 220B(1989)413. N=0: SL(2,C)/Z, ~Sp(2)~SU(l,l) S(Z, Z) = f SWd1 (--=), K{z, z) = Bi N=l: OSp(l| 2)a SO(l)x Sp(2) s(z,z) = |vTna,(--L), if(z,^) = (DA)' = a,z + se.s N=2: 0Sp(2 I 2) =SO(2)xSp(2) S{Z, Z) = | v7/? [D, D\ (-J=), * (*, Z) = (/>*)(£?) = &f+±( fa.?+fa«) N=4; SSU(1,1| 2) =SU(l,l)xSU(2) i i S(ZtZ) = -kD a 7iT(Z, Z) = \Jdti{DB)dtt(b~e) = i(J90l»)(fl /t) = 5,1 + i/a^o"' + U'd.L — 107— -108- Lie Algebra Co ho mo logy and N=2 Superconformal Field Theory $2. N=2 Super-GKO construction In this section we introduce the N=2 super-GKO construction formulated by Kaaama Shinobu HOBO nof and Aldhiro Tsnchiya" and Susnki[4] in the form which is suitable for our later arguments. Let g be a simple Lie algebra over C which has a root space decomposition; Department of Physics, Nagoya University, Nagoya 464, JAPAN o = t©^j7 , (2.1) "Department of Mathematics, Nagoya University, Nagoya 464, JAPAN a where t is the Caxtan subalgebra and A = A+ U A_ is the root system. We normalise the Cartan-Killing form ( , ) :gxg-» C by (9,0) = 2 with 8 being the maximal root. Consider a reductive subalgebra hCg which containea the Caxtan subalgebra t. Then h has Abstract: a decomposition into its abelian part ho and the simple parts hj (1 < ; < Jf) as h = h ®hie-*>®hjv . (2.2) We study a cahomologicaL sttuctuxe of the N=2 superconibnnal field theories con 0 structed by Kuama and Susuki. We note that the N=2 snpexconformal field theory can According to this decomposition of h, the Cartan subalgebxa t also decomposes as be formulated precisely as the affinisation of the Lie algebra cohomology theory for the t ="t0 e ti © - - - © Ijv . (2.3) classical Lie algebras by Kostant. A general theory far writing down the Poincaxe polyno In addition, the root system of h has the following structure; mial for the chiial ring is also presented. A(h) = A(h,)U-.-UA(hN)CA . (2.4) As a simple consequence of our definition of h, we have 0 = h©m+ffim_ , (2.5) SI. Introduction [htm±] Cm* , (2.6) It is known that N=2 sapeiconformal field theory(SCFT)[l] has a variety of remarkable where ffl± = (A \ A(ti)) n A±. We denote the affine Lie algebra of g, h and m as g, h and properties owing to the complex sttnctnxe in its (snper-)algebra. One of them is the non- A, respectively. lenormaliiation theoxem which we obtain when we describe the N=2 SCFT by means of the The basic ingredients of the N=2 super-GKO construction are the g-integnfale module Landau-Ginsbnxg effective Lagrangian(2]. This theorem makes a beautiful correspondence 1(A) with the highest weight A 6 P+ (dominant integrable weights of g ) and the fermion between the partition fnnction of the N=2 minimal (c < 3)SCFT classified by ADE type Lie Fock space F, (e — 0 or | according to the Ramond or the Neveu-Schwarts fermion, algebras and the form of the snperpotential classified by the same ADE in the singularity respectively ). As for the g-module part, we denote the g-current operators acting on £(A) theory[3]. Another point, which is the main subject of this paper, has a representation as ^(n) (n € Z) using some basis of g. The integrable module L(A) admits a unique (, up theoretical meanings: A wide class of the N=2 SOFT recently formulated by Kasama to constant, ) positive definite hermittan innerproduct with Jj(n) = JA(—n), For later and Susuki(4] via super-GKO[5] construction admits a precise parallelism to the theory by use, we choose the set of basis {JA}AGA "eh that it decomposes, according to (2.2), as Koslant for the generalised Boxel-Weil theorem[6]. To elucidate this point, we shall reveal {JAU*A = {JMJMC* ® • • •« {J.*}.**** © {J«}aeA(m )UA(m_) i (2-7) the cohomological structure behind the Kasama-Susuki model contrasting to the (classical) + Lie algebra cohomology theory. As a resoll, we shall be able to parametriie the non-trivial where {Ja<}aiMi foxms a set of the basis of hi (0 < i < N). We construct the Fock a a cohomology element in terms of the affine Weyl group. The Pobcaie polynomial for the space T, in usual way utilising the complex fermion oscillators i> {r),i>a(r) = i> (r) (a € chiial ring defined by Vafa ct al [7| shall be discussed also. A(m+), a € A(m_), a & -a, fGZ + f) and the vacuum vector |0) such that The program described above has already done by Vafa ei di., including some inter esting conjectures. Though our formulation overlaps their arguments tn same places, H will give new Insights tor the problem* and conjecture* addressed by them. j«(r}[0} = 0 forr>0anda€A(m+) , a ' Work supported in part by JSPS No.01790194. «V (r)|0) = 0 forp>0andd€A(m-) . -109- -110- Again we can introduce a nniqne (, ap to constant,} hermitioan innerprodact in ?, by Definition 2.3. (Fennion current operators): setting *•(*)»=*«(-*). Now we define the cnrrent operators JA{*) and the fermian operators j>a{z) hy the formal Laurent series on a variable t\ »„(*)=«•(*) s: «>•(*)*,(*): («6A(m+)) (2.15) /,(*) = E ».(*) (2.16) WsSJi(»)«'"'1 . t2"9) .€A(n4l ,T(r)== 2 *-(r)«—* . (2.10) Now we can write down the operators which realise the N=2 supersynunetry in ^*. ® £(A). Definilion2.4. (T(x), C*(i), J(«) ) In order to tame the infinities associated with the infinite snm, we define the following normal orderings; rw=27i4^r£ :^(«y"w: + j E :••(')•"('): V r'-t€/4 »6a(»+) J (m)/B(n) (m < n) { 4 -^EEjA.wi-w:, (2.17, M^MJB(n) + JB(n)JA(m)} (m - n) (2.12) G+w a •to(»)J*(m) (m > n) . = \lrr:<- E * WJ.w-; E W- **(«)^(«Wr(») >}(»•»> In the following, we construct operators acting on the space F & £(A) ( with A = fcAo + fi t •«(-) .it;, A, A is the classical part of A ). G-(x) = {i7t(.))' , (2.19) Definition 2.1. ( it-current operators ): a 7 l + J I a J J(«) =r^E ' <( ) '( ) ' "'=< - ') i«6%) J Using the structure constants fAB° of g ».t. [JA, JB] = Eo fAB° o , we define the diagonal action of the hj (0 < t < N); J..t» s/.,(«)- £ /../'**«*.<*)' - <2-13> ft = 5 E « • t"i) a€A(m+) where the subscript a* belongs to the index set A\ (0 < i < N) With these definitions of the operators) Kasma and Snsnld showed that: Proposition 2.2. Proposition 2.5. ( N—2 snper-GKO construction^]; Kasama-Snsnki[4]) fif-current operators (1 < t < JV) satisfy the following operator product expansion (OPE); (i) The Virasoro operator T(n) (n e Z),lhe snper-cnrrent G*(r) (r s Z + c) and the n(l)-cnrrent J(n) (n e Z) defined by T(i) = £„ T(n)«—*, G±(s) = 2 £„ G*^)*-'-*'' and J(r) = £„ Jtn,*-"-1, respectively, satisfy the following '*)-6S^Ew«"i • <"> N=2 snpereonformal algebra. (SGA): • For e = J (NS); where o^fcj.cj belong to the index set At and 0( is the maximal root for each h< . The central charges, Va, we given by * + g' - hJ&gA with g' and V'» t>«™g the [ T(n), T(m) ] = (n - m)T(n + m) + £(n» - n)*. „.„ (2.22) dnal Coxlel numbers of g and hj (1 < > < tf), respectively. As for the abelian pact + e ho, we aarame f^ltf t* ^ 0 and A" = 0 in the above formulas. ( T(n), J(m) ] = -mj(n + m) (2.23) -Ill- -112- ± i [T(n), G (r)|=(!-r)G («+p) (2.24) J3. Chain complex structure on Tt ® L{\) (J(n), J(m)] = jnf.+m.o (2.25) + In this section, we define the chain complex induced by tne supercharge G (0) on the [J(»), 0*101 =±0*(n + r) (2.26) space FQ ® -L(A) . A paialleliBm to the classical Lie algebra theory by Kostant[6] will he + pursued. {G (r), G-(.)} = 2T(r + •) + (r - .)J(r + •) + |(r' - J )''+•,» (2.27) We first note that the supercharges G*(0) hare a property {G+(r), G*(')} = { [*(0),G*(0)]«±G*(0) , (3.1) • For c = 0 (H)j + 3 [T(»), G*(,)l = (^±i-,)G±(n + r) , (2.23) in addition to G (0)* = G~(0) = 0 and G"(0) = G+(0)» . It can be eauly deduced that these properties defines a chain complex on F0 ® £(A); + {G M, G-(.)} = 2T(r + .) + (r - . - l)J(r + .) + jr(r - l)S,+.,0 , (2.30) Definition 3.1. ( G*(m+1X(A)) ) t with the other (anti-)commutatiou relations remain the same as in the NS-case. We denote as C'(ra , L(A)) the (co-)chain complex on ^r ®£(A) whose coboundary For both eases, the central charge is given, with a,/3,7 6 A(m+), by + 0 operator is given by G+(0) and degree * is determined by the interger eigenvalues of the fermion number operator J/(0). The main subject is to determine the cohomology group B*(m+IL(A)) Cor C*(m+fL(\)), For this purpose, we first note thai the spaces J3"(m+,X(A)) as well as C(m+,I(A)) (u) The abore operators of N=2 SCA commute with the diagonal action of the h; admit the diagonal n-actian. This is because the supercharge G+(0) commutes with the •/•t(n) (0 < i < JV) as shown in Prop.2.5. Owing to this property, we may consider the [ T(»), i.(m) 1 = [ G*(r), J.(m) ] = [ J(n), /.(m) 1 = 0. (2.32) decomposition C(m+, Z(A)) with respect to the irreducible integrable modules of h as The eohomologieal interpretation for the N=2 super-GKO construction introduced C-(m+,£(A))= Y, C'(m+,X(A))( , (3.2) abore is one of our main subject in the latter sections. Before closing this section, we remark one point that seems to bare been overlooked so fax in the titer times, where P+{h) means the set of the dominant integrable weights for h with the fixed central Remark charges ki (0 < i < tf) given in Prop.2.2. The spaces C"(m+,I(A))< are equivalent to the modoles of the branching coefficients Bn,t in the GKO constrnction Ft 9 £(A) = At first sight, it might be seen peculiar that the N=2 SOA changes its form depending £ a tf,(0). = (T,e> (T6f)} A(m+) whereas the usual normal ordering in the N=l SOFT would say J {^ (0)V'c,(0) + a i>a(Q)4> (0)}. This difference of the normal ordering is the origin of the changes inch as (2.29) and (2.30). It will soon become clear that our choice of the definition for the normal This branching coefficient inodoles BA,< constitute the (, not necessarily irreducible,) N=2 ordering (2.11) is the natural for our cohomological interpretations. SCA-module whose characters close among themselves under the modular transformations. Now we calculate the Laplacian {G+{0), G~(0)} on each irreducible component. —113 — -114- Proposition 3.2. Then W decomposes into a product as 1 Consider an arbitrary state v\t( £ £A,C then the Laplacian acts on this state as W = Wt • W , (3.7) + J a l {G (0), G-(0)}OA,{ = ]^{|A + p| -|4' + ^| K( , (3.4) thai is, arbitrary element u G W has an unique expression as w = wi« , wt € 1 l Wu O € W. a where ? = fcAo + £ + const.* , £ S P+{h) and p = g*AQ + J £a6A+ - Proposition 3.4. (proof) The seros of the Laplacian (3.4) occur if and only if there exists a 6 W s.t. As an element of ^*0 <9 £(A), t>A,( can be expressed as , e = HA + p)~p . (3.9) "A,e = E c?1'.r*5fi:;f^«'(-rl)-"*"'(-',)"-*'(-«t)-"*''(-'.)l<)) (».»),W,<) l If t? belongs to the set W, then *;* = tr(A + p) - p mod ZA0 + Z5 £ J*+(h) gi^es the highest weights of the integ-'-ble modules Lh(() with the levels fc,(0 < i < N), /•6Q+ given in Prop.2.2. 1 d The action of T,(0) and ft(0) on this siate m ('jffiff) + f) «" 'liUffi' The proof of this proposition is a simple application of the prop.2.13 given in ref.[10]. (« = iS,6i+(b)7,) respectWely. Since T,(0) - \J[0) = (T,(0) - !/,(<>)) - ^ , Complete proof will be given elsewhere[12], there we will also give a precise relation to the this part of the Laplacian operates on v\>( as a scalar Witten index calculation. To proceed, we mention some definitions. For each w 6 IV", let $£ 2 w(A_) D A+ and {$6) = Etf€»# ¥>• Then the relation (#,;) = p — Qp is a simple consequence of the properties of the (amne) Weyi groap. Recalling the definition Wl in (3.6), we can convince 1 l l z d ourselves that if w G W then (#*.) =,3- &p e E«§eA(m+) >° * where Ihe relation [ 5/(0) - j J(0), «>"(-r) ] = rV°(-r), in onr definition of the normal ordering, mnst be tued. Summing op the all eontribntloni in {G+(0), G"(0)}SA,J, we Proposition 3.5. obtain the expreuion for the r.h.i. of (3.S) with C = AA0 + f - (AT + mt)S. Q m l i) The mapping tr (-* E (m+,L(A.))itt is a bijection of W onto the set of all According to the Hodge theory, we can characterise the cohomologj element! u the irreducible components of #'(m+, 1(A)) as the it-module. And the multiplicity Eeros of ihe Laplacian: of ,ff*(m+,Ir(A))(, is one. , ii) The highest weight vector |A,(») has a form, as a representative harmonic J"(m+,j;(A)) = H (m+,£(A)) cocyclein ff"(m+,I(A))f,, = {» 6 ^o ® £(A) | O+(0)« = O-(0)» = 0 } In onr case, this problem is reduced to an algebraic equation dne to Frop.3.2 . To write |A,(»)= II +»(-')|0)®ff(|A» • (S.10) down the solutions of ihe equation, we describe some properties of the Weyl group: Lemma 3.3 iii) The degree of the cocyele |A,&) is giren by Ihe fermion uimber '(*); Let W be the afflne Weyl group for g and Wi be thai for the subalgebra fi satisfying t +1 ) l i'(*)s#{peA |y.g*»}-#{ (.SA! - |^6*,} , (3.11) (2.2)-(2.4). Define the subset W CW s.t, + ( t ( H"s{wS!VMA-)nA+C A(m+)} . (1.6) with A +*' = {7 S A+| 7 = ±/3 + mS, 0 6 A+, m € Z}. -115- -116- Prop.3.4 and Prop.3.5 are the analogue of the Theorem 5.14 in ret [6] obtained for the P reposition 4.1 classical Lie algebra pairs g and h. The extensioui of the Kost ant's theory to the case of g and a classical h hare been done in re&.[10][ll]. Onr arguments are for the pairs of the Define the following automorphisms of the Cllifbrd algebra of the fermiona and the affine Lie algebras g and h. (Feigin and Prenkel[9] hare introduced the modified length affine Lie algebra g: function /(«) for W.) The resolts by Kostant for the classical Lie algebras can be xecorered i) For the Clliford algebras, let + by taking the "classical limit", i.e., G (0) -» Q, G~{Q) -» Q' and T(*) -• l{tr) with »{(+») = *•(*+<»,«)) , (4-D with v &' and (r,a) e Z for Va € A(m+) U A(m_) orZ+j for va € A(m+)UA(m»). ii) Fox g, let respectively. Remark i) The automorphism a[ is a special case of the well-known BogoliuboT automorphism. u) The automorphism (7 with 7 € AT (the long xoot lattice of g) coincides with the 1 $4. Spectral flow and Paincare polynomial translation part of the affine Weyl group. In this case i7 acts on the highest weight vector |A) £ 1(A) as MIA)) = |f,(A)) 6 1(A). In contrast to this, when we extend In the last section, we have described the cohomology element in terms of the subset the situation to 7 € P and 7 g M, then t7(|A)) docs not remain in £(A). We can of the affine Weyl group, TV1. According to the results obtained there, N=2 SCA-module formulate the action t,, 7 6 P by relating it to the automorphism[14] which originates l in the symmetry of the extended DynHn diagram. B\,t has a harmonic eocycleifand only if £=s (t with some element Consider the diagonal action o*7 = tri +i7 defined on F®L\ with va — (7, a), 7 e P the GKO construction for the Viiasoro minimal seriet[5]; *"u(2)k =mraad Jl ^'+t,»+l ®^((^'+lJAo + '^i), (0 <*^fci0 -117- -us- The property (4.5) enables us to perform the field identification program explained in §5. Summary and discussions the first part of this section. We have revealed a cohomological structure behind tbe N—2 SCFT associated with Now let ns restrict onr attention to the tt=2-module whose primary fields belongs to the affine Lie algebra pairs (§, h), by formulating the problem as the affinisation of the the nontrivial cohomology elements. Then we can describe all such states as the elements of theory by Kostant. As a result, all the nontrivial cohomology elements (, the chiral primary F ® Li, due to Prop.3,7- Furthermore by solving the automorphism (r formulated above, l 7 states,) proved to be expressed by the knowledge of the subset W of the aJfine Weyl group we can know how we should construct the irreducible theory by the diagonal modular of g. The Poincare polynomial defined for the N=2 SCFT of the diagonal modular invariant invariant partition function at least for the part that contribute to the Witten index. partition function can be written after the general field identification program. Then the Poincare polynomial P{t) is defined by the summation over the primary fields of the theory weighted by their charges Qi; In the case of the minimal(c < 3) SCFTs, it is known that tbe Poincare polynomials have the following form Thus we have established a general procedure to write down the Poincare polynomial. Explicit expressions of (4.8) for some characteristic models will be reported in the future «-«D!-7> • <«> publication^2]. Here we only mention a lemma which, is useful for writing down the Poincare polynomial. where ^'s are the charges of the chiial primary fields and c = j^pj, d = A+2. Tne fact that Lemma 4.3. the Poincare polynomial has the form (5.1) is originated in that the chiral primary fields have a ring structure defined by the polynomial ring divided by the gradients (, equations l The set W defined by (3.7) can be expressed as of motions,) of the snperpotential of the corresponding Landau-Ginaburg Lagrangian[2]. For the general coset models by Kasama and Susuld, however, it turns out that there are l x W = lb P< W , (4.9) many theories whose Poincare polynomial does not have the form (6.1) and (5.2)[7][12]. We should think seriously what this fact means. At present it seems very interesting to 1 where To s {t\ \ A g IQ} and IV is defined for the classical Lie algebra pairs (g, ft). search the geometrical meaning of the Foineart! polynomial for Jlr > 2. For the case of h ~ 1 and (g, h) corresponding to the hermitian symmetric spaces (HSS), Vafa el al[7] Thus the order of the set Wl is infinite. However we can see that this infinity arises showed that P(i) coincides with the Poineare" polynomial for the cohomology ring of the associated with the u(l)-current algebras[12]. As a result, we can determine the Powcare flag manifold QjB. It is probable that the Pobcare polynomial for k > 2 is related to the polynomials from the knowledge of the set Wl for the classical theory. quantum deformations of a certain classical polynomial associated to the flag manifold. From the view point of the representation theory for the Lie group, Kostant'a theory is the starting point for the BGG (Bernstein-Gelfand-GeUand)[18] resolution. The BGG resolution has a bosonic picture in contrast to that the Lie algebra cohomology has the fermionic picture as constructed in sect.3. It is very curious to connect these two pictures for the case of the affine Lie algebra. Especially the two cohomology structures, the Lie algebra cohomology in the fennionic picture and the BUS structure for the vertex operators[l7][9] in the bosonic picture, should be unified in the geometry of the infinite dimensional flag manifold. -119- -120- References [1] W.Boncher, D.Friedan and A.Kent, Phya.Lett.B172(1986)316; RDiVeccU, J.L.Petersen end H.B.Zhtng, Phys.Lelt.B162(1985)327; S.Naum, Phys.Lett.B172(1986)323. [2] E.Martinec, Phy».Lctt.B217(1989)431; "Criiiealilf, catastrophe, and compactifica- lJofunin the Kniinik memorial Tolame ( eds. L.BrinJcs et af., World Scientific (1989)) C.Vnfa and N.P.Wamer, Phys.I,ett.B218(1989)51. (3| VJ.Amold, Singularity Theory, Lond. Math. Soc. Lecture Series #53. [4] Y.Kaiama and H.Sttinb, Nuc].Phys.B321(1989)232; Phy».Lett.B216(19B9)112. [6] P.Goddard, A.Kent and D.Olire, Commun.Mnlh.Phys.l03(1986)105; V.O.Kae and T.TodoloT, Commnn.Maih.Phy».102(1985)337, [6] B.Ko«tsnt,Ann,ofMath. 74(1901)329. [7] W.Lerche, C.Vafa and N.P.Wamer, Nucl.Phys.B324(1989)427. [8] A.Schwimmer and H.Seiberg, Phys.Lett.B184(19B7)191. [9] B.L.Feign and E.Fxenhel., Representations of Amne Kac-Moody algebras and Bosonoia- tion, Representations of AfHne Kac-Moody Algebras, Bosonisations, Aifine Kac- Moody Algebras and Semi-Infinite Flag Manifolds , Preprints(1989) [10] H.Garland and J.LcpowsU, Inr.Math. 34(1976)34. [11] S-Kumar, J.DuT.Geom. 20(1984)389. [12] S.Hosono and A.Tsachi;a, in preparation [13] D.Qepner, Phys.Letl.B222(1989)207i Princeton preprint, PUTT-1130, May(1989) [14] K.Hasegawa, Pnbl.RIMS, Kyoto Vnir. 25(1989). [15] A.Tsuciiya and Y.Kanie, Tanignchi Symp.PMMP, Katala(1985)385. [16] LN.Bemstein, I.M.GelJand and S.I.Gelfand, in Lit Group and Their Repreaentaiioni, ed. by 1-M.Gelfaud, WUey New York 1975. [17] G.Felder, NnclJ>hji.B317(1989)215, D.Bernard and Felder, Preprint ETH-TH/89-26. -121- November 1989 clements of Weyl group), where (Torus) is defined by the root lattice with arbitrary Parafermios as ZN Orbifold * radius and Weyl group is the group generated by monomials of the Weyl reflections, and level 3 SU(Z) Kac-Moody Algebra ** We want to study the parafermionic structure of orbifold models, e = 1 models axe already classified well and named c—l rational torus models [8,10]. In this paper, we Alrira FUJITSU discuss simple models, AN-I/ZN, which are abelian orbifold models. We interpret the Graduate School of Science and Technology, Kobe University, JVada, Kobe 657, Japan representation space of orbifold models as that of Zff parafermionic current algebras. All the other orbifold models, involving nonabelian ones, will be studied successively Abstract in the future. In Sec.2, we explain our language, a correct quantization of a closed string on We Btndy quantum field theory of bosons on torus and oibifold. When torus is in torus/orbifbld. In Sec.3, we discuss crural algebraic structures and the representation special moduli, representations of the theory are equivalent to those of some rational space of rational torus models. Sec.4 is devoted to discuss parafermionic current conformal field theories. We show that there aze parafermionic current algebra in Zjf algebras of Aft-\jZs orbifold. orbifbld models. 2. Quantization of Bo torn on Toru$/Orhifold We quantize bosons on torus/orbifald correctly, and get correct cocycle factors 1. Introduction naturally. The action principle for free bosons, or say coordinate fields of bosonic Conformal field theories [1] are recently studied by many people to understand closed string on torus is critical behavior in lattice models etc. and first developed as a main language in string 1 ab IJ J J theories. S = jdr ~ J'*" dtr^d^dtX + C B daX dbX }t (1) We have an idea to produce a realistic four dimensional superstring theory. The where $ is a parameter which indicates the storting point in tr. BIJ is a constant back idea is to introduce c = 9, JV = 2 superconfonnol models as a model of internal space. ground antisymmetric tensor field. The equation of motion (d —6> )(d +d )A'(r,o>) = Then many authors tried making models [2,3]. Some of the models are successfully T ff r ir 0 tells us that bosons separate into left- and right-movingports , interpreted as Calobi-Yan manifolds. In any models, it is important, we think, to understand that models have a * = ^{*>to+ ?(*)}. (2) structure of (Sl)<8(PaTafermion). This structure is also seen in every Kac-Moody where j = e' * We report only this part of the talk, This report is the same as our preprint The total momentu.nm is canonically defined as entitled 'Parafermionic Current Algebras of Torus/Orbifold Models' [11]. ,+T 6S 1 ** This part is based on the collaboration with M. Tabnse. Please see it in Ref. [10]. / -123- -124- IJ ,1 J These results don't depend on the parameter J for any T . We take s = 0 for where (BpH)' = B p R. By torus condition X ~ X + 2TA, P 6 A* dual to A. By closed string condition X(tr + 2») = X( IJ 7 rJ We here introdnce a tensor T formally which is usually taken to be sero. in W'-sector. Using eq.(ll) and the commutation relation [Q'.i/* ] = iS % we derive Now we will decide the form of tensor Trj. [X',QJ} = -ii*l{l-W>)-1]". (12) If we suppose the interaction of the three strings in the covering space where the configurations of the string states are presented, it is very natural for us to have an This result has been already discussed by Itoh et al. [7]. idea that the starting point of the string configuration does not move when the two Correct cocycle factors are realized naturally in the vertex operators by the rela strings link up to become the other string. We accept the idea and use it as a principle tions eq.(9) and eq.(12) in untwisted- and twisted-sectors, respectively. for the quantitation. Let ns consider the string state of the winding numlrer L = 0 and at the position 3. Wave Function* on Torut/Orbifold and Ckiral Algebra of the center of mass coordinate X — *. If another string state oiL~l and X = a + W We discuss the wave functions on the torus. connects with the state at the position a, the linked string stale is not of X = x+x+xl The center-of-mass coordinate field and the coordinate field conjugate to the bnt of X = z + */. We can represent il in the level of the first qnantiiation. It winding number L are fQ follows thai when we operate e' to the state \X = e, L = 0 >, we get a state *(«.«) - ^W«)+ «»)]. ,.„ |X = *+xl, L — I >. So the operator e"' moves position in X space of the stale, and l «(*.*) = j;M*)-m +BP{z) + B^i)]. "' the operator Q a not commote with X. This also leads that e'*x moves Q —> Q-np. We now get the physical wave functions 9 on torus with the momentum p € A* Therefore, we have a result and the winding nnmber I € A: T" = -ix«". (9) 9(p,l) = e*p[t(pX(r,x) + lt)[x,i))] By using these relations, we calculate commutators of gt, — 125 — -126- We want discuss finite theory, say rational tons models in this paper. We explain The finiteness of this model is stated in the following. Dnal lattice A* is made of conditions for the finiteness of the model in the following. the composition of root lattice A: If p -1 - Bl ~ 0 for a ¥(p, /), then *(p, /) is independent of z and we call it chiial A" = {0 + A}U{A,+A}U{A, +A}U"-U{Ai + A}U"-u{AW-i+A}. algebraic structure J{z): J(z)=ezV(i\f2lV{z)). (15) We assign Zit charge k for lattice {A* + A}. The current given in eq.(15) has positive integral conformal dimensions Let a, 6 are the vectors in the lattice {X\ + A}. If we take p = a+6,1 = o — o g A, then d = l-l = l{p-Bl) = l-p£Z+. (16) «(p,/) = «p(i-j|j>)raK>^p) (18) It is well known that simply-laced level 1 Kac-Moody algebras are realized in the case is primary up to J and J. Let us introduce nonlocal current algebra as of d = 1. Similarly, J{i)'s are defined in p +1 - Bl = 0 cases. «M = «p(<>W). M9> [ If we introduce the equivalence relations of J(z) ~ Jgj&nd J{z) ~ lt then we ^-•»*<>$«»)). ' can classify {$} into the equivalence classes. We can take a representative of each which have Z charge (2fc,0) and (D,2fc) and conformal dimension (^# ',0) and equivalence class so that it has the smallest conformal dimension. Then we express the N (0,"J^" '), respectively. This wave function corresponds to eE^fclfc) a*a*e m *ne set of the representatives as {¥}/{/, /}. If there are the physical currents j ^ id w l When we consider torus H ~ /A, where A is the direct sum of simple Lie lattice conformal dimension ("^57^,"^j^). This corresponds to spin state «,',,) iu the A* with a radius Ri, A = ®Aji{j, p — / — Bl = 0 are expressed as parafermionic system. If we take p = h - U 6 A, I = I, S A, then mi = (fl(it,(a(-o/) + BJ;]ny, (17) 9(p,I) = elp(,-^p)«p(,-=2iji!lip) where p = A(mjiJ|"1, J = aimRi, B'J = BifR^Rj'XjXj and e»( is a simple root and ~ /(-'.)*(»!, 0). Aj is a fundamental weight and nj, mi 6 Z. These equations are constraints on By. $(Ii, 0) is primary up to J and J. This wave function has Zn charge (0,0), Zn neutral, We are now at the stage to study special cases, .Ajv-i torns models. We consider n and conformal dimension (•}•,•{•). This corresponds to ZN nentral state, e[0|0) i ' "" the simplest case, R = 1 and B = 0. In the case, we have chiral algebraic structures parafermionic system. To summarise these three statements, we conclude that there are not the other J°(z) = ezp(i\fla-V(z)) ?pes of wave functions in the theory. for all root vectors a. -127- -128- a /. Parafermionic current algebra of AN-I/ZN orbifold We have no ZN invariant U(l) current because trivially Y = 0. Every para-fermionic We are ready to discuss ZN arbifold models by virtue of the lost section. Let ns current V^ has its energy-momentum tensor Tj which is introduced by the operator staxt from an explicit definition of ZN oibifold. The torns AN~\ is defined by the product expansion i elation ii(*M)H») = faHjf-M .... ,1 l X ~ X + 2TA, = jr^{»+('-») rii(-)+«(«—)•>. ' h I t, where A is the lattice generated by the simple root system of AN-I, «ii • ",orjv-i- wheie A, equal to o\ = ^ ^' \ is the dimension of ^ measured by energy-momentum Weyl reflection Vi is the reflection with respect to the hyperplane orthogonal to the tensor 7} and c is the central charge of the Virasoro algebra of Tj. We have an explicit simple root a,-: form Tj(z) = ^{(W + £ fP»J{°) + B-ipNJ(-a)}. (29) r,(,) = ,-fc!a( (22) a>Q An element of Weyl group W — TiTj • • • TN-I transposes vectors as For each j, energy-momentum tensor satisfies real parafermionic Virasoro algebra of central charge c = ffifj3. This means that the parafermionic cnrrent projected W : ati -*• a,+l (ajv = 0:0 = -atx aW-i), (23) into every one direction and every phase is just like ZN parafermionic algebra. Bnt totally the system is of the energy-momentum tensor, |(iflp)* with c = N — 1. Al so that W is the order N discrete rotation. The wave functions of the string living in most of parafermionic cnrrent algebras are not closed except for ZN invariant sector, the Zff orbifold must be invariant under this rotation. In the untwisted sector of ZN orbifold, we apply the same quantisation as in torus We next discuss the twisted sectors in Zji orbifold. A closed string aronnd the model. Fall contents of the representation of ZN orbifold are obtained through ZN fixed point on orbifold in W'-seclor (J = 1, • • •, N-1) satisfies the boundary condition projection Jfy of all wave functions in torns model, {4//}: X( N sector as where W = X.. To show parafermionic structure, we introduce ZN projection of nonlocal para fermionic currents and 17(1) currents. ZN projection into ZN dgen states is defined as *'w=»- »(r —j+-E drfr -"-""'• («) Then we get vacuum energy of the primary states o-j, which is the twisted boson's ^ = i£V«IV', 9 = eV. (25) " 1=0 vacuum, So we have parafermionic currents! *L=P}, fc Yi = /^(A. • v,), WY> = ely*. (27) vertex operators as *(p,/)-. ¥(p,Z) = z~ £-*ezp{ip-X{xtz)}exp{il • -129- -130- Chiral algebras in the twisted sector, make it possible to realize the cocyde factor in the vertex operators naturally and to discnss translation of Axed points in orbifold. We have used the expression of the J(a) = *-*Mp(iV2ot*¥>). vertex operator in the twisted sector, z~l ^c'1*, because of absence of zero modes in the twisted sector. canse discrete translation of t,.e fixed points, X -* X 4- zy, ,, where z. . is a solution p f P As we Btated in the paper [10], we can construct some chiral algebraic structures of the fixed point equation, Zf, , = W'*/.p. + 2wa. p other than parafermionic current algebra as orbifold models. If we consider orbifold We have the twisted parafexmionic current models such as (Torus) x (Para/ermsons), then we can get nontrivial cnrrent algebran by linking the torus sector with parafermions through the antisymmetric tensor B, a # = ^PW.V"* •*>(*))• (") shift vector embedding of W and selecting radius for each torus. This produces energy-momentum tensor in the twisted W' sector We conclude that the ZN orbifold model, except for orbifold models denned bya discrete shift, is a model of some kinds of parafermionic current algebras. General Weyl orbifold models, which have nonabelian structures, should be investigated in the course of classification of couformal field theories in the future. where e(j, 1) is the phase factor generated by the twisted oscillators. Vacnnm state of the twisted boson is at the fixed point z/.p.. Then we have Acknowledgments: The author is grateful to M. Tabnse for helpful discussions, and to the members of particle physics group of Kobe University. ^ fi \ - - '*p('V2°'•*/•?•) nil < J(a) >vacuum= "j lJI>J References and conforms! dimension, measnred by I), of the vacnnm state is [1] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nud. Fhys. 241 (1984) 333. (f)f = 7fe{2t., + E.>o ^«>J>(«V2<« •«/.,.)<(;. 0 (36) [2) D. Gepner, Nucl. Phys. B280 (1987) 757. +p e-tcxp(iV2a • «,.,.)«(;, 0~1}- N [3] Y. Kazama and H. Suzuki, Nucl. Phys. B321 (1989) 332. Totally, [4] A.B. Zamolodchikov and V.A. Fateev, Sov. Phys. JETP 82 (1985) 215. £tf)° = jfr- ("» [5] L. Dixon, J.A. Harvey, C. Vafa and E. Witten, Nuel. Phys. B281 (1985) 678; B2T4 (1986) 285. We conclude that in the twisted sector, spin structure of the every parafermionic [6] M. Sakamoto, Kobe University preprint KOBE-89-02. current algebra is not equal and very complicated. [7] K. Itoh, M. Kato, H. Kunitomo and M. Sakamoto, Nncl. Fhys. B306 (1988) 362. 5. Conclusions [8] H. Dijkgraaf, E. Verlinde and H. Verlinde, Coram. Math. Phys. 115 (1988) 649; We have shown the parafermionic structure of R = 1 and B = 0 AN-\/ZN orbifold models. In this work, it is important to introduce a correct quantisation. For P. Ginsparg, Nucl. Phys. B295[FS21] (1988) 153. [9] D. Gepner, Nucl. Phys. B200(FS20] (1987) 10. instance, the commutation relations [10] A. Fujitsu and M. Tabuse, Kobe University preprint KOBE-89-03-Revised. [X', QJ] - -i*8'J, in untwisted sector, J [11] A. Fujitsu, Kobe University preprint KOBE-89-11. [X',Q ]= -iirKl-W')-'}" in twisted W'-sector — 131 — -132- 1 Introduction c=3d Algebra * In the search for four dimensional realistic string theories, space-time supersymmetry and world-sheet supsrconformal symmetry are the most important requirements and internal sector is restricted severely by them. If internal sector is considered as the non linear a model on certain manifold, from the study of the low energy supergravity, N ~ 1 space-time supersymmetry implies the manifold must be a Calabi-Yau manifold [1,2]. Satoru Odake* This manifold possesses a unique covariantiy constant spinor and an (anti-)holomorphic d-form, where d is the complex dimension of the manifold and, in the case of four dimensional string theory, d is equal to 3. On the other hand, world-sheet N — 2 Department of Physics, University of Tokyo superconformal symmetry and U(l) charge quantization are equivalent to space-time Bunkyo-ku, Tokyo 113, Japan N — 1 supersymmetry[3,4,5,6,?,B]. The N = 2 superconfonnal algebra(SCA) has an automorphism (so-called spectral flow) due to the V{\) Kac-Moody subalgebra[9], so that Neveu-Schwarz(NS) and Hamond(R) sectors are mapped onto each other by the spectral flow which are considered as space-time supersymmetry transformation. In previous papers[10,ll], we studied the extension of the N = 2 SCA by adding the flow generators which generate the integer-shift spectral flow. Its representations are Abstract invariant under the integer-shift spectral flow because such flow corresponds to twice We define a superconformal algebra with the central charge c = 3d, operation of space-time supersymmetry transformation. In this context the covariantiy which is the symmetry of the non-linear a model on a complex d dimen constant spinor corresponds to Ramond ground state and the (anti-)holomorphic d- sional Calabi-Yau manifold. The c = 3d algebra is an extended superconfor forms correspond to the spectral flow generators. mal algebra obtained by adding the spectral flow generators to the N = 2 In this talk we generalize the previous d = 3 result (the c = 9 algebra) to arbitrary superconformal algebra. We study the representation theory and show that d case (the c = 3d algebra). Since space-time dimension of string theory is 10 — 2d, d its representations are invariant under the integer-shift spectral flow. We more than three case is not relevant to string compactification. However, study of d > 3 present the character formulas and their modular transformation properties. case is interesting because it is the symmetry of the non-linear a model on a complex We also discuss the relation to the JV = 4 superconformal algebra. d dimensional Calabi-Yau manifold (i.e. manifold with SV(d) holonomy, i.e. Ricci-flat Kahler manifold) and finding modular invariant partition functions will give one method to study the properties of Calabi-Yau manifold itself[6,8]. The N = 2 SCA, which is "Talk given st KEK Workahop on Topology, Field Theory ud Snpentring Theory, the symmetry of the non-linear a model on a Kahler manifold, is invariant under the Nov. Uk-lOth, 1989 spectral flow but its representations are not so. We want to find the extended algebra 'Fellow of the Japan Society for the Promotion of Science. which representations are invariant under the integer-shift spectral flow. For c > 3, — 133— -134- its representation contains infinite many representations of the N = 2 SCA because (GY)(z) + \{dIX)(z) - (TX)(z) = 0 (2.3) leptesentation of the N = 2 SCA never comes back to itself under the integer-shift (cVjrjf)O) = 0 (i = 0,l,---,d-l) (2.4) spectral flow in contrast to the rational case c < 3. The N = 4 SCA, which is the (tvjrrjM = (pyyKz) = o (; = o,i,-..,d-2) (2.5) symmetry of the non-linear 2 Structure of the Representation Space Combining these relations with various I proves this lemma.D We will present a proof of the following structure theorem: From this, the next proposition holds. Representation space of the c = 3d algebra is a direct sum of representation spaces of Proposition 1 the N = 2 SCA, which are mapped from the highest weight state of the c = 3d algebra by the integer shift spectral flow of the N = 2 SCA. IP 1" From this theorem, one obtains character formulas of the c = 3d algebra as a sum of X-XY-Y'X-XY-Y \h, (?) character formulas of the N =» 2 SCA. We will prove following lemmas, propositions and J Z{N = 2)X---XY-'-Y\h,Q) m>0 theorems by induction and using the operator relations coming from the associativity J ntr = 2)X~-X?---?.\h,Q) m<0. of the c = 3d algebra: I -in Here we abbreviate -¥_„, -• • X-„t to X ••• X etc. Roughly speaking, these lemma and (/*)(*)-a;r(*) = o (2.1) proposition say that X and Y have "flow charge" +1, X and Y have flow charge -1, (/;-)(*)-&T(*) + i(e5jr)(*) = o (2.2) — 135 — -136- T,/,G and G have flow charge 0 and this flow charge is compatible with the c = 3d From this we get algebra and its representation. EaaJU._,_tKJ-J-.j!n> = 0, (2-12) j=0 whTe at, = rir=o(j + "o - J _ r) C S 'iJ S 0 » * regular matrix because, in 1. massive representations. general, a determinant of a matrix A = (ai,)a X^_J.iY1.l.„.H = 0 (0 f[y.kU-i)-9\KQ) m>0 M = >:± (2.8) Let j = /, we get desired results Jf_^_,_i|m+ 1) = 0 (0 < / < k - 1). Similarly, operating -(n + no) mode of (#YY)(z) (=0, 0 < ( < k - 1) to \m), . i-> J where 117=™ * = *• a = (6 yy)_„_jm) (o<;<*-i) Lemma 2 X_„, y.„|m) = 0 (n < A(m+ \) + Q, m > 0). = £ n(p-^yi-r)K.,n-„-„|m). (2.14) na We have only to show X-„,Y-,\m + 1) = 0 for n0 < n < no + k, where no = (proof) i(m + i) + Q. Operating -(n + n0) mode of (&XY)(z) (=0, 0 < I < k - 1) to N, (1) m = 0 OK. (ii) Assume that m case is OK. 0 = (8fXY).„.„,\m) (/ = 0,1,...,*- I) Let no = k(m + 1) + Q. By assumption, J4„|m+l) = rA,y-„.)|m) (n>0, A = TJ,G,G), (2.17) — 137 — -138- so we will consider only 0 Squared norm of |m) is given by recursion formula For massless representations, we only mention the statements. (m + l|m + l) = (-iy(m\[y-„.,Y„}\m) 2. massless representation with Q = 0. = ^+f + |'", + (;r + (?MHm>. (2.1s) In the first line of this equation, we have used X„, Y„\m) = 0 (n > no, m > 0), We define the states which is proved by induction, and in the second line we have used the formula eq.(2.9) of [13]. CJ \m)-. |0,0) m = 0 (2.20) ff?-M3-|)-i^-||ll.0> m<0. Lemma 3 X.n, Y.„\m) = (N = 2)|m + 1) (n > k(m + 1) + Q, m > 0). (proof) Operating -n mode of {GY){z) + \(3IX){z) - (TX)(z) (= 0) and (TY)(z) - Lemma 4 A'-,, Y-„\m) = 0 (n < fc(m + \) + 1, m > 0) ; (n < j, m = 0). 8Y(z) + \{GX)(z) (= 0) to |m), we obtain Proposition 4 |m) are Me highest weight states of the N — 2 SCA with squared norm = nr-» ?(4 - * - &+ *tf - i+*)') > o (m > o). (hm + \Qm)X.„\m) = [-£, G,Y-n-p- E V--P Lemma 5 X-„,K.„|m) = (JV = 2)|m+ 1) -sEfo + WU-p/h) (n > k{m+ J) + 1, m > 0) | (n > f m = 0). (n-9„-|)K«N = (E '^-.-P + 5 E Let no = t(m + J) + Q. By definition, V_M |m) = |m + 1). From eq. (2.19), X^,,.^\m), y. ,_i|m), X_„_ j|m), V-„,_a|m), • • • are expressed B 3. massless representations with Q > 0. as(JV = 2)|m+l)inturn. D We define the states From these the next proposition holds. m> 0 Proposition 3 X• ••XY---Y\Sl =(N = 2)|m) (m > 0). ni'-K.-D-fllffl) m \m): (2.21) Il^-^o-i^-l+glfl?) "»<0. We can show the similar results for \m) (m < 0). Therefore we obtain the theorem: -139- -140- Lemma 6 A"_ni Y„n{m) = 0 {n< k{m + \)-rQ, m> 0). Acknowledgments Proposition 6 |m) ore the highest weight states oj the N = 2 SCA with squared norm I would like to thank Professor T. Bguchi for useful discussions. This work is sup 0 ported in part by the Grant-in-Aid for Scientific Research from the Ministry of Educa -nr.li(¥-1-Sr+iu-§+*n> (»^°)- tion, Science and Culture of Japan No.D1790191. Lemma 7 X_„, K.n|m) = {N = 2)|m + 1) (n > fc(m + 1) + Q, m > 0). Proposition 7 *• ••XV •••K|0) = {N = 2)lm) (m > 0). References [1] P.Candelas, G.Horowitz, A.Strominger and E.Witten, Nud.Phys. B2S8 (1985)46. LemmaS ^„,>Ln|m) = 0 (n < H-m + £) + l ~Q, m < 0) ; (n < j-Q, m = 0). [2] M.Green, J.Schwarz and E.Witten, Superslring Theory 2, Cambridge University Press(1987). Proposition 8 |m) are the highest weight states of the N = 2 SCA u/i*A squared norm = n;:U(i¥-$-'a§r£+£(j-*-V)I)>° ("•<<»• [3] W.Boucher, D.Friedan and A.Kent, Phys.Lelt. 172B (1986)316. [4] A.Sen, Nud.Phys. B278 (1986)289; B284 (1987)423. Lemma 9 *_„,?_|m) = {N = 2)|m - 1) (n > i(-m + J) + 1 - Q, m < D) ; (n > 4 - Q, m = 0). [5] T.Banks, L.Dixon, D.Friedan and E.Martinec, Nucl.Phys. B299 (1988) 613. Proposition S J?-y •••?.10) = (AT = 2)|m) (m < 0). [6] D.Gepner, Nud.Phys. B296 (1988)757; PhysXett. 199B (1987)380. —m [7] N.Seiberg, Nucl.Phys. B303 (1988)286. Theorem 3 V. = © v£g. © V™„. m<0 m£0 [8] T.Eguchi, H.Ooguri, A.Taoimina and S.K.Yang, Nucl.Phys. BS1S (1989) 193. Similar results hold for massless representations with Q < 0. [9] A.Schwimmer and N.Seiberg, Phys.Lett. 184B (1987)191. [10] S.Odake, Mod.Phys.Lett. 4 (1989)557. [11] S.Odake, "Character Formulas of an Extended Superconjormal Algebra Relevant to Siring Compacti/ication", Tokyo Univ. preprint UT-543(1989) (Int.J.Mod.Phys. to be published). [12] T.Eguchi and A.Taortnina, Phys.Lett. 210B (1988)125. [13] S.Odake, "c = 3d Conjormal Algebra with Extended Supersymmctry", Tokyo Univ. preprint UT-551, Sep. 1989. — 141 — -142- iV = 2 Superconformal Symmetry based on N = 1 Supersymmetric Non-Compact Group Current Algebra* 'W<»>~(rV (i) Shin'ichi NOJIRI others ~G . National Laboratory for High Energy Physics (KEK) The current J3(z) is bosonized and parafermion field VOO is introduced: Tavkuha 305, JAPAN J3(2) =.y|a«r), abstract (2) J±(z) =VX^.±(i)e±'VF*(') . N = 1 supersymmetric £70(2,1)717(1) coset model is analyzed. We show that this model By making boson field t$> decouple, we obtain a theory which contains parafermion contains unitary representations under some conditions. Furthermore, it JB shown that this field only i.e. SU{2)IV{\) model. Here (7(1) is generated through J\z). By algebra is equivalent to Dixon, Peskin and Lykken's N = 1 superconformal algebra (c > 3) which adding back a new free boson *•W conformal algebra can be obtained from SU{2) current algebra, whose operator The central charge c has the folowing form: product relations are given by, 3k ' h + 2' (6) The parameter k should be a positive integer or SU(2) current algebra does not * This work was collaborated with Mihoko M. Nojiri. have any unitary representations. The unitary representations of jV = 2 super- -143- -144- conformal algebra (c < 3) are obtained from representations of SU(2) current representation. It may be natural to expect that the coset model obtained from algebra. the non-compact group current algebra gives new unitary conformal field theories. In fact, Bars analyzed general coset models constructed from non-compact group. In case c > 3, we start with the OPE in 50(2,1) current algebra: While he has not shown the full unitarity, his theory may compensate Kazama and Suzuki's coset model based on usual compact group current algebra. [z — w)- z — w In this report, we analyze JV — 1 supersymmetric 50(2,1)/I7(1) coset model. J\z)j\w) (z-wy (7) It is shown that this model has unitary representations under some conditions. Furthermore, we show that this algebra is equivalent to Dixon, Peskin and I lW) ~ i ^ z — w Lykken's N = 2 superconformal algebra (c > 3) which was obtained from non- others ~0 . supersymmetric 50(2,1) current algebra. The sign differences from 517(2) current algebra (1) tell that any representation TV — 1 supersymmetric current algera can be regarded as a combined sys of this algebra is not unitary and contains negative norm states. In a procedure tem of bosonic currents {Ja(z) : a = l,---,JV} and free fermions {j°(z) : a = analoguous to SU(2) case, we obtain a parafermion theory by subtracting a free li" *' i N}i which are decoupled to each other. In case of JV = 1 supersymmetric boson tj> which generates negative norm states: 50(2,1) current algebra, currents {J^z), J3(z)} and fermions {^{z), j'(z)} satisfy the following OPE, «(J)«I») = + III(J-UI). (8) 2 Dixon, Peskin and Lykken showed that, while 50(2,1) current algebra cannot (Z -U)) Z - W have unitary representation, the parafermion theory i.e. the coset 50(2,1)/J/(1) •^W^J-fT^' (io) can be unitary. N — 2 superconformal algebra is obtained by adding a new positive-norm boson. The conformal anomaly c is given by, J"(,)7*(„)~±^), z — to others ~0 , It should be noted that here k can be any real number greater than 2. We obtain unitary representation of N = 2 superconformal algebra (c > 3) from (U) representaions of 50(2,1) current algebra. Continuously infinite number of rep iWM~f5i. others —0 . resentations exist for fixed c. An interesting point in Dixon, Peskin and Lykken's theory is that the coset The factor (-2) in Eq.(ll) is the value of the 50(2,1) Casimir operator? The 50(2, !)/[/(!) can be unitary while 50(2,1) current algebra cannot have unitary minus signs in OPE of two .73's in Eq.(lO) and two j3's in Eq.(ll) tell that this — 145 — -146- theory contains negative norm states. The energy momentum tensor Tso(2ll)(z) And T "<>> is given by • . , 10 is given by, imiaim) i l , , T-m-'(z) =- : K{z)K{z) : -^ : J3(z)J3(z) : r5°(2,1,W = -rl-z : {l(J+(z)J-(*) + J-{:)J+{z)) - J3{z)J3{z) (18) k-1 2 (12) + 5(j+(z)rW + r(z)j+(z)) - j3(z)j3{z)} : . Here we used the following equation: The N = 1 supersymmetric 50(2,1) current algebra contains N = I supersym- metric U(l) subalgebra, whose energy momentum tensor is given by ^TT : U+(*)r(*) + j-(^)j+(^)) := 5 : K[z)K(z) : . (19) 2(i Tm(') = -~z : {(J\z) + K(z))(j\z) + K(z)) + i\z)j\z)} • . (13) f^W1 is decoupled to ftt>(W>/(7(l>, i.e. Here K(z) is defined by, iii J jso(2,i)/y(i)(,):f, ^ (u,) _ 0 t (20) + + K^) = --zj^ZT)-.u {z)r^)-r{')i U))- • (n) because Then we can obtain the energy momentum tensor corresponding to coset space TSO(2,1)/I/(l)(r)j3(uj) _, rSOfSJl/ffO^yi^) ^ 0 , (21) 50(2,l)/£/(l)9: SO(2,l)/(/(l) _ j.SO(2,l) _ U(\) _ ( ) T T 1S Therefore any representation of this algebra is a direct product of the representa tion generated through T soM)/uW{z) and that generated through f "W (z). In the following, we show that the coset model Eq.(15) has unitary represen Dixon, Peskin and Lykken showed that f s°(2i1)/"t1)(r) has unitary representa tations under some conditions. For this purpose, the following decomposition is tion when useful, k > 2 . (22) 1 rSO(J,l VU(1) = jWO(5,l)/I7(l) + g^'flff' ,16j Therefore, due to Eq.(16), we obtain the condition that N = 1 supersymmetric Bars gave this kind of decomposition for general coset models. Here fSopi'WO) 50(2,1)/I7(1) coset model has unitary representations if we cosider the unitary is the energy momentum tensor of non-supersymmetric 50(2, l)/f/(l) model: miwnu condition for T "HI (r). In the following, we consider the condition. imvwii) y|1) 3 3 The energy momentum tensor T corresponds to coset space U(l) fSO(2,imi){!] =_L_ . {l{J+WJ-[t) + J~(z)j+(Z)) - J (z)J (z)} (generated through J3) times U(l) (generate-! through K) divided by U(l) (gen 3 3 + i : J (z)J (z) • • erated through J3 + K). This coset model becomes another U(l) current algebra -H7- -148- model, which is generated through the (/(l) current M(z) which is a linear com G*(z) are given by, 3 bination of J {z) and K(z): G±{:) = h-^^w • (28) ± ± z 3 J (z) decompose into parafermion fields V' ( ) and free boson $(2), M{z) = J {z) + J-(z) = v^Me*^'-"' • (29) The coefficient o is determined by the condition that M{z) should be orthogonal 3 t0J (z) + K(z): Here d>(z) is defined by, M{z)(J3{w) + K{w))~0. (24) j*(z) =e±'5*A''-) , OPE of two M(z)'$ is then given by, Z (31) iK(z)=2i Jdwli(w) . M{z)M(w)~ -^. (26) 7 (7^ Then we obtain G±(--) =s/k,l>±(z)rf f WM-^M) Therefore M(z) generates unitary (positive norm) Fock space when (32) =Vh/,:t(z)e:fl /''M{M • k > 1 . (27) Therefore we can regard M{z) as 1/(1) current in Ar = 2 superconformal algebra. We analyzed N = 1 supersymmetric 50(2, l)/t/(l) coset model and it was This condition (27) is identical with the condition (22) that non-supersymmetric shown that this model has unitary representations when k > 2. This condition is 50(2, l)/U(l) model has unitary representations. identical with the condition that non-supersymmetric 50(2,1)/W(1) coset model has unitary representations. Furthermore, we have shown that N = 1 supersym In the following, we show that M{s) is in fact the 1/(1) current of N = 2 metric 50(2,1)/U(1) model is equivalent to Dixon, Peskin and Lykken's N = 2 superconformal algebra. superconformal algebra (c > 3) which was obtained from non-supersymmetric Because the coset space 50(2,1)/C(1) isahermitian symmetric space, N = 1 50(2,1) current algebra. The compact group correspondence, i.e. the equiv r 50(2, \)/U{l) model has N = 2 superconformal symmetry. The supercurrents alence between A = 1 supersymmetric SU(2)/U{1) coset model and unitary -H9- -150- discrete series of *V = 2 supercomforraal algebra wliich can be constructed from non-supersyrnraetric SU{2) current algbra ' , has already been well known. The equivalence car. be understood from the matching of conformal anomaly c be cause we have only minimal series in unitary N = 2 superconformal field theories with c < 3. But in case c > 3, such an argument is not valid because c varies continuously. We are especially indebted to Professor K. Higashijima and Dr. R. Nakayama for useful discussion. We would like to thank all members in theory group in KEK for discussion. REFERENCES 1. Mihoko M. Nojiri and Shin'ichi Nojiri, KEK-TH-234 KEK preprint 89-91 (19H9) 2. C. Hull and E. Witten, Phys. Lett. 160B (1985) 398 3. A. Sen, Nad Phys. B27S (1986) 289 4. T. Banks, L. Dixon, D. Friedan and E. Martinec, Nucl. Phys. BZ99 (1988) 613 5. L.J. Dixon, M.E. Peskin and J. Lykken, SLAC Preprint, SLAC-PUB-4884 (1989) 6. A.B. Zamolodchikov and V.A. Fateev, SOY. Phys. JETP 63 (1986) 913 7. A.B. Zamolodchikov and V.A. Fateev, Sov. Phys. JETP 62 (1985) 215 8. I. Bars, University of Southern California Preprint, USC-89/HEP02 (1989) 9. Y. Kazama and H. Suzuki, iVud. Phys. B.'.-il (1989) 232 10. V.G. Kac and T. Todorov, Commun. Math. Phys. 102 (1985) 337 11. W. Boucher, D. Friedan and A. Kent, Phys. Lett. 172B (1986) 316 12. Z. Qia, Phys. Lett. 188B (1987) 207 -I'll- Spontaneous Deformation of Maximally fact, the numbers of the chiral superadds ($ and $) in 27 and 27 of Ee and also Symmetric Calabi-Yau Manifolds of the complex structure moduli and Kahler class moduli superfields(C and R) and in 1 of Es are given by the Hodge numbers(np') of the compactified manifold'" A Realistic Four-Generation Model* . . Although the compactification on the Calabi-Yau manifold guarantees N = 1 space-time supersymmetry at tree level, Ricd-flatness of the compactified manifold is not generally maintained for higher order corrections'" . It is necessary to impose conformal invariance on the theory so as to be fully consistent. From this viewpoint Takeo MATSUOKA Gepner has constructed the consistent theory algebraically by tensoring minimal Department of Phytia, Nagoya University JV = 2 superconformal field theories . Recently the structure of fundamental Nagoya i6i-01JAPAN link between Gepner's models and the geometry has been clarified by using the Landau-Ginzsburg theory and the singularity theory . It has been recognized that there are many possibilities of the consistent com pactification. At the present stage of superstring theories, important clues to select Abstract out a true compactification from them are possibly provided by phenomenological We propose the spontaneous deformation of a conformally invariant Calabi- requirements. The realistic superstring theories must solve the phenomenological Yau manifold with a maximal discrete symmetry. This spontaneous deformation is issues such as the proton stability, quark/lepton mass spectra, smallners of neutrino triggered by the soft supersymmetry breaking. A realistic four-generation super- masses and so on. Falicularly the proton stability requires that the compactified a string model is derived on the basis of this mechanism. In this superstring model, manifold possesses a specific discrete symmetry which leads to the absence of ($$) - 3 large masses of leptoquark chiral superfields g and g are induced by the existence and (dj4) -terms in the superpotenlial W. ls 16 of the large intermediate energy scale Mr = 10 ~ GeV. This result guarantees In this paper we take up the four-generation model brought about by the com the longevity of protons. Furthermore, only one family among six families of Higgs pactification on Calabi-Yau manifold K with the maximal discrete symmetry. The sector(fc and k') remains massless at energies below the scale Mj. muiifold K is constructed as K = KB/GJ, where JTo is a simply connected Calabi- Yau manifold and Gj = Z5 x Z's. In the algebraic approach, Ka is made up from the five copies of level 3 superconformal field theory, denoted as 3s-model' . Gep ner's 35-model has the discrete symmetry (5s x Zs*)/Zs and the gauge symmetry 1. Introduction Ei x 17(l)2rtrn- The extra-f/(l) gauge symmetries are peculiar to Gepner's model. In the heterotic superstring theories, the Calabi-Yau compactifications are phe- Geometrically, Ko is defined as the hypersurface in CP* with the specific defining nomenologically promising. The gauge symmetry, matter conte.;'s anci Yukawa polynomial £)*f = 0, where *»(*' = 1 ~ 5) are the local coordinates in CP*. The interactions in the four-dimensional effective theory are closely linked to the topo complex structure moduli parameters cjfj ~ 0 ~ 4) are all vanishing and then the logical and geometrical structure of the compactifted manifold. As a matter of manifold K is maximally symmetric. The maximal discrete symmetry gives strict ) This work is done with C. Hatlori and M. Matsuda. (DPNU-S9-56) t About details of the four-generation model, see Refa.5 and 6. — 153 — -154- selection rules for the superpotential of the effective theory which lead to important eigenvalues(or',a') with respect to the transformations B and S defined as phenomenological implications. For example, the selection rules bring about the B; i-.a2,,r (as = 1), (1) large intermediate energy scale M[ = 10ls~16GeV, which is consistent with the I i longevity of protons * . S; z,--z,+1. (2) We now propose the spontaneous deformation of the maximally symmetric In the expression Tj,, j means the generation index. Tja{j = 0 ~ 4) correspond to Calabi-Yau manifold mentioned above, which is caused by the soft supersymmetry Aj which consist of kj, h'j and Sj in 27 and to Cj in 1. Tj,(j = 1 ~ 4) for the breaking. The introduction of the soft supersymmetry breaking induces sponta cases i = 1,2,3 and 4 are assigned to the fields ( ken spontaneously as G -> G' with preserving G,, = St7(3)c7 x SU(2)L x 17(1) Tw-Jillcc-'zfzi+txi., (3,0,1,1,0) a a' Au Cx at the intermediate energy scale Mi. This symmetry breaking is attributed to Tu = V^Ea""^i+2^-l (1,0,2,2,0) c! a' A,, Ci the nonvanishing vacuum expectation values of the G,<-neutral chiral fields 5 in o3 a' T . = ^E"-"'i'hi^ (2,1,2,0,0) A3, C3 27 and 5 in 27 and the complex structure moduli fields Cj in 1. Throughout 3 2 (1,3,1,0,0) a« a' At, CA Tu = v/iE«-"^+tf-i this paper we consider G = SU(3)c * SU(2)L x £17(2);! x P(l) x ^(l)*,.,™ and G' = S0(3)c x SU{2)c x SV{2)n x 17(1) as a realistic gauge hierarchy. Within 5 Table 1 . The relevant quintic polynomials and their Zs -charge assignments. As this framework, the present analyses for mass spectra indicate that all leptoquark for the latter, the sum should be taken over the cyclic permutation. The chiral superfields g and g, four families of Higgs superfields h and h' in 27 and one eigenvalues under the transformations B and S defined in the text are family of those in 27 gain masses of order Mj and that only one family of h and also listed. h! remains massless at energies below the scale Mi. Next we construct the superpotential W which reflects the structure of N ~ 2 superconformal theory. In the M"1 expansion, W at the perturbation level is 2. Selection Rules and the Superpotential generally expressed as The chiral superfield nomials TJOU = 0 ~ 4,« = 0) and Tjt(j = 1 ~ 4,J = 1 ~ 4) which have where the generation indices are omitted. We now neglect the contribution of -155- -156- gauge-singlet superfields from the adjoint of Et because they gain masses of order terms of the suppression factor Mc from world-sheet instanton effects . The A(C) and n'"'(C] depend on complex structure moduli fields C, but A does not ! . Hereafter we redefine the fields to e=(fl)exp(-{it)), (8) be dimensionless dividing by Mz for simplicity. In the four-generation model the non-renormalizable terms ($4)" take the values only for n = 4,9,••• due to the in which (R) represents the size of the manifold K. The Yukawa interaction selection rules containing the .^-charge conservation A(C)$3 is free from the world-sheet instanton effect, whereas A$3 and the non- renormalizable term W^R are subject to this effect . As a result, the lowest term m ^g/ + 2 = 0 mod5 (4) of WNR becomes the eighth-order polynomial with respect to Sj, S, Cj and e due J to the selection rules. Here we take a simple solution in which only So among Sj{j = 0 ~ 4) possesses a nonvanishing vacuum expectation value. Therefore, we for the term of m-th degree in W. If we write down the generation indices explicitly, neglect the contribution of Sj(j = 1 ~ 4) and abbreviate So to S in the superpo- the Yukawa coupling in £ *iit(C)Si4j*i(i < j < k) is tential. Then WNR is expressed as A^iCC) = A'J> + £ AS;',I..,IC,,...C,1+.... (5) W = b(SS)i + b ^SSfCo + b3C,{SS)'{Cot+a-,C,C3}, (9) {1,-1,} fflt l 1 This also comes from the selection rules. The sum of generation indices should where 6,'s are constants of 0(1). It is noteworthy that WNR does not include C\ be zero in modulus 5 from B-invariance. Moreover, among the maximal discrete and C\ at the eighth-order. As shown later, this fact enables us to predict the symmetries one can introduce the transformation relation (Co) ,(C,), (Cj). (S), (S) < (C>, (C,) = 0(1). (10) Y; I,-.* (6) Under the action of Y, Tj, is transformed into Tt-j 3,. The Y-symmetry of the manifold demands the invaiiance of W under the interchange; *j *-> $3, *i <-> *4, 3. The Scalarpotential and Spontaneous Deformation Cj «-» Ci and C\ <-» C«. Using the JV = 2 superconformal 3s-raodel we obtain the where a is lowest order couplings to be A§5j, = a" ,X0% = a and X^u ~ °"'" i The scalarpotential is described in terms of the superpotential W and the defined by Kahler function K. Dixon et al. have derived the important result that the Yukawa coupling %ijt(C) and some part of the Kahler function K are interrelated with each other through a common holomorphic function of C1'" . Making use of these argu ments we can calculate the .F-terrn of the scalarpotentia], which is given by The Kahler class moduli field R does not appear in W at the perturbation level but affects W through the world-sheet instanton effect. This effect is described in Vo S 2S*t\ + t'S'f'g + bla-'e'S'iC', + C}) (11) —157 — -158- under the approximation of Eq.(lO) with with non-zero values of the c-j's as seen in Eqs.(16) and (17) triggered by the soft su persymmetry breaking. This is exactly the spontaneous deformation of maximally 4 2 2 2 fA = 46iS + 3&jeS Co + 2&3< {C + 2 /fl = 655 + 263£Co. (13) 4. Yukawa Couplings and Mass Matrices To minimize the scalarpotential the U-term contribution should vanish in the We are now in a position to calculate the Yukawa coupling on the spontaneously scalaipotential. This means that (5) = (S) and then we put S = S in Eqs.(ll), deformed manifold obtained here. At a special point cj = 0(j = 0 ~ 4), the Yukawa (12) and (13). couplings for g-g sector, Higgs sector and quark/lepton sector have already been As a further step let us introduce the soft supersymmetry breaking term V,,t obtained by several authors . For the cases of non-zero 3 l8 of the maximal discrete symmetry. This is an essential point of the spontaneous where p stands for M,jMz {M, ~ 10 GeV and Mc ~ 10 GeV) . The full scalar- deformation mechanism. The Yukawa coupling receives the modification from the potential V is of the form terms other than the first one in Eq.(5). Here we confine ourselves to the second V = V + V„ . (15) S 0 h term 0(C ). As mentioned previously, the £5'-charge of Aj is exactly the same as that of Cj. Then it is efficient to write down the eighth-order polynomial consistent By minimizing V and taking a Y-symmetric solution, we obtain the set of complex with the selection rules; structure moduli parameters CJ defined as {Cj) !! co, c, = c3, (S)~pi ~10- , (16) ii(C) = uCa{C?C< + CiCl)+qCa{CiC3Cl + ClC,C})+MCo(ClClCl + ClClCl) provided that c ~ p1'. In this solution the size of Calabi-Yau manifold determined by {R) = MrilMc is set to be ~ ln(Jlfc/Af,)/6 ~ 6. In the lowest order there is no +tCoC,C3ffi + Cj) + IMCJC? C\ + V2Cl(C\CiCt + C1C3CJ) restriction on the value of (C\) and (C4) but the C\- and (^-dependence of Wf/n from higher order terms leads to +v0C&Cl + C\) + --'. (18) ei=ei = 0(l). (17) The relations among the coefficients in Eq.(18) are The intermediate ene:gy scale AT/ is given by M x (S). Starting from the maxi c 2 , 2 , 5 , 2 . 2-JE v'S] 24 _ mally symmetric manifold with CJ = 0(; = 0 ~ 4), we ace led to a stable manifold -au=-*q = -*. = -at= —av, = —av, = „o = ^• V-. (19) — 159 — -160- where the constant a is given in Eq.(7) and ps is defined by where the parameters are defined as 1 J 4 W = / (*i*a*3*4*s)8. (20) fi'=i[(a +ct )+2(« + <« )}S JK s'^^+s^ + ^J + sfa+ «<)}» In Eq.(lB) we explicitly pick up more than the fifth-order terms of C\. and/or Ct 9 = "^{3 + 2(a2 + a3) + 4(a + a4)} q with at least one Co because later we replace three of C/'s by Aj's and take account (23) 9" = ^{4 + 5(o2 + a3) + 8(a + **))§ ofEqs.(16) and (17). i' = 1{2 + 3(a2 + a3) + 4(a + o4)}« The Yukawa coupling So-A;-fcJ is obtained through the replacement of Co by 5 lb 2 3 4 and two of d by M in Jj(C). By choosing eigenstates of the Y-transformation as ? = ^{4 + 2(a + a ) + (a + n )}(". the bases, we find the mass matrix for Higgs sector in 27 In the maximally symmetric limit which implies A(C) = A'0' due to all (Cy) = 0, (*<• ^ ^ ^ *^) two of four g and 9 sectors are massless and the other two get masses of order Mi. Since a proton can decay fast via the light leptoquark g and g exchange, this solution cannot interpret the stability of protons. While in the present model /o5+«o \/2«i \Z2oj 0 0 \ I h'° \ all g and 9 fields gain masses at the intermediate energy scale Mj = 10l5~I6GeV \/2»l 2u 2rr 0 0 from the mass matrix (22) as a consequence of the spontaneous deformation of x<5) V^«j 29 j + i + a 0 0 *& (21) Calabi-Yau manifolds. This fact satisfies the phenomenological requirement that 0 0 0 0 0 >!-*! the masses of leptoquark fields g and 9 responsible for the proton decay should be V2 ls 1 0 0 0 0 i-l-a) {^J heavier than O(10 GeV). where the barred notation means to multiply c\ to each coefficient in -Fe(C). From the mass matrix (21) we can say that only one family (fti-ft^/v^ and (h\-h'K)/y/2 of Higgs sector is massless and the other four families of Higgs sectors are all massive S. Conclusion at O(Afr). One family of the Higgs sector in 2? also gains mass of order Af/. In conclusion the spontaneous deformation of maximally symmetric Calabi-Yau For g and 9 sectors we can calculate the mass matrix similarly. The results are manifold plays an essential role in constructing realistic superstring models. This mechanism is caused by the introduction of the soft supersymmetry breaking. A re f'\ alistic four-generation superstring model has been presented explicitly by studying a" a' 9" 91 the dependence of the superpotential, containing the world-sheet instanton con (91 9* 91 Si) (S) (22) <7 9" O + P 93 tribution of Kahler class moduli superfields R, on the complex structure moduli 9" 9' -.< a + PJ W superfields C. In the present model we obtain the following results: —161 — -162- a) one Higgs family h and h' in 27 remains massless and other five families in 27 REFERENCES and 27 get masses at the intermediate energy scale Mr of order 10ls~l6GeV and 1. P. Candelas, G. T. Horowitz, A. StromingM and E. Witten, Nud. Phys. B258 (1985),46. b) the masses of all four families of g and g leptoquark superfields are given by E. Witten, Nucl. Phys. B258 (1985), 75; B268 (1986) 79. the order of Mi. A. Strominger and E. Witten, Comm. Math. Phys. 101 (1986), 341. Especially the latter fact guarantees the stability of protons. These results are 2. M. T. Grisaru, A. Van de Ven and D. Zanon, Phys. Lett. 173B (1986), 423. favorable to unify low energy gauge groups SU{2)i and SU(Z)c at the compact- M. Freeman and C. Pope, Phys. Lett. 174B (1986), 48. ification energy scale within the framework of renormalization group analysis D. Gross and E. Witten, Nucl. Phys. B277 (1986), 1. D. Nemeshansky and A. Sen, Phys. Lett. 178B (1986), 365. 3. D. Gepner, Nucl. Phys. B286 (1988) 757. 4. E. Martinec, Phys. Lett. 217B (1989), 431. C. Vafa and N. Warner, Phys. Lett. 218B (1989), 365. B. R. Greene, C. Vafa and N. P. Warner, Nucl. Phys. B324 (1989), 371. W. Lerche, C. Vafa and N. Warner, Nucl. Phys. B324 (1989), 427. 5. C. Hattori, M. Matsuda, T. Matsuoka and H. Mino, Prog. Theor. Phys. 82 (1989) 599. 6. M, Matsuda, T. Matsuoka, H. Mino, D. Suematsu and Y. Yamada, Prog. Theor. Phys. 79 (1988) 174. 7. M. Dine, N. Seiberg, X. -G. Wen and E. Witten, Nucl. Phys. B278 (1986), 769; B280 (1987), 319. 8. J. Distler and B. Greene, Nucl. Phys. B308 (1988) 295. 9. D. Gepner, Phys. Lett. 199B (1987) 380; Nucl. Phys. B311 (1988) 191. D. Suematsu, Phys. Rev. D38 (1988) 3128. P. Zoglin, Phys. Lett. 228B (19B9) 47. 10. L. J. Dixon, V. S. Kaplunovsky and J. Louis, SLAC-PUB-4959 UTTG-19-89 (1989) 11. M. Harada, C. Hattori, M. Matsuda and T. Matsuoka, DPNTJ-89-57 (1989) — 163 — -164- In this letter, we shall prove that any compactified doted boiontc tiring theory • on an abelian tymmetric orbifold with a gauge group H it equivalent to that on a Torus-Orbifold Equivalence lorut if the rank of 3 it equal to the dimenjion of the orbifold. This statement has in Compactified String Theories already been pointed out by several authors [1-4]. However, any complete proof has not appeared yet to onr knowledge. An orbifold T"/G is obtained by modding out of a torus T" by discrete rotations which are the symmetry of the lattice defining the torus [5]. In other words, the iden Makoto SAKAMOTO tification of the point x1 to U'JzJ [U e G) gives an orbifold, where U'J is a rotation matrix which transforms the lattice points into themselves. Then, strings on such an Department'of Physics, Kobe University orbifold obey various boundary conditions, i.e., untwisted and twisted ones. Since we IVada, Kobe 657, Japan shall restrict our considerations to abelian orbifolds, it will be sufficient to examine 2 Zw-orbifold. In this case, the elements of G are given by G = {1, U, U , • • •, U"''} with V = 1. Let us first consider strings in the untwisted sector whose boundary condition is the same as in the torus case. Let us introduce the left- and the right-movingstrin g coordinates and momenta, Xi(z) = zi-ipi^z + iYJ-"l^-' , *B(*) = 4-V£ 1*S + 'Erk.1"'' > 1 Abstract Pi(*) = .fl,xJ(*) = 5>i,v*-'- , nt-Z 1 It is shown that any compactified closed string theory on an abelian orbifold is iftz) = .-aIx£{f) = £aj,,,ir»- , (i) nCZ equivalent to that on a torus if the rank of the gauge symmetry of strings on the orbifold is equal to the dimeniiion of the ozbifold. Oar proof clarifies the correspondence of where ar£0 = p£ and a'Ra & pj, (/ = 1," •,!?). The left- and right-moving eenter- operators as well as state) between two theories. of-moas momentum (p£,Pn) lies on a (D + C)-dimensional lorentsian even self-dual lattice [6,7], The quantisation conditions for the non-sero modes are given by M [«L.»k] = ** *™+..oH<»iL.«J.] • (2) For the quantisation conditions of the r-ero modes, we follow the prescription discussed in Ref.(8]. Let o be a simply-laced Lie algebra of the rank D, AR(C) (A|y(o)) be a root (weight) lattice of g and art (/>') (•' = 1> • • •, -D) he a simple root (fundamental weight) — 165— -166- of g, where a< is normalized such that (or*)1 = 2. We assume that (PX,,PH) lies on the Here, Bu is an antisymmetric constant background field which will appear in the first following (V + I?)-dimensional Lorentziau even self-dual lattice; quantized action [7,8]. Note that the vertex operator defined in (5) satisfies the correct cocycle property, that is, the product of two vertex operators V[ki, fa; w)V{k'Li kR; r) A.e, = { (rf.rft) = £>,W) + »'(<»f.O)) I "!.•,»'• 6 Z } . (3) for \w\ > |i| has thesame functional form as its reverse order V{k*IiikR\z)V{kiikR\w) 1 = 1 for \z\ > \w\ due to the nontrivial commutation relations of the zero modes [8]. Since p£,p^ £ Ajv(ff), the affine Kac-Moody algebra g © g can be constructed in the A la Frenlcel and Kac, we see that the operators P[(z),VL(a;z) = V{a, 0;x) and vertex operator represent at ion a la Fienkel and Kac [9]. The generating functional i^(z), VH(O!; X) S V(0,a;r), where a1 is a root vector of g, satisfy the following of VitaBoto operators are given by operator product algebra of g © g: 'H'l'iw ; »-*)'T • P[(v,)VL(a;z)=-£-1VL{a[z) + -~ , [ C 1 1 „ , , where : : denotes the normal ordering with respect to a£n and aRn. Both L Ln and + , Pi( )+ - , if a -0 = -2, (»-«)» M-I'" - ' " LJJI satisfy the Virasoro algebra of the central charge D. VL{C,;*,)VU0;:) = if a/3=-l, To realize the afiine Kac-Moody algebra g © g in the vertex operator representa li) — Z tion, we introduce the vertex operator as follows: . ••• , if a • /3 > 0, V{kL,kR;:) = --'*p{ikt,-Xd') + il'R-XR(z)}: . (5) (£-«) , (8) Now we have where • • • denotes regular terms and tiwtiw = (—}?+••• . <6-J) e(a,/J) = txpi-ila'B"?} . (9). Let us introduce the operator i^D) which induces the Z/y-transform at ion, i.e., J i'iHV(*i,tjiiJ) = -^-K(i1,tH;i) + .-- , (6-J) iJ(0)(^(^),Jfi(i))K -„S = (0-"^ ( ),»-"^(i)) • (10) 10 — Z ( I Ph{w)V(kL,kn\z)=-^;V{kL,kR;:) + -.- , (6-4) We require that the ground state 1 0 > defined b) V{k ,kn;w)V[k' ,k p£|o> = p{t|o>=o , (ii) where • • • denotes less singular terms in the limit of w -* z (in fact regular terms in is invariant under the Zjv-transformation, i.e., R[ )\ 0 >= | 0 >. (6-1 )-(M)) and 0 Let go® go be the snbalgebra of jffi j which is invariant under the Z/v-transform a -tion. Since every physical state of strings on the Z^-orbifold must be invariant under IJ + kHS" - fl")*]/ - kR(6 + B")k'i)) . (7) the Z/v-transformation, the gauge symmetry associated with the algebra gQ © g0 will — 167 — -168- appear in the spectrum bat not g ® g [5]. In the following, we shall restrict am A,a, be the set of the basis vectors of the lattice A„9o . We will have the oper considerations to the case that the rant of go is equal to that of g, i.e., the rank . In other words, we shall choose the Zjv-transformation to be an inner automorphism V(Jfeji, kR\ z) and which satisfy the following relations: of g. Then, by suitable linear combinations of P/(r), Vc(a;z) and -Pfl(z), VR(IX\ Z) , PL'W{qi.,qR;z) = -£-V{, , ,z) + --- , we can construct the operators P^[z) and Pj[{z) (7 = !,•••, D) such that L qR P'd^)V{qL,qB;z) = ^1V[qL,qR;z) + --. , RmiPL'M,itf(-'))V) = f^1''p«<*"» • (12) l / and ^(Hyto.gfl;') = (Jl'),y'(gx,Mia) + ^-7fl>y'(ot,te;z)4---- , #( ")#(«) = , ., +regular tenna , 6" r"(jj;,oji;u>)V',(--«L,-ojt;z) = - -3——;TT- + -" , Pg{w)Ptf{z) = ^—r+iegul It follows that (V'te, ««:*))' = (i)" (i)'" V(-M,-mii) . (18) «(D)(^,<)iI - i=(«'i «l{.) • <») ( G 1 Then, one can show that in terms of the new basis of operators V'fgt, qR\ z) is written and as [•&..•&] = m«"*m+„,o = ML, #(.)•£_•&*—' Xjftz) = z'l - iri In z + i £ lai'.,- , nlEZ flf|i)BE«4r-> . (is) ngZ ^(z-jH^-.^t.^.^ia^z-- , (J=1,...,B). (20) In this basis, we'define new Viiasoro operators by Here, z'/ and •'£ are introduced as the canonical conjugate operators to p'[ and p'R. L 4%) = \-APi!U))"^'E i°y"-' • It is easily verified that the ground state | 0 > defined in (11) also satisfies 0, 17 4 (*) • I: wft*))': • E iff*"""' • < ) «j?.|0> = aj{JO>=(l . fom>0 , P,/|0>=pi{|0>=0 . (21) where ° ° denotes the normal ordering with respect to a'[„ and a'J,n . Since both l'l„ and ifl° satisfy the same Virasoro algebra of the central charge D as LLl and LRl , The whole Hilbert space in the untwisted sector will be constructed by successively by qnantnm equivalence theorem [10] we find that X£°'(z) (ij, (z)) is equivalent to multiplying the ground state by the vertex operators V'(gL,qn;z) (±(9£I9A) £ A0a0) G, . Therefore, the allowed eigenvalues of {p'l,p ) in the untwisted sector will be of the 4 («) (*S?(*)) • R form! To determine allowed eigenvalues of the momentum operators p'[ = a'la and (PI>PK)6A„, . (22) p'£ = a'l0 , we will need to construct vertex operators in this new basis. Let — 169 — -170- 1 s Since the operators J^'(i), Pj{(i) and the ground state | 0 > are invariant under The center-of-mass coordinate x ~ j(z£ -I- x R) is called a fixed point [5,11]. The the Zw-transformation, one can show that in terms of the new basis of operators the quantization conditions for the non-zero modes are operator i?(o) is simply written as him,, 7ii,] = miS"Sm,,v., , ,R(a) = e*p{>2x(Pi -VL -PR-VR)} , (23) [7£»,,7j;L,] = "•/*"*,»„,., , otherwise zero . (29) where (v[,i^) is a constant vector and will be specified later. The physical space which describes strings on the Zjv-orbiibld is given by the Zjv-invariant snbspace of For the quantization conditions of the zero modes, we follow the prescription discussed the Hilbert space. This means that only the states otR = 1 survive as physical ones. T m in Ref.[ll]. The Virasoro operators in the [7 -sector are defined by Therefore, in the new basis the allowed momentum eigenvalues (PiiPjO of physical states ale restricted to x«r>W ^ I nm (HWH(*) - TT^F) - E *&-"-* • 3 (ris'.rfDsA,,, withpi-»t-pl,-tfl6Z . (24) MM = \ umf (HADUM - ^JJJ) = E ^'n*"""' • ( °) We shall next consider twisted closed strings in the Ur-sector (r = 1, • • •, N — 1) . ExpUcitly, 4''(*) ""1 4r'(*) It will be convenient to introduce the following symbol: For any number c, we define are given by [c] by 0 < [c] < JV where c = [c] mod N. Since U'J is an orthogonal matrix, it can be diagonalized by a unitary matrix M: 1 MUM = Udi„ . (25) ^W-i:^!))':^^-^)^. («) Since U" = 1, we may write Note that the c-number terms appear in the right hand side of (31) and are consistent tfi.. = «"«" . (26> with those obtained by use of the (-function regularisation [12]. It is easily verified N that both I*H and Ir^ satisfy the Virasoro algebra of the central charge D. where 0 < // < N (h 6 Z) and in = e"'l . Now introduce the (untwisted state emission-) vertex operator by The left- and the right-moving string coordinates in the {/'-sector (r = 1, •• •, N- 1) will be given by V{kiMi') = C{kLtkR;z): «P{iAt • XL{i)+ikR-XR(z)} : , (32) *£(«)=••£ + ' E ^"fL,*-'- «"&,*") , where (*£, k'R) G AflSfl and *i(I) = *'H + ' E rlV'-A.,!-' - «"TJL,*"'} . (27) ^ ' ^ n=l k 1 Then the left- and the right-moving momenta, P[{x) s id,X[{i) and Pa[z) = x(l + c<'ul-")' """ " . (33) idX (x), satisfy the following (/'-twisted boundary condition: f R The vertex operator obeys the following boundary condition: (P/(e'"z),Pi(e-"'z)) = ((F')»P/,(f;')»^(f)) . (28) lw, V{kL,kR\' -) = V{V-' -kL,U-' •*„;.-) , (34) — 171 — -172- tk m tk am l tr i 1 where we have used the relation e *' *+ *- = e '*f '-i+' '* ''"- [U]t It ^ As in the case of the untwisted sector, let us consider the operator V"(gx,g;i;-) r S not hard to show that Pl{z)tPR(z) and V[kL}kR\z) in the J7 -sector satisfy the {MILIIR) ^O9J) which will be constructed from i^(i), P^(f) and V(fci,fcn;r) and same operator product expansions (6) as in the untwisted sector. The operators which satisfies the relations (IB). Then, as before one can show that in terms of the new 7 Pl(*)iVL(a\z) 5 V[a,Q\z) and P^(z)tVR{a\z) = V[0,a\z) , where a is a root basis of operators V*'(gx,g;ft;z) is written into the form (19). It should be noted that vector of g , satisfy the operator product algebra of (twisted) g © g whose singular V[ As in the untwisted sector, we introduce the operator R(r) which induces the (PLIPR) "* detail. Ground states in the E7'-sector axe defined by %N- transformation: 7i»/|0;»!>=7H„,|0iJ!>=0 , for nj>0 , (40) J RM(Xfc),X'K(W{r) = W"Xi(*),V' XHz)) . (35) where z1 are fixedpoints . It is easily verified that the ground states satisfy Ii follows from the relations (28), (34) and (35) that in the [/'-sector any operator or/„|0;i>=a^„|0;i>=0 , for n > 0 . (41) 0(z,z) constructed from P'L{z), Pfe) »ud V(*LI*H;*) will satisfy 3 A suitable linear combination of \0;z >'s will form an eigenstate of (p'£,p%): (it(r))'0(z,i)(J?,r))-'=0(e "z,e-"'z) . (36) Let us consider the operators P£'(i) and Pj{[z) (I = li • • • > D) which are denned J>LI»«0',*<»> >' = »I(r)l»I by the same linear combinations of P[(z), Vi{a\ z) and PR{Z), Vn(a; z) as done in the Pfll»L(-).»i!(-)>' = »fl(,)l,'M')>,,il(')>' • (42) untwisted sector. Then they will satisfy (13) and The eigenvalue ("i(t)i»J(,)) has to satisfy *M(#M. tfOWw = (#(»). PSW) - (J - 1. • • •, 1>) • (37) It follows from (36) and (37) that )*$*{*) and P\[(x) satisfy the same boundary con dition as in the untwisted sector, i.e., because LJJ' = L$ and £„'„' = I^. The whole Hilbert space in the !7'-sector 1 1 (-piV"*), p ^-" '))=(#(<). p£m • (as) will be constructed by successively multiplying the state |«£(r), vj?(*) >' by the vertex operators V'{qi,qR\z) {±(qi,q' ) € A j). Therefore, the allowed eigenvalues of Thus, P'i(z) (PR{Z)) can be expanded in powers of z (t): R oS (PiiPjj) "> ""s V'-sector will be of the form: i.€Z (pi,p«)eA„,+ (»{,„,»/;„,) . (44) *] = Ei^,"' • (39) n£Z We will now show that in which a'[n and a^n satisfy the commutation relations (15). New Virasoro operators (»£(,)."£(,)) ^("I."))) ">°d A,9„ , (45) L'^'\z) and L'n'(z) in this basis are defined in the same manner as (17). Since both L'l'J and L%J satisfy the Virasoro algebra of the central charge D, by quantum equiv where (t/£, ti^) is one of momentum eigenvalues of ground states in the ff-sector. We alence theorem [10] we find that £][''(*) (i#'(J)) is equivalent to J#'(«) (££'(*))• first note that from the structure of the operator product expansions the operator —173 — -174- V'iqLiWs) (Mli'^) e &a®a) w^ict satisfies (18) can sector-independently be de In the previous sections we have shown that H is equivalent to fined by suitable linear combinations of P£(r), PR(=) and V{kj k ;z) by appropriately Jt R •H = {a'L,~°'L~-\lti.JR>' |n,m,-6Z>0, (rf.rf) € Aj,, } , (52) choosing the basis vectors of Afl9fl. Then nnder the Zjv-lransformation ^'(oXiSfli-) will transform in the same way in every sector and we find where w-i RMV'{qi,qR]z)R-) =«"'(«•«-«-«) V'(qz,qR]z) , (46) Ai., = { (rf.rf) 6 \J{A-,e. + '("iyR))\PL^L-p'R-VRSZ} . (53) where (v^,u^) is a constant vector and is independent of r. It follows from (19), (36) It follows from the properties (49) of (v^v^) that A^ep is a Lorentzian even self-dual and (44) that lattice if ABe„ is. Therefore, the total physical Hilbert space of strings on the orbifold is nothing but that of strings on the torns associated with the Lorentsian even self-dual (%))'V'(0l,?B;i)(iJ(,))-' = V'iqcqn-y'z) lattice \'gQ,g- The X'[{z) and A"JJ(£) in the new basis can be regarded as the left- and = e»»(w»itr)-««-o«i.))y'(ttl,J,;S) . (47) right-moving string coordinates on the torus. Comparing (46) with (47), we have tlie relation (45). Taking r = 1, we conclude that Finally, let us examine one-loop partition functions. The one-loop partition func ("ii^jl) can be identified with one of momentnm eigenvalues of ground states In the tion of strings on the Z^/-orbifold is given by [13] {/-sector. From the relation (46), we deduce that in terms of the new basis of operators N-\ i!(,) is written as Z{T)„HSM= X>(r,*;r) , (54) *(») = «p{>2f(pi • »L - v'n • "R)} • (48) where q = eiTT and The phase ambiguity which would appear in the right hand side of (48) actually N liB B vanishes as we will see later. Since (R(,)) = 1 and (v'Llvn) is one of momentnm Jf(r,.|r)-ili,„[(Jl{,))V -*«" -*] • (55) eigenvalues of ground states in the (7-seclor, (v[,v! ) must satisfy R The trace in (55) is taken over the Hilbert space in the f/'-sector. The equivalence of the partition functions of strings on the orbifold and the torus is now obvious because Ar(» ,.«ii)eA „D, r 1 T in terms of the new basis of operators (54) can be rewritten as J ^(r)„ ,„ = Tr'[gJi"-^j"i»-^] , (56) iK)' = ^) = jE(l-|)^ . («) H u 1 Every physical state in the C*-sector mast obey the condition iZ{r) = 1 because where the trace is taken over the Hilbert space (52) and it must be invariant nnder the Z#-transformation. Thns the allowed momentam 1 ™ *i gen values {p'l,p'^) of physical states in the {/"-sector are restricted to ^LO = 7 -175- -176- References [1] K.S.Narain, M.H.Saimadi and C.Vafo., Nncl. Phys. B288 (1987) 551. [2] K.Higasbdjima, private communication. [3] K.Kobayashi, Phys. Rev. Lett. 58 (1987) 2507; K.Kobayashi and M.Sakamoto, Zeits. Phys. C41 (1988) 55. [4] R.Dijkgiaaf, E.Verlinde and H.Veilinde, Utiecht preprint THU-87/17 (1987); P.Ginsparg, Harvard preprint HUTP-87/A068 (1987). [5] L.Dixon, J.A.Harvey, C.Vafe ud E.Witten, Nncl. Phys. B261 (1985) 678; B274 (1986) 285. [6] K.S.Naiain, Phys. Lett. B180 (1986) 41. [7] K.E.Narain, M.H.Sarmadi and E.Witten, Nncl. Phys. B279 (1987) 369. [8] M.Sakamoto, Kobe pieprint Kobe-89-02 (1989), to be published in Physics Letters B. [9] LFrenkel and V.Kac, Invent. Math. 62 (1980) 23; G.Segal, Commnn. Math. Phys. 80 (1981) 301; P.Goddard and D.Olive, Proc. Conf. on Vertex Operators in mathematics and physics, ed. J.Lepowsky et al. (Springer, Berlin, 1985). [10] D.Priedan, Z.Qin and S.Shenker, Phys. Rev. Lett. S2 (1984) 1575; J.F.Gomes, Phys. Lett. B171 (1986) 75. [11] K.Itoh, M.Kato, H.Knnitomo and M.Sakamoto, Nvcl. Phys. B300 (1988) 362. [12] C.Vata and E.Wilten, Phys. Lett. B159 (1985) 265. [13] N.Seiberg and E.Witten, Nucl. Phys. B276 (1986) 272; K.Inone, M.Sakamoto and H.Takano, Prog. Theor. Phys. 78 (1987) 908; A.Moroiov and M.Olah&netsky, Nncl. Phys. B299 (1988) 389. (14] K.Inone, S.Nima and H.Takano, Prog. Theor. Phys. 80 (1988) 881, -177- Scattering Amplitude of String on Orbifold* tit -J T orbifold ± A'/( AW^^a^w -* its] A€G, W 6 A Abstract CO|i: orbifold ±Tli# G ©SIcfcUE Lfc boundary condition £#-»*: Polyakov's path integral *$<,<£ orbifold ± a ,Ar( ,) V!(-V(.-,)) = F,(fl^(i())«!" ' - (2.2.2) — 179— -180- ;;t Fi{dX) It, d„X' © polynomial Tibio X'[z) Co^TOa#lifl-»® T*i. S-.T, -P 4 "=3¥ It classical solution X0t, (2.6.4) >£i8fc + J; 5 tt V boudaty condition (1.2) acfcllELT, •f'C-Cffl i lcr3i.1T IC-3 11-CC2 B;S«) Laplacian © determinant., -tftK Y J J X'(U,z) = Rf X {z) + nWf i = 1 ~ N (2.3) *r*5. *i\t,=.->r>tttoi»tUf (2.1) SCttltJTC £ 3 <0T*4 #, * ixli iftt fl, 6 G, Wf 6 A JtHiS-citt'itt.Sfctt-eiiteu,, iCT, S-fXo *if-5-p^. TSJiS^iMiat- SSIft-ffcOlco^Tff^o £ C~C, Viz T; interacting point *,- ffl||!IS-gitJ 2-2 X0 om& £ W/ ©«*&fctr*ta:i*h.5bttTMittU£V>-3*T*5o -eniiif-5 1.^* 2-2-1 Extension of world sheet A-^SIBSnajl-il^ t *R[IB«/&K branchpoint 4ittl> i ^ 5 *fl=T?* 1. l.> classical solution Xo Sftoltfcl'><0'Po53i', Xa <0 boundary condition 4> * z ^PBiKlt/S *i (i = 1 ~ N) * branch point kti cut (lunotn •5„ *OiS*5)S J A"=.ffl58lT & zjv G = {Tl = l,r,,"-,7),--1Ti} (2.7) X(z) = RN (RN-! (R„-7 • • • (Rt(RiX(z) + irW,) ;;t,CI ©7E®1H»« iiLt, ftR, § 7) KflpSL-C £ fflffl CPl surface si 4WAL, &•>- I- «i ±1= branchpoint zs £•£, RO*-*B Lfc^iTft + ffWj) • • • + *Wt,-l\ + ffWV-i) + tWN (2.4) &!» = »! (2.8) JZ/^Ajv-i ---/li = 1, (2.5.1) + tt*S «i J:®^i a"=>ffiSLT .-j o**)!)S-®**>Si SM J: 03816-fa* z KSJit+5 J: •> Kt 5c ::ri|) £li G OS 2)j K*fIStfc^- KO»"C* £Vw, = 0, (2.5.2) 0, Ty ST,y=T,ft TS*Lfc E(T, BIU A i ttEtt*«H*#Ki"8o US, (=1 0 1 R' = RNRK.i'--Ri+i. T,„=T,rm T„,„-. =T„7)T" . (2.9) x OTIC, X (jt) * classical solution ffl8B#£*ffl[§|t) 3 = 1_L + f;!^iL£ (2.io) flafloXD' = 0, (2.6.2) Xl{U,i) = R{JxHi) + wW{, (2.6,3) V'(V,*) = R{JYJ[z), (2.6.4) i = 1 ~ N. Rr,' = 1 (2.11) — 181 — -182- £19$. CftiC9**ffll>TtaW, -ttfc 7Z iOffgB 1-form/(71) (cSF) Ti ±(D, 3b i 1-form T,/(R)|„=/(K)|,n (2.17) ¥'(7l)| 3dJr '(z). (2.12) 1 D tsa+ntf *i5. 0" lc*f LT si II i, ic£sa>y C 71 otdfiSOSa) T,/(»)|„=T,"*J(»)|„ £/i*'(R)|i = *'(K)|, -5 J it z, o @ ; yi¥ (7I)|l = .Sf-'r'(7i)|l 2-2-2 Classical solution HBrWCI* *(fc) &tf *(7l) ffl dual**(7l) It St^T9;CDSSrf#So (2.13) *'(K)|j = .#•'¥•'(71)1, ¥(7t) = S.Xodi + a,XDdy, (2.19.1) «tf (71) = -d„X dx + dsXody, (2.19.2) S ilc 0 (x + iy = »). T?«tK t»W5= (Uf £< ©Ms iKS2H®E/ tt«3S+5CtB + 5o) 8£^.T 2 d(*¥(7l)) = (fl'Xu + a, X0)rf« A dy (2.20.1) = («Jft)"*''(7! d(¥(7l)) = (a«5„Xo - d,d,X0)dx A = R>0-,y'(7l)|J T?*5At, (2.20.1) SO*aii Jfo i1 classical solution •P*S£!.i5*a"J7l± •e fe 5 *• ii, fflV'fcS i C J5T?5= — ^ (2.20.2) S d(«*(7l)) = 0 on 71 (2.21.1) (2.15) c((¥(7J)) = 0 on n-{z,} (2.21.2) (SB *(7J) li Tl-{zi) ± — 183 — -184- Cfci-C r(7J) fcER-fhtf, r(7Z) |* ft-{.-,} ±TiEHi|tt l-form(3i5 3ae>T- **#£) T"S5o 2-2-3 r(») to-mi «(K) = Rer(K) (2.23) T*0. HBrWKtt T Xaa (2.31) Sr?it.ft5Ci*-i3*-.t. T(K) = 28.JM* (2.24) / r(K) = *r/r,/>Jwj T-*5„ ST, ^ »i ±4"Jtt)3ieLT zt aM>)Z rj Mi t> 5 m»* C,[j) iSSt St. C,U) li W±i?iiS1ffllSittto-cK4rfJT't<0BSt;«ai«* X>i(j) its. * r,r(7i) = T,r(7i) / T(Tl)= [ *{K) + i [ **(R) (2.25) rooa*ffitt r(7t) O-HJBI* •/CiO) «iW) -/ciO) eia©3fr.jgit stokes osacj; 5 r(7i) = r0(K) + u(7l) (2.32) i I *9{Tl) = i / <*(**(«)) = 0 (2.26) iS5;iiHfc*5, CCT- r0(7l) li (2.29) i (2.30) SSSfcTh MT« (3 w(tt) li TC ± i.\ fc 5 £ C h S JEM «S holomorphic 1-form "C :« (2.21.1) KSffl-,*:. -Sm-511 *(») OttHiO r,u(K) = T,w(7J) (2.33) / *(7?l = irrvT,;>jWj. (2.27) SSJfc+fcfflTftSo i«SCt4'fci>l, il CI?. Pj \t Rj -invariant projection matrix $ if u(7l) is o ^T^i J; 9 o —Ittls genus tfi g «D !l - •» vBJsO harmonic 1- 1 ,'"1 form liff-RS® complex vector £|ffl£a-*-i> 6, *fflS)£*w«(K) (o = l,'",ff) itSo £®B*, T,w (K) fe harmonic l-form «4> 6«„(K) ffl«M*S &?*>!**„ •* fc=o a (/})' = /}, ^^ = ^ (2.28.2) r,«.(K) = / r(7l) = irrjT,P/lV). (2.29) <'., li w„(ft) atjfcSftlf, *ne*JJ6LT»*S«T*8. -* w(ft) £ u„(7l) •WiO) •CBHLfcS; itt 0. r(7l) li > = ij, j = 1,'-,JV IcSK irrjT|P,W/ ffl simple pole 4j#-j ui(7l) = irV.u.(fc) (2.35) TV»»:iK44a *fc (2.18) SUO OfflaicT, SffifflStiT-PO (2.33) Si (2.34) S Sffl 5 £ T,r(7l) = T,r(7i) (2.30) , irV.t'alwl(«) = irTiV.u.(«) ittSo (2.29) £ (2.30) i l^Z-3ffl«fl=4«fctr(7lZ-3 — 185 — -186- tttSo cnnv, 1:306is.s (2.41.1) a<0*li r(7I) V. = i^ri-iCtli. (2.37) tttt^o Lfc*^T, »:«>at»5 1=1 (1 - fl,)(l - ^)Jr0(*i) = TT(1 - P,)Wi (2.42) L CtvJ: 0 »W = 7£T,-,C,(i„(li. (2.38) (1 - Pj)XoM = x(l - P,)QiW, (2.43) otfE r0(7i) EoKTMttLJtio r0(W) 11(2.29) St (2.30) a*fflfc+** Gi = -»£*«,)*(1-Ji) (2.44) J' i=0 SSS (2.43) a*«fc + J;iE Xo(po) *C„ ItftA&ftfctftt&ttH,, ££D«tt 71 Kfc. ffiSffl-jSp i ?*B&UTJE|in*,p i ? E*fttfhSK (2.43) aoStRS^-tT* J: ?„ B*eAi: fly *« invariant *aiB*8fctt li-la simple pole *fifo harmonic l-form 7)[p, g](7l) 4'gffi-f 5 = nlBS> BBS Pj = 0 (2.45) CO vlP.91(72-) *fflHT t*J«S**i5. CO)H&(2.43) SI! To(K) = ^t *W^[w(').Po('™)l(«) (2.39) X0(zi) = ir(l-iJJ)-'WJ (2.46) 4 LT-pnifCftli (2.29) t (2.30) *fflfc + *atfcfr4,, '. CTpj(l) 11* »i J: k tt**l. C ttli j SBro string O i = r> tti^-s*o, Po(/m) ili », J:»*5ja-C*50 (2.39) Slcfett* (0 string O fixed point (= j giffl string ©fi£>) fflttE Xo(z) <0 * = tj(— r0(7l) A< po(M B pole *&fcttl»#feft+*i«-C* 6. *SR r(ft) 0-ft»tt, interacting point) TTOfflNM* L < 4*14 V* SSffife •* i «•«*#** LTHS»*s f § 3 Discussion r'(K) = -^E £ xTj -'W/,bj('),Po('m)l + ^^C£''L"- («•*») JsU.msl f=l orbifold ±ffl string O path integral Kot>Tj|JSt> *cB«f"Pif}R t tt* ittSo CC-C—3SS»*«iJJKT!*6. Jt®»fiEf> (2.31) a * it TKTbfr 5 J; 5 classical solution m — HJftttfilliRffiBol'vTIIIft t, fc„ B. a»S» *o(po) * <7. 4l»-=fcff*tt««&**lTH*. L*»L*t*«4.:*l& *f, *££< *>*>&ttucfc*<**. *hi*.,§3©*iSE&«A: sassa e«gBot>ni»i. -3* B, *fl= (2.43) OTT, Xofoo) * 0, #*)•»*<& r (2.6.3) at ^ EA-.rroaaB^itT'p*. »4.nS*>4*iraHi!*So A#«Effl#tt«T;*htB* (1 - Pi)XQ{Uj2) = R,(\ - Pf)Xo(r) + ir(l - P,)W,. (2.41.2) K B Mfc boundary condition Sft,TU4»i, — 187 — -188- * tin S « !£ trivial tiWittWCb h ? ifi, * fttfr S nif gB*>-8tJ*-C S 5 i> i, operator formalismTli3tJ*4 -189- Operator Formalism on Higher Genus Riemann Surfaces' We can see a definite correspondence between their formulation and the path-integral formalism. However, the correspondence between their formalism and a conventional quantum field theory in the Hamiltonian fonnalism is not clear, and it is difficult to define the vacuum unambiguously in their formalism. SHUICHI OJIMA In this report we want to demonstrate a new formulation of field theory on Department of Physics, Osaka University Riemann surfaces with arbitrary genus. We will make use of the formalism re Toyonaka, Osaka 560, Japan cently developed by Krichever and Novikov (KN). The KN formalism has been extensively applied to various problems. ~ One of the remarkable aspects of the KN formalism is that a global definition of time is possible on a Riemann surface of arbitrary genus. Introduction of time renders us to define the Hamilto ABSTRACT nian properly as the generator of time translation on the Riemann surface. As a We formulate quantum field theory for b-c systems on higher genus Riemann consequence one obtains a field theory on the Riemann surface in conformity with surfaces. By using the fonnalism recently proposed by Krichever and Novikov, the conventional canonical formalism. We will see that the so-called KN algebra we can define Hamiltonian properly, and derive the Ward-Takahashi identities. plays an important role on the general Riemann surfaces with genus greater than or equal to one. It seems that KN algebra has a great advantage for investigating the field theory on arbitrary genus Riemann surfaces. 1. Introduction This report is organized as follows. In §2 we bruty review the KN algebra and introduce K-functions associated with b-c fields on a genus g Riemann surface. Conformal field theories have been developed along two different approaches, We develop quantum field theory for 4 — c lystems on the Riemann surface in i.e. path integral and operator formalism. In the path-integral formulation, §3. Fermi fields b and c are expanded in terms of the KN bases introduced in Eguchi and Ooguri,'" and Verlinde and Verlinde ' calculated various quantities §2. The Hamiltonian is defined and various commutation relations are found in such as Green's functions, correlation functions etc. on arbitrary Riemann sur §3. The Ward-Takahashi identities arc derived in §4, §5 is devoted to discussions faces in terms of theta functions and prime forms. On the other hand operator and summary. techniques have been effectively applied to string theories. Field operators are chosen to be representations of the Virnsoro algebra, which plays a central role in two dimensional CFT. The operator formulation on higher genus Riemann surfaces may be possible by sewing lower genus surfaces.w However, explicit computations are actually very difficult. • This talk is based on the collaboration with R. Kubo, II. Yoahit and Samir K- Paul ( RRK 89-25, to be published in Ptog. Tlleor. Phys.). -191- -192- 2. Review of Krichever-Novikov algebra and bases u,/2{z)=u,(z), (2.3/) In this section we give a brief review of the Krichever-Novikov(KN) algebra where u(z) denotes the third Abelian differential defined on M. and meromorphic differentials denned on higher genus Riemann surfaces. For a The vector fields e;'s satisfy the following KN commutation relations: more detailed survey of materials introduced in this section the reader is referred to Ref.[7) and [8]. 2A [ei,ei]= Y. cfa+i-' ( ) Let M be a compact Riemann surface with genus g. We choose two generic 5=-SO points f+ and P-. On this Riemann surface we can define the basis for \- with Cfj being the structure constants, hence the set {e,) constitutes the KN differentials, which are holomorphic outside P±, the so called KN basis. For algebra. The central extension of (2.4) is defined by the commutation relations, integer A ;£ 0,1 and g > 1, the behavior of the KN basis is given by Jo , , X !t ( , e,-,ej]= Y, Cj e,-+j-, + !((ei,ej), (2.5) /i (--±) = ^ -±"' * (l + 0(.- )). (2.1) i ± I a=~ga where where x(ei, e>)'s are cocycles, which vanish for |t + j\ > 3g. "(A) = | - ACff - 1) . (2.2) It is important to note that there is the third Abelian differential u holomor phic on M \ {P , P_} with residue +1 at P+ and —1 at P- and only imaginary Here z±(P±) = 0, where s± are local coordinates at Pj.. The index i takes on + periods. By making use of the peculiar property of u, we can define (proper) integral or half-integral values according as g even or odd. We also denote bases time on M by of meromorphic vector fields e,-, functions /,, The behavior of the meromorphic functions /, and the Abelian differentials w, near P± is given by (2 2 J ft /3 /i(--±) = af.-£-' (l + 0(r±)], (2.3a) In particular, T -» Too as P -> P±. We have a set of closed level lines (Cr) on +3/2 1 M, where «,•(=*) = /3f4' - [l + 0(.-±)| , (2.36) CV = {P 6 M\ T(P) = T 6 R} . (2.7) with a, and ft being certain constants. For i = -g/2,..., g/2 - 1, one has The level line Cr could be interpreted as closed string configuration at time r. /,(*_) = ar*:i-'/J-l[l + 0( r-)], (2.3c) J The meromorphic bases introduced above satisfy the following duality con + 2 ditions: u{{z.) = 0-zl " [l+O{z.)], (2.3d) in consequence of the Weierstrasa gap theorem. In particular we choose Let us next define "A' — functions" on Mt which correspond to "D — /,/a(*) = 1 . (2.3e) — 193 — -194- functions" in conventional quantum electrodynamics. Let The action for this system is given by (+, S[6,c] = 5 / £zJZ?,b{z)V*c(z) , (3.2) Jf(HI, in') = JT (ui,w') + K^\wt w') , (2.9) where V = p-1(r, i)9j. The stress-energy tensor t(z) is found to be where t{z) = -A6(z)V,c(z) + (1 - A)(V,6(--))c(r) . (3.3) KM(w,w')= Y. eifaWw') ,r > T' , (2.10a) l=-jn+2 The fields b(z) and c(i) are expanded in terms of the KN bases as -Jo+I JT'-'(u;,u;')= £ e|(w)fi,{"''),'•<'•', (2-106) l=-00 *(*) = I>/iA,(*) . (3.4a) A'(u>,ii/)c, =2*Htr-o'). (2.10c) t <:(=) = I] q/,"-A)(-), (3.46) Also if/;, AU and AV are defined as fallows: where 6jt and c? satisfy the anti-commutation relations, A'JJKU/) = -fl(r - r')A'(ui,ii)') , (2.11) {6jt,c;} =6k+ii0 , (3.5a) A*A(<",»»') = 8(T' - T)A'(UI,U)') , (2.12) {6t,A,} = {ct%c,} = 0 . (3.56) (+1 KF(u>,w') = 8(r - r')A' ("'V) - 0{r' - T)A''-'(II>,U') . (2.13) Here, k and I may take both integer or half-integer values according as the genus is even or odd. 3. Field theory for b — c systems The stress-energy tensor l(z) can be expanded in terms of the KN basis for quadratic differentials, ft(z)t as We develop quantum field theory for b - c systems on twice punctured Rie- mann surface M of genus g. A metric now takes the form: t(r) = £ A-„n„(.-) . (3.6) n da2 = p{z,z)dzdz . (3.1) We are moat interested in the case of A = 2, which corresponds to the ghost system in the bosonid string theory. In this case We choose two differentials, 6 = 6(i)(dz)A and c = c(z)(dz)l~x , where b(z) t(I) = ^llfi_t(r), (3.7a) and c(s) are conjugate to each other with conformal spin A and 1 - A, respectively. i -195- -196- c(r) = 5>e|(*) , (3.76) (violcj = 0 /or J < -so + 1 . (3.12d) / where {e/(i)} is the KN basis for vector fields. The KN operators K„ are ex Let us now introduce a new coordinate system parametrized by pressed as A w = T(P) + io{P) (3.13) '" = E E Clnb_k_a+,ct, (3.8) fc J=-ffo where use has been made of the duality relation (2.8) . It is easily shown that with T(P) given by (2.6) and the operators K„ satisfy the KN algebra Bo ff(P) = i/ («-*). (3.14) [A'm, tf„l= Y, C'mnKm+n-, + *,Ymn • (3.9) We attempt to formulate quantum field theory for the b — c system in this new We next define the ghost vacuum |i#o) by the conditions,' coordinate system. The parameter T{P) plays a role of time on a Riemann surface, hence one can define time-ordering products for any fields on M. As I 6(r)«(r) |fSo) = 0 , (3.10o) a consequence we get more clear-cut insight into space-time structure on the Riemann surface. I c(z)q{!)\t0) = Q, (3.106) JCr First of all we define the Lagrangian density on M by where v{s) and g(j) are a vector field and a quadratic differential having poles only at P+, namely, £ = 2>>(|: + ''£)cW' (3-15) "(*) = {«j(*)!J<-ffo + l}, (3.Ua) The momentum conjugate to c{w) is given by g(z) = {fi;(*)!J>-So + 2}, (3.114) 3 16 From eqs. (3.10) and (3.11), it is immediately follows that f = 2>>> <- > bj M = 0 for j > g0 - 1 , (3.12a) from which we find the Hamiltonian density, Cj M = 0 for ) > -jo + 2 . (3.126) * = ^ = 2>1?. Analogously we can define the dual vacuum <^o| by means of where c = dc(a, T)/8T. The Hamiltonian H is obtained by integrating H over (flS0|6j =0 for j — 197 — -198- the level curve Cr, Substituting (3.7) into this expression, one obtains H = I Hda = — / doh(a,T)—c(o,T) . (3.18) A>(tiMi>') =0(r -/) Y, etMnjfcC"') Jcr 2ir JCr da -So-H According to the conventional procedure for canonical quantization, we set up -fl(r'-r) ]*T et(wMw'). (2.13) l=-oo the equal-time anti-commutation relation, This satisfies the differential equation, {c(i»), 6(i"')}r=r' = 2TT6(.7 - a') , (3.19) (^- + »-=-)A'f (io,u)') = 2*62{w - w') . (3.23) or oa where a and a' are taken on the same level curve Cr. The Heisenberg equation Also substituting (3.7) into the anti-commutation relation one finds implies the holomorphy of the fields, {c(w), *(«;')} = A"(i», iff') (3.24) d^(w) = (^- + i-^Ww) = 0 (3.20) with oo A"(u),io') = 2ir £ «t(iB)nt(tii') . (3.25) where (ji{w) stands for b{w) or c(ut). The momentum of the system is defined by t=-oo Taking the limit r —> T' in (3.24) we have the equal-time anti-commutation P = lij dab( The stress-energy tensor t(w) is not a covariant tensor, but it satisfies the The Hamiltonian if and the momentum P are conserved quantities. Here, it differential equation, should be emphasized that the integrand in (3.18) and (3.21) are obviously regular functions except for poles at P± and that H and P do not depend on r, the case V"i(u>) = 0 , (3.26) different from those obtained by Bonora et al' and by Saito et al. * In their that is, t(w) is holomorphic on M. case H depends on r through 2g zeros contained in the third Abelian differential Commutation relations involving t{iv) are obtained as follows: By making u>. In our case, however, the integrand in (3.19)does not contain w, hence H does use of (3.24) one obtains not change its value when Cr passes through the 2g zeros of ut. |!(»),*(»')l = k d K(w,w')Mw') + K(w,w')8* -B(.r'-r){fo\b(w')c(w)\<),0) • (3.22) and ^jt(io) stands for lt{w) or c(w). Here we have omitted regular terms, which -199- -200- are expected to appear in the right-hand sides of (3.32). Equal-time commutation + ^2 (rWi(u>i) • • • 0»-i(«H-i)[tW, 0t(u»l)l*t+i(« [t(ui), Corresponding to the KN commutation relations (3.9) one finds the commutation where relation between t(w) and t(w') in the form, -2 for b{wt) for c(w) {1 t [t{w),t{w')} = - ndlK(w, w') - 2dwK{w, u/)t(ui') and we have used (3.26) and (3.31). Integrated form of this identity is found to + K(w,w')dm,t(w') , (3.30) be from which follows the equal-time commutation relation, (T(i(u»)#i(u>i)"- h + 2irS(a-cr')dwtt{w') . (3.31) + Y^,{ k3wJ 4. Ward-Tnkahashi identity for 6 — c systems Now we define the Teichmuller deformation of the correlation functions in terms of 3g — 3 Teichmuller parameters yj and a partition function Z by In this section we derive VVard-Takahashi identities for energy-momentum insertion in b — c correlation functions. We consider the quantities such as (T(t(u;).M>ui)''' 'PNi'HN))) ™d {T{t(,x)t(w)^{w,) • • • {T{Ki = (T(elI,((ui)0i(u>i)---*jv(ii'jv))) -201- -202- where is independent of this singularity, because the Hamiltonian (3.18) is independent of the metric, aid so are commutation relations. (A-J) = AlogZ. (4.6) We have shown that the Hamilton formalism is possible for the 6 — c sys Substituting (4.6) into (4.4) one obtains tem in a conventional way on the Biemann surface. The equal-time commu tation relations between the stress-energy tensor ((ID) and the 6 — c fields are (T(i(iiiW (ti)i)-..^(iii r))) 1 A derived automatically. Operator product expansions in conformal field theory JO-2 g = (t(«i))){r(^i(uii)-"^(u;/»)))+ X! fyMg-(IWu'i)-"<*iv("'w))} are replaced by the equal-time commutation relations. We have also derived the Ward-Takahashi identities for one and two insertion of the stress-energy tensor to N correlation functions, in correspondence with the results obtained by Friedan, and Eguchi and Ooguri. It seems that our approach is applicable to the free x(Wi(u;t)---M'»Ar))! • (4J) bosonic theory. Moreover it may be possible to construct the scattering theory since we could provide the usual Hamilton formulation. The Ward-Takahashi identity for double insertion of the stress-energy tensor It becomes clear that the KN algebra plays an essential role on Riemann can be obtaind in a similar manner. su^Bces. We believe that by further extensive investigations of the KN algebra we ii get a better insight into physics on Riemann surfaces. 5. Summary We have considerd the operator formalism for quantum field theory for b — c systems on a general compact Riemann surface of genus g. As a concluding remark, we would like to mention about a new space-time coordinate system HI = r + iff, in §2. The relation between ui and z is given by (3.13), i.e., dw{P) = u(z)dz(P) , (5.1) where u(«) is the third Abelian differential holomorphic outside P±. As a conse quence, one has the relation between metric tensors, p(zj) = p{w,w)\u,\2. (5.2) Since w has %g zeros as well as two poles, this equation tells us that p{m,w) becomes singular at 2g zeros of w, if p{z, S) is finite. However, our formulation -203- -204- REFERENCES 11. L. Mezincescu, R. I. Nepomechie and C. K. Zachos, " (Super)conformal Algebra on the (Super)torus ", preprint, UMTG-144, 1. T. Eguchi and H. Ooguri, Phys. Lett. B187 (1987) 127. R. I. Nepomechie, " Conformal Algebra on the Torus and its Supersym- 2. E. Veilinde and H. Verlinde, JVuci. Phys. B288 (1987) 357. metric Generalizations ", preprint, UMTG-145, L. Bonora, M. Martellini, M. Rinaldi and J. Russo, Phys. Lett. B206 3. L. Alvarez-Gaume, J. B. Boat, G. Moore, P. Nelson and C. Vafa, Phys. (1988) 444, Lett. B178 (1986) 41, G. Konishi, T. Saito and W. Takahashi, preprint, " KN Superalgebras with J. B. Bost and P. Nelson, Phys. Rev. Lett. 57 (1986) 725, Vertex Operators ". L. Alvarez-Gaume, G. Moore and C. Vafa, Comm. Math. Phys. 106 (1986) 1. 12. L. Bonora, M. Matone and M. Rinaldi, Phys. Lett. B216 (1989) 313, R. Kubo, S. Ojima and S.K. Paul, Mod. Phys. Lett. A4 (1989) 1423. 4. T. Eguchi and H. Ooguri, JVuci. Phys. B282 (1987/308. 5. A. Cohen, G. Moore, P. Nelson and J. Polchinaki, Nucl. Phys. B281 (1987) 127, G. Moore, Phys. Lett. B176 (1986) 369. 6. L. Alvarez-Gaume, C. Gomez, G. Moore and C. Vafa, Nucl. Phys. B303 (1988) 455. 7. I. M. Krichever and S. R Novikov, Funk. Anal. Prii 21 No.2 (1987) 46 ; 21 No.4 (1988) 47. 8. L. Bonora, A. Lugo, M. Matone and 3. Russo, Comm, Math. Phys. 123 (1989) 329 , A. Lugo and J. Russo, JVuci. Phys. B322 (1939) 210, J. Russo, Nucl. Phys. B322 (1989) 471. 9. L. Bonora, M. Bregola, P. Cotta-Ramusino and M. Martellini, Phys. Lett. B205 (1988) 53, T. Saito and K. Wu, Phys. Lett. B220 (1989) 93, G. Konishi, T. Saito and W. Takahashi, Prog. Theor. Phys. 82 (1989) 162. 10. L. Bonora, M. Rinaldi, J. Russo and K. Wu, Phys. Lett. B208 (1986) 440. -205- -206- Renormalization Invariant Effective Action theory, such as symmetry of the theory as well as the structure of the vacuum of the (7-Model String Theory we need a foundmental formalism for effective action which ought to be not only enough simple and elegant but also be directly obtained from two dimensional nonlinear er-model string. In lower order of a' there is a well known relation between 5eyy and 0- Y.X. CHENG functions of graviton and dilaton Seff = J[dX]jGe-™P* (2) Department of Physics, Osaka University Toyonaka, Osaka 560, JAPAN where 0* = /J* - iG*"/3 Considerable endeavors have been devoted to solve the conjecture that this formulation may be hold to all order in a'-expansions as a foundmental one of effective action. Before a complete formulation of string will be found we have to use indirect In present note we shall discuss how to realize this conjecture. We take the methods to construct the effective action for string theory with current string renormalizable property of the effective action as a starting point. The renor technology. One approach is based on the calculation of S-matrix elements. An malization group invariance will be imposed on effective action as a principal other possibility is to use the /3-function of the nonlinear Because this is a converse procedure it becomes very difficult to determine a determined by setting scale parameter in a designated value. general formulation of the effective action corresponding to the ^-functions con As we shall show the requirement of renormalization invariance of the effec taining higher order corrections with respect to string slope parameter a'. tive action will demand a consistent subtraction of mobius infinity appearing in On the other hand, field-redefinition ambiguity of the effective action is a nonlinear cr-model string theory. Then it is automatically reduced to Tseytlin's well known fact, there exist infinite many equivalent effective actions whcih are ansatz that the partial derivative of the partition function of the nonlinear a- related with each other through field-redefinition, and give same dynamics of model closed string with respect to scale parameter can be identified with the physics. Therefore to use the effective action to encode some properties of string effective action S,[[ of closed string. -207- -208- To simplify main proof we first review some well-known relations between the The chsractoristic of the closed string theory is that there appears moWus effective action and S-matrix elements generating functional Z( that there is a effective action St/j written as infinity arising from degeneracy of integral regine on surface. To see why nonlin ear a-model string y?-functions can be identified with equations of motion derived l Sel! = -j *&<)> +Sif^ (3) from effective acction Seff obtained by (8) we first consider following formulation dZ_ and a classical field 4>ci is defined to satisfy equation v v 30f| t -1_,^ mM^. ' •• •**"™^/n^MIJH-"IJ " \ * *' -vJ)/Sos nt 8S* (9) A*cl + -^i = 0 (4) where {ci1*1} is a set of backgound fields in which closed string is propagating. In order to subtract correctly the poles we may consider the degeneracy occur- Z(9) = St!l\U9)\ (6) 2 where we have defined a converse transformation by mean of reduction formula |nd^Jr'...^»^vm,(2rt,,)...vrhM(zrt,M^ -EfirSpi'*' •••#*"«:.•-SEMI, •»*(*! *».fc.) 2 8 £ini(¥>) > P^ °f Z(v) beginning from three-points amplitude. Therefore in (-2ir/ne„) / JJ d i„ S.fi(M = £[s>(4«i)l (8) M x(-2wlm )j f[ d2z -209- -210- IU^I/UUI is a combination factor which describes the possibilities of N-points de following equation generacy inside M-points operators. Singularity (—2irlne) arises from integrating az A out scale parameter /J which describes short-distance anomaly in operator prod • == ^F*"(£ F**" ' -*'""'*• • • ^"* * *«rl*l-•••.**.-*••*„(*i- ••.**. )-*.) J ) i M 1 v=l • ' (13) uct expansion [z,- - zn = /JUIJ, kilMi = 2ir/f' dnii"~ '~ = -2-KITK + ...]. In X H< (v*,(r)Vft(0)^ equation (10) the coefficient function A%t m„ has been defined by The term in parenthesis is nothing but the conformal anomalies which appear Al «„(«:,,...,*„,-*=«) = ZlZZ • • • ZZA"m> mw(*i,•••,%.-M (11) in tr-model calculation. It is defined as ^-function A (ti N p w = Y, •&**> *« kN,-k„)r'...r (i4) and A%, mH(ki,...,kN,-k„) is the (N + l)-points on-shell scattering ampli tude. we then reexpress (13) as Thus by virtue of formulas (10) and (11) in (9) we obtain equation ||If-/J"W)G..(« (15) where Gmn( M GW*)s|r|«Afc(r)Vi(0)\ (16) ., (-2Wn«„^|) / ' ft rf2-*d^«rf3^.(v*(-*)V»(.-») vs*~,< ,=„+1 Which is also called as metric of string field space. , x V mK+,(irt„+|)...Vmw(«tiiM)) Now we naturally identify /J*^) with a classical equation of motion if we / So (12) set 0 at a critical point ning of this note. But there remains another problem. Because Stu(tj>) must be /v=i renormalization group invariant we are in turn obliged to check whether so iden tified effective action Z[if{if)] respects the requirement that it should be invariant under renormalization group, i.e. it satisfy the Zamolodchikov-like equation S(2ir/nc) p w oty* (17) Now there appears an one-pole divergence (-2Wne„). However we can use moWua volume to cancel out this logarithmic divergence while we fix two points For following discussion here we have used the fact that the partial derivative of operators Vm and V„ In (12) by setting »i = zm = r -» 00,12 = rn = 0, of renormalized Z( and noting that/of = -27r/nemitiu,. Thus identifying („ = —211 — -212- To pro\*e equation (17) we can write Z{4>) in such formulation that there ap function we should take derivative of it with respect to /ne, instead of naive pears two-poles degeneracy, in TV-points and remaining (jtf — JV)-points operators division by. This can be achieved by simultaneously. Using the formulas (10) and (11) in calculation we have ,. (-2Wnem)(-27r/n<;„ «—lim< (—2ir/ne juw) m (19) 9(-2ff/ne)*,*„.*: , . =2(-2ir/ne) here i = nijn^obius. u x **»•'... r AlNt fflM(AAr+I ku, -*«) This is a very interesting and reasonable conclusion. Therefore we evidently x (-2Wnerfl)(-2irMe»)^jpr^7 JtPi*#i -5 £ jffi**1-**"< *,(* **."**) ZW = (-2Wn«W*(«)/»*(«Gfl,.(^) xEjBi*1"-/*^: .,(*i */»•-*.) jv=o • and (-2irfaem)(-2irfae»)|_| / , ,.,„,„,\ < T/ = fwrwc w (20) - ;'1tiil"^'''^^'0''W S - " 2 (-2irm«mjii11J) (18) Before we extend our analysis to higher genus topology we give a summary The factor | in final two steps of equation (18) is arising from a double-count for above discussion. The reguirement that Z[f((j>)\ be renormalization Invariant of the ^-function product. Here we specially marked the logarithmic divergences demands a consistent subtraction of the mobius infinity, that is, we should have 1 with notations m,n and motriua to indicate that it results from the m-degeneracy, 1/VS£(2.C) a d/d(2nlnt). This is just Tseytlin's ansatz '"*' and thus results in n-degeneracy and mobius volume, respectively. the equation (2) Now we compare this equation with (17). We see that if we take naive The problem in higher-genus extension of our discussion is how to subtract the division of (-2ff/n« )(-2!r/n«„) by (-2ir/ne 5N„,), there would be a difference m m moduli logarithmic divergence and write it as mobius volume-like forms. We like of the factor \ between (18) and (17). To eliminate this discrepancy we notice to emphasize here that the above picture plays an important role in determining a ourselves that it is unnecessary, in fact, is impossible, to distinguish which tnc- consistent effective action describing the loop-corrected S-matrix of string theory. factor in multiple intermediate poles corresponds to the mobius one. We must The simplest way for this extension is to adopt Schottky parametrization form. deal with them on an equal footing as pointed by Tseytlin in Ref.6. This implies It has the advantage of subtracting mobius volume clearly. that when we try to subtract the mobius logarithmic infinity from the correlation -213- -211- The nonlinear where PJJLQW) has been given by (14) of section 2. The factor 2 in front of right hand of equation (25) arises from taking partial derivative of Z^\ with respect (21) to (2-xlne). It is a crucial factor. Now we see from (23) and (25) that there is where H is a handle operator and given by no renormalization group invariance in torus level due to factor 2. But when we consider Zp=i(^) as an one-loop correction to effective action Stff — Z(^), we + + (22) '*=^ ^^ '- find that the equation (25) will guarantee the renormalization group invariance of the one-loop modified effective action ( This becomes more obvious when we Vi and V3 is tachyon and graviton emission operator respectively. consider the renormalization of nonlinear c-model string by a'-expansion ) Therefore for graviton component we get immediately &//(**) = Z,=o + Z.=i+— (28) That is, there is -~jtL — n. Another required term for this discussion arises from 5 K n di>* 7 27r(/mr) VSL(2.C) genus-two calculation wh ch is 1 d v x Jd z0d'z«(v = C|r|^Vf(r)V*(0)) It can be obtained by choosing weight of the two-loop Polyakov integral. This remind us of the interpretation of the small handle divergence in terms of an anomalous dimension, where it is suggested that the Virasoro algebra for higher by Betting JJ = zm = r -+ oo and sj = rj = 0 in rrioMuj volume. Here C is order does not act on a surface of a given genus g but on all surface with genus cosmological constant of torus which is calculated by < g. Thus based on approach used above we get an unified picture to deal with the higher loop contribution to effective action for nonlinear u-model string theory. J 2ir(ImT)1 As a conclusion we have discussed Tseytlin's ansatz in direct calculation and The equation (23) obviously gives an one-loop correction to graviton ^-function found that this approach gives an unified picture to obtain a consistent space- which is well known as Fischler-Susskind mechanism. time effective action for string, specially in loop level. We show obviously that — 215 — -216- the consistent subtraction of the mobius logarithmic infinity from multiple poles REFERENCES arises from the requirement of the renormalizability of string theory. The con sistent subtraction of the mobius logarithmic infinity is to take partial derivative 1. P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl.Phys. B2S8 of the partition function Z(ij>) with (2iWne), or equivalently, to scale parameter (1985) 46, E.S. Fradkin and A.A. Tseytlin, NucLPhys. B261 (1985) 1, C.G. Callan, D. Friedan, E. Martinec and M.J. Perry, Nucl.Phys. B262 t after renormalization. Thus we have Tseytlin's ansatz Sefj = g^^fi °n the sphere. (1985) 593. It is worth seeing another fact that this approach is compatible with the 2. C.G. Callan, I.R. Klebanov and M.J. Perry, Nucl.Phys. B278 (1986) 78. conformal invariance of the string theory. In free case Z vanishes on sphere due 3. A.A. Tseytlin, Phys.Lett. B194 (1987) 63, H. Osborn, Nucl.Phys. B308 to mobius volume. Physically, this indicates that the space-time cosmologies! (1988) 629. constant vanishes at tree level. When the massless backgrounds are attached to 4. A.B. Zamolodchikov, JETP Lett. Vol.43,(1986) 732, A.M. Polyakov, Phys. surface, the principle we shall insist on is still possible to explain the quantum Scr. Vol.T15, (1987) 191. component of Z{ip) as a cosmological constant of space-time. This is a dynamical 5. L.D. Faddeev, in Methods in Field Theory,eds.R. Bali an and J. Zinn-Justin issue. Since string theory is a theory of gravity we should have eventually the (North.HoUand.1976.) effective action 6. A.A. Tseytlin, Phys.Lett. B208 (1988) 221. 7. D. Friedan and S. Shenker, Nucl.Phys. B281 (1987) 509, J. Polchin- hence we may set F(4i) = [quantum component of Z{ The author would like to thank Prof. K. Kikkawa for useful discussions in 8. R.S. Das and B. Sathiapalon, Phys. Rev. Lett. 56 (1986) 2664, C. Love preparatory stage of this work. This work is partially supported by Japan-USA lace, Nucl. Phys. B273 (1986) 413. International Cooperative Researches in JSPS Grant. 9. R. Brustein, D. Nemeschansky and S. Yankielowice, Nucl. Phys. B301 (1988) 224, Jevicki and C.K, Lee, Phys. Rev. D37 (1988) 1485. 10. T. Kubota and G. Veneziano, Phys.Lett. B207 (1988) 419. 11. H. Ooguri and N. Sakai, Nucl.Phys. B312(1989)435, 12. N, Seiberg, Phys.Lett. B202 (1988) 81. -217- -218- Symmetry of Cutoff Theory and Symmetry of String Theory cutoff symmetries were denned by naively extending the symmetries possessed by the free cutoff theory to the interaction one. However this method is not Kenji SAKAI always well-suited, particularly, in the case where the gauge field exists because Department orPhysics, Osaka University, Toyonaka, Osaka 560, JAPAN its existence changes nontrivially the symmetry of the theory. The purpose of this paper is to derive the cutoff symmetry from the classical We relate a classical symmetry to that of the corresponding cutoff theory, which is an one by considering the cutoff theory to be the effective one of the continuous the extension of Wilson's renormalization group equation. The rulpotent BUST symmetry ory. We apply this method to the BRST invariant string theory in order to obtain of the string theory can generate the string field equation with a gauge symmetry. When the string field equation with a gauge symmetry and make clear the relationship thin gauge symmetry is fixed, the equation turns to be equivalent to the condition of the between the two dimensional symmetry and the string field symmetry. Weyl in variance. 2. Classical and cutoff symmetry 1. Introduction As explained in the introduction we regard the cutoff theory as the effective The cutoff theory is defined by the action S^(A) with a momentum cutoff one of the corresponding continuous theory. We start from the continuous theory A. The cutoff theory is an effective theory obtained from a field theory with with the free action the large energy scale above A; the lattice, continuous theories etc. It is ex fl 1 pected to produce the same expectation values as those in the same theory with sm = _/>*, d x2 ^x.jG;; ,,,,)^.,) = fujOf,',,*,,, (2) a different cutoff. Then the cutoff action must flow according to the Wilson's 3 and an interaction action SUM. Its partition function is renormalization equation"' ' . SfW + } Jd"x d°y d*(«, y)wfifsssH A) Cje-AM+We-AMl (3) D D •/ -lJd zd yGHx,v)1faSJ-{\)jfasH\) = 0. -/n%,<-**i|(„. Here and below the dots on the cutoff Green function ( 1 1 equation of motion. Some people' "' have tried to derive the string field equa p %] - exp (l J d x, d«n G,,,,,,)^ g^ J = «p(jG{n)4m«m). (5) tions from several Wilson like equations related to the cutoff symmetries. The -219- -220- H L H Let's separate the Green function G = G +G into the high energy part G Tci depends on n fields then Ti^i and Tmt contain n+1 functional-derivatives and the low energy one G^. We define the low energy effective interaction action with respect to the field because of the bilinear differential of the £[£;"]• In by general these cutoff generators contain short distance singularities through the i high-energy Green function G? .. In the next section we give some examples S; = -In£[c„]eXp(-.S7), (6) with the finite generator. As in the classical theory the symmetry restricts the and we rewrite (3) in the following form as the cutoff partition function: form of the interaction action, I/ci gives a constraint on that form in the cutoff theory. Z=JV' fvc/,e-s'-s'. (7) Next we consider the algebra of the cutoff generator. Suppose that the clas Sg is a free cutoff action including the inverse of the cutoff Green function GL. sical generators satisfy the algebra Suppose that the classical action S = Sa+S/ has a symmetry with a generator k [Td,Td] = f' Td. (11) Td[M], i.e., TM,S] SM = 0. (8) If the structure constants /'' are field-independent then the cutoff generators If the generator has only single functional derivative S, the partition function is Tfcut and Tmt also satisfy the same algebra because of their definitions; invariant under the following variation : 1 s , S PLA] = e-* %»]e «[i;,.Ij)e-*E[G H]e ' = P'^Li- (12) s s AZ = JV4iTd[ s s Tlcl{M = i 'Tdl 1 1 BRST symmetry should be satisfied by the free cutoff action. But his BRST T/M,[*, S] = Brc^T/dfe, i]^ ,,, = S[0ir|e*Tde-*f!(o (f)1 (10) transformation dose not have the nilpotency. 5 Te„.[M] = e^T/catlM^ = e-^Eio^e^Tde-^E^e ''. In the classical string theory there are two types of the BRST transformation. T i is a symmetry of the cutoff theory corresponding to the classical one. If cn One is the off-shell transformation and the other is that of the on-shell. Here we t Here we think the symmetry of not the theory but of the action. If the symmetry of the derive the cutoff BRST generator from the off-shell one in the classical theory in theory is considered, then we must take account of its anomaly, which arise from the measure of the fieldi n the path integral formalism1" . contrast with Redlich. — 221 — -222- The BRST invariant free string action ° is given by The BRST generators Qmi and Qicut constructed in our method have the particular properties; the nilpotency and the higher derivatives. The nilpotency ab 1 2 So = J Ah {g^daXBiX + Bab(g - /') - b^Vfc'}, (13) (Qcut) = (Q/cu<) = 0 is guaranteed by that of the classical BRST generator due to the general arguments about the algebra of the generators. Since the classical where we used notations; jj = Jggab and Vfcc(x) = g"dc^+g'":d c'i-d {ccg"i). ai c c BRST generator includes the field-bilinear forms, the cutoff generator contains This action is invariant under the off-shell BRST transformation generated by the differential operators through the third order. Qd = J A [-S3.X-L _ ^±i + ^ + iV-c^] As in eq. (9), the BRST invariance of the cutoff string theory restricts its interaction action, which is identified with the equation of motion for the string (14) field, We can check easily the nilpotency of this generator. S S e 'Qlcute- ' = 0. (17) Since on the sphere the metric jjf', and the multiplier B„|> ^ can not propa This equation has higher interactions up to the forth order because of the higher gate and do not cause any short-distance singularity, We should treat the matter rank derivatives in the generator. For the nilpotency this equation has the fol fields X?xy the ghost field tX. and the anti-ghost field i„j (rj in the cutoff theory. lowing symmetry: Let us denote the high energy parts of the Green functions of the matter and the ghost fields by G*(zi,X2jff) and G^ c(x\\Xi\g) respectively. We define a -tf _ -s'< = e-sf + Q n -sK (18) differential operator UH corresponding to fV;"] m etl* (3). e e Icat e then the infinitesimal variation of the interaction action is U„ = exp( JO^,*,,,*,,, + \0«h\n)ScmS^). (15) 1 5 Using this operator we can obtain the BRST generator of the cutoff theory; AS/ = -ln(l+ «*'' This infinitesimal gauge transformation contains inhomogenious terms with re spect to Sj1 similar to usual covariant string field theories""' . This gauge symme - W^)WGfn)]h^m + *(W0(i>G5(u)*(i)«?») try arises from the freedom of the gauge fixing condition of the two dimensional + 5°ii(33)t9<=G"z)] *(1)6(2)S°3) metric g°b(x) because in general the change of the action by the BRST transfor - JG2W«AD*(»)««I) + JI*OS!u)]«fn*ra)^i) (16) mation corresponds to the change of the gauge fixing condition " . Next we show the relation between this string field equation and the former - \Gal,\n)[dcGffd{li)] *(J)*(V'«'> one derived from the Weyl invariance. To do this we fix the gauge symmetry of the string field (interaction action) by choosing a particular gauge condition of the two-dimensional metric. The simplest metric condition is joi = n°', which is i G l5 i l5 X {il'jrit(l) (.H)] (3)*(4) + 3[jaS(l)Gu(34l] m(3)* * }• -223- -224- originally chosen in the continuous theory. Thus i?0j must be a multiplier field for lowing manner: the gauge fixing of the metric which is already contained in the free cutoff action. G A ThiB fact implies that the interaction action is independent of this field. For Qic« - -AcP(i){4n + Ki7ln pi)]V(j)} - - (c)P(i)^ii- (23) the BRST transformation of the anti-ghost field, the interaction action cannot The condition (17) turns out to be equivalent to the condition of the Weyl in- include &„&. Furthermore since the ghost number of the interaction action is zero, variance for the matter partition function. The matter partition function can be the ghost field is also excluded from it. Consequently the ghost and anti-ghost written in the form; differentials in the BRST generator become trivial and this generator acts on the s s interaction action as follows: Zmali„[p] = J^Xe- '- ' {M) L s a c Qlcttl = - (C 3a-Y)(]|4(l) + (VJ C )(i)l5joi(i) = E[G ]e~K - H[(C"a»)(l) + (^V.) + |(^C)(3)'W(3)]G(i2,}<5(')'S(2>- Then, as is given in ref [5], the Weyl invariance of the partition function is expressed in the form This interaction action is the functional of the matter fields X(x) and the metric gab{x) or gab(x). A Z s P(D W) "»»«'-W = %i]Ap(i)W(„e- ' = 0. (25) Since the full Green-function of the matter fields is the inverse of the Lapla- ab cian dag &b\ it transforms as a scalar under the reparametrization, i.e., If any dynamical degree other than Sf is not introduced, this condition will give the equations of motion and the arbitrary gauge conditions for the mode fields in [(ffl.)(u + (fa.)(2, + kivtfhfyampm = c (2i) the string field. This arbitrariness corresponds to the degree of introducing total divergent terms into the r.h.s. of (25), but such a degree is not a dynamical one. The generator Q[c«i becomes regular because it contains only the low-energy Green function. Let's consider to make the string field equation (17) regular. We should keep the gauge invariance (nilpotency of the cutoff BRST generator) and the gauge On the other band we can construct the low energy Green function and fixed equation (23). We modify the cutoff BRST generator by dropping out all the interaction action so that they arc also reparametrization invariant but not dependence on the full Green function from the cutoff BRST generator (16). New invariant under the Weyl transformation . Therefore they consist of not the cutoff BRST generator Qi t is defined by metric s°6 but j0' = pjoi. p is the Weyl factor of the metric and transforms m under the reparametrization by Qu.t = ui;'Qc,uL. (26) A(()p = daCp ~ i'daP. (22) The replacement of the differential operator UJI by Uj}1 obviously dose not change Consequently the generator acts effectively on the interaction action in the fol- the algebra (nilpotency) of the generator. The same gauge fixing can be applied to the string field equation obtained from this generator and we can derive also t In the more general cnse we must reconstruct the BRST generator for not only the repara- metrizetion but also the VVeyl transformations, as used in ref. [10]. the same gauge fixed equation. -225- -226- 4. Discussions nilpotency; Qlcut = U^Q U , (29) Our generator depends on the continuous theory as the original (large scale) cl L one of the cutoff theory. We can give the more general form of the cutoff gener where Qcj is the on-shell BRST generator of the classical free string theory. The ator; equation written in terms of this generator may correspond to the covariant string field equation because the gauge symmetries of the both equation utilize only the *]%*], (27) nilpotency of the free on-sheli BRST generator and have no two dimensional field theoretical meaning. To show their correspondence explicitly we need further where T„is[$, 6\ is a generator independent of the cutoff parameter, which sat investigations about physical and auxiliary components, their transformations, isfies the appropriate algebra. Torjj[0, #1 may be specified by the original theory S-matrices, decompositions of the Riemann surfaces and so forth. of the cutoff theory under consideration. This form is supported by the follow ing arguments: If the cutoff theory has a symmetry, T/CU| gives a constraint on 5. References the interaction action. The cutoff theory must be subject to the Wilson's equa tion. In order that the two constraints are compatible the cutoff generator must 1. K. Wilson and J. Kogut, Phys. Rev. 12C (1974) 75. commute with the Wilson's operator defined by 2. J. Polchinski, Nucl. Phys. B231 (1984) 269. 3. T. Banks and E. Martinec, Nucl. Phys. B294 (1987) 67. 4. J. Hughes, J. Liu and J. Polchinski, Nucl. Phys. B318 (1989) 15. The generator satisfying this condition must take the form as eq. (27). 5. U. Ellwanger and J. Fuchs, Nucl. Phys. B312 (1989) 95. 6. A.N. Redlich, Phys. Lett. B213 (1988) 285. In this paper we treat only variations of the action or its exponent by the cutoff generator. To investigate the symmetry of the theory, however, we must 7. K. Fujikawa, Phys. Rev. D21 (19S0) 2848. study further the transformation of the field. Then we will meet with some 8. L. Schafer, J. Phys. A:Math. Gen. 9 (1976) 377. problems related to the anomalies. 9. M. Kato and K. Ogawa, Nucl. Phys. 3212 (1983) 443. We gave in the last section the string field equation with the gauge symmetry. 10. E. Witten, Nucl. Phys. B268 (19 ,) 253, Since we used the off-shell BRST generator, the equation contains more degrees H. Hata, K. Itoh, T. Kugo, H. Kunitomo nnd K. Ogawa, Nucl. Phys. B286 of fields than the on-shell one and the covariant string field theories. In order to (1987) 433, make clear the relations between the renormalization approaches and the usual A, Neveu and P. West, Nucl. Phys. B278 (1980) 253. covariant string field theories we must construct the on-shell cutoff BRST genera 11. T. Kugo and S. Uehara, Nucl. Phys. B197 (1982) 378. tor. It is difficult to construct this generator in contrast with that in the classical free string theory. Instead of it we can define a cutoff BRST generator with the 12. A. Das, Phys. Lett. B205 (1988) 49. -227- -228- where a , Cm, b are annihilation operators of string coordinate, ghosts and anti- Canonical Quantization of Witten's String Field Theory m m in Mid-point Time Formalism ghosts, respectively. The bra vacuum ( -i- j is the Fock vacuum annihilated by negative frequency parts of ctm, c„, bm and the ghc^ zero mode CQ. In this letter, Masakiro MAENO r we drop Lorentz indices when no confusion occurs. The zero mode ajj = p Q of Department of Physics, Osaka University, Toyonaka, Osaka 560, JAPAN string momentum is a derivative with respect to center of mass of the string r, and it can be considered as a generator of translation of the string. This interaction is W? carry out canonical quantization of Wit ten's string field theory in mid-point time Tor' a non-local interaction because the centers of mass of three strings sit on different maliam, A divergence which arises in kinetic term can be regularized by discretizing the string. places from one another. As a result, (V3I contains infinite orders of derivatives, The equal time commutation relation of string fields can be calculated in this manner, and the especially, time derivative. Clearly, in so far as we stick at the center of mass theory ia shown to coincide with the one which is expected in a formal Lagrangean path integral time formalism, we can not apply the canonical quantization procedure to the quantization. string field theory. However, this non-locality is spurious, because all mid points XJ,(a = |) of three strings are connected locally v.i V'.'i*ten's three string vertex. Quantization of manifestly covariant string field theories are studied by many people. Most of these works are based on a formal Lagrangean path integral If we express the three string vertex in mid point cojrtl-nates -Y£(ff = j) instead formalism, in which, one simply mimics the perturbation method by regarding of the center of mass coordinates, the vertex no longer contains any derivative the kinetic energy as an unperturbed term and the interaction as the perturbed with respect to -Y^(o- = £). This rewriting of the vertex is first introduced by 1 one. In such an approach one can not ensure consistency of theories with the Morris'*' and further studied by Manes'' and the author.''' Hence, the difficulty canonical formalism. Some attempts in the canonical formalism are performed concerning non-locality can be avoided by employing mid-point coordinates. for free string. However, the canonical quantization for interacting string has not been carried out in a satisfactory fashion. The difficulty lies in the non- In the following, let us first summarize the mid-point coordinate formalism. locality of interaction. The Lagrangean density of Witten's string field theory is In the center of mass coordinate, an independent set of canonical variables of written as string coordinates and momenta arc £ = {v2|*)Qfl|*) + ^{v3|*)|*>|*), (i) coordinates: XQ and i = —=-(a„ - Q_„), where (V„\ ia an n-string vertex, and QB is a BRST charge operator. The second m \/2n term of (1) represents an interaction of respective string fields, which describes momenta: p0 and pm = ~p=(a„ + a-„). ' ' connection of three strings as shown in Fig. A. The three string vertex can be {\om,"„}=S „, , [xo,Po]=i) expressed in a Fock space representation: m+ 0 -229- -230- Let us turn our eyes to BRST charge in the kinetic term coordinates: xj and zm = -y=~(an — d_n)i : a momenta: pa and pm = —~{an + Q-n). Q = -, Yl m<*n<:-n-m • +5 £(n - ">): c_„c_ &„ : +co. (7) v2 B 2 ^ ' "'""" '" ' ' 2 m +m m,n m,n ([5m,dn] = <5m+t«.D » b/.Po]-*) The symbol : : denotes normal ordering. In mid-point coordinates, it can be In terms of the new coordinates, the normal mode expansions of string coordi rewritten as nates become QB = cjL + Mba +K + cjbo J+ - &oc/J"_, (8) X(a) = i/ + v^y^Xnfcosna — cosn—) (4) where P(ff) = - 7rpo*(ff - -|) + \Z2^pncosn<7 2 9' L=^(0)a + /*9(1 + I\ n>0 ;"=i^5gcosn^, Similarly, the normal mode expansions of ghosts and anti-ghosts become 0) 1 - L1 = - - 2J : a-nQn : - X,: nc-»4» ; +1' a c [a) =ci + ^d„(cosn c'(o-) = - i^2 c„smmr, /<• =P8„ + A-', 5 sinn sin n iV)=— £&„sinn<7, (5) *"=- * E " f E"« 2 A'' = ~ 5 E E "o-mOmC-. 6'(IT) =-Ur4oi(o'- -) + y^incosnff . "*'$1 (io) + m : c c ({im,c„} = «„,„ , {60,e/} = l) 2 E E (" ~ ' -» -min+m !i r n-ftn*o where bm = &m — &o cos m§ with 6m being the normal mode operator in the — : >Jcncosn—£':, center of mass formalism. Now the three string vertex no longer contains any derivative, i.e. W nc ainn CmSinm 11 3 oo = E » f E 2 t ) -231- -232- coefficient 6(0) in L means a divergence of one-dimensional i-function. i.e. The action has a gauge invariance, which can be fixed by using Siegel-like gauge condition i>o$ = 0. In terms of component fields, this condition can be 0 2 2 deduced to i/>' ' = 0. Although this equation is same as the ordinary Siegel gauge J^ cos n- = Yl sin n- = TT6(0). (13) in expression, the meaning of 6o is not the derivative with respect to co but one with respect to c\. Remarkably, owing to vanishing of "mid-point of ghost" c/ This divergence of the kinetic term comes from the ^-function in P(ff). Note that on three string vertex, the ^-component of string field does not present in the QB contains a term ir /„' dr7C(a)P((r)'2. interaction term. Now we can write down the gauge invariant action of string field theory in In free case ( g = 0 ), the gauge fixing procedure can be performed in the mid-point formalism. In the following, we expand string field as |$) = &o|aM0)^ •+• similar way in Refs. [9][10]. First, we add the gauge fixing and the Faddeev- |V-(0'). Further, the fields |^(°!) and |V>l°I) are expanded as follows: Fopov ghost terms to the gauge invariant action. The added action has the same-type gauge invariance. To fix this invariance, it is necessarily to introduce i^-Ew)*!?' (u) infinite series of BRST auxiliary fields listed below: A ghosts |^(B)) (n>0), and I0W> (n>0), (17) W-EW*?' (15) anti-ghostB |^(">) (n < 0), Nakanishi-Lautrap fields |B(n)) (n < 0). The index A runs through all Fock space vectors, The component fields /A and ip ' have no ghost number JV,, which is indicated with the superscript (0). Since A If we assume that the procedure can be applied in the interacting case, we can total fields |*) have JV, = -J, the state \A) in (14) has JV, = +j. Similarly, the obtain a gauge-fixed action state \A) in (15) has JVS = —1, By using this expansion, the Lagrangean density turns out to be 0 Sfi„d =(V2|i0|*>£W - {VAI^'/MltM )) + 2 (0) t0) + j(V3|fco|#)W>W> + 2(V2|M^ )(^-J+)l* ) (16) where (n) n) , ) In the above expression, we have carried out some partial integrations. The i*) = T; i* >. i*>=T: i^ ). IB> - "E I« - ). no-oo nesO n=0 interaction in (16) is local with regard to the mid-point, in particular, to the mid-point time coordinate. Now let us consider a divergent coefficient of the kinetic term which is written -233- -234- around the mid-point is described by A"'(^ir) = X(fiir) — -Y(f). Then the integration measure is equal to -J(0)|c3 VX = dx,VX'. (23) Potting, Taylor and Velikson"" regularize this divergence by ^-function method and set 6(0) = 0. However, it is unsatisfactory since their regularization changes where the dynamics of string field theory from Klein-Gordon type to Dirac type. VX' = "f[dXX^r). (24) In order to regularize this divergence, we apply the discretized approach of strings. A discretized string becomes a one-dimensional lattice with N + 1 In the second choice, we can choose center of mass XQ = -j^rr^X^w) as a points. (See Fig. B) Then, integrations are replaced to summentions, and 6- zero mode. The fluctuation around the center of mass is described by X(^ir) = functions to Kronecker-S symbols, >'. c. X(Jyir)—xo- Because all -X^ir) (0 < n < N) is not independent, we can exclude A(f) from Xs. Then (22) can be written as + 1 *J * „=o (20) D VX = (N + l) dx0VX (25) 'Sijjir-^x) - (iV + l)*»... where Accordingly, the divergence of kinetic term is regularized as rcr=n «*(£»). (26) -^(fl)8« - -(N -rl^ctycty. (21) where D{= 26) is space-time dimension. Since we can easily confirm VX' = VX, these two integration measures ( dxo and dxj ) differ by a factor (N + l)° which In this discretizing approach, the string field is expressed as a function having arises from the Jacobian. The factor N + l becomes iro"(0) in continuous limit. N + l arguments, i.e. $ = 22 VX = [J <**(£*)• ( > in center of mass coordinates turns out to be n=0 Jl/p (TJ m(l (28) We have two choices of zero mode of string coordinates. We use the term "zero / *'[-(jv+V-'fo '- ''-^ mode" for the coordinate which indicates the location of whole string. In the The tildes mean change of orientation of string. Let us quantize the action by first choice, we can choose mid-point i; = -V(|) as a zero mode. The fluctuation assuming the time-component of xj as a time. The momentum conjugate to 4> -235- -236- IS We have quantized Witten's string field theory in the canonical formalism II(x/,JT') = l Q^datixjtX') + ( no derivative term) (29) by using the mid-point time variable with a discretized regularization and ob tained the same results as the formal Lagrangean path integral quantization. We Calculating canonical equal time commutation relations with use of Dirac bracket, emphasize here that the ordinary center of mass time formalism, due to the non- we can get locality of the vertex, does not define a simple canonical momentum associated with the string field $, while in our formalism the canonical momentum can be , l s W*/,jr ),aWw,y')]|I.=,? = W + f~'(*i - viW - *")• oo) defined as (29)because the vertex is local. The path integral measure for string field * can be then well defined. Starting with this ( mid-point time) canonical where l.h.s. contains a divergent factor. However, this factor can be attributed formalism and changing the representation from the mid-point back to the cen to the Jacobian factor due to the transformation from mid-point S-functional to ter of mass normal mode expression we have reproduced the perturbative string center of mass A-functional. i. e, theory which have been conventionally used. To apply this quantization formalism to other string field theory is a highly (JV + l)fl-'S(x, - y,WX' - Y') = S(x - )6(X - Y). (31) 0 m non-trivial problem. In the close string, we do not know how to decide zero mode since closed strings have no special point as the mid-point. The interaction Eventually, we can nbtain a commutation relation of string field in center of mass of closed string may occur at any point on a string. Especially, in the n-string coordinates as vertices of the recently proposed non-polynomial theory for closed string,11 the number of interacting point is not one. However, the absence of canonical quan M -0,*),cW( «„J!')]| ; ,; = «(*0 - W)6(X " H (32) i ! I = tization of closed string field theory is a more serious problem which should be resolved. This problem seems to be a worthwhile issue to investigate. Now let us come back to the continuous theory. The above commutation The author would like to thank Prof. K. Kikkawa for valuable discussions relation leads to the free propagator and careful reading of manuscript, and S. Sawada for useful discussions. (*(PW»)> = 2^5*(P + «>- (33) If we substitute dm = am — po cos m|, L[p) can be written as UP) = -5P2 - 5 H: a-»Q": + £: nc-^n '•+1- (34) If we replace in with i„, (33) turns out to be the same form as the one in the conventional Lagrangean quantization approach of string field theory. -237- -238- REFERENCES 1. E. Witten, Nucl. Phya.B276(1986)291. 2. H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D34(1986)2360.; Phys. Rev. D3S(1987)1318. 3. I. Bengtsson, Phys. Lett. B172(1986)342. 4. G. Siopsis, Phys. Lett. 195B(1987)541. 5. T.R. Morris, Nucl. Phys. B297(1988)141. 6. J.L. Manes, Nucl. Phys. B303(198S)305. 7. M. Maeno, Phys. Lett. B216( 1989)81. 8. W. Siegel and B. Zwiebach, Nucl. Phys. B263(1986)105. 9. H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Nucl. Phys. B283 (1987) 433. 10. C.B. Thorn, Nuci. Phys. B287(1987)61. 11. R. Potting, C. Taylor and B. VeHkson, Phys. Lett. B198(1987)184 12. R. Giles and C.B. Thorn, Phya. Rev. D18(1977)366; C.B. Thorn, Nucl. Phys, B263(1986)493. 13. J. Bordes and F. Lizzi, Nucl. Phys. B319( 1989)211. 14. M. Saadi and B. Zwiebach, Ann. Phys. 192(1989)213; T. Kugo, H. Kunitomo and K. Suehiro, Phys. Lett.226B(1989)48; preprint KUNS 988. M. Kaku, preprint CCNY HEP-89-6; OU-HET 121. -239- Non-Polynomial Closed String Field Theory* we take 2ir) to any closed string, it is shown that one has to add n-string contact interaction vertices for arbitrarily high n and the theory becomes non-polynomial. They have found that the patterns of string overlaps are given by the polyhedra Kazuhiko SUEHIRO having always three edges at every vertex, and explicitly determined the n-string contact interaction vertices specifying the integration regions of the lengths of the Department of Pkysics, Kyoto University edges of n-faced polyhedra for n =3, 4 and 5. Unfortunately, however, they were Kyoto 606, JAPAN not able to specify the higher n-string vertices and the question is left unsolved whether they can be specified at all by some simple conditions. A non-polynomial closed string field theory la presented In a complete form, in the framework recently proposed by Saadl and Zwieboch. All the string Interaction In the first paper, we have completely determined the n-string vertex for vertices possessing string overlapping p&ttema of polyhedra are specified by very simple arbitrary n and show that the Feynman diagrams constructed with those vertices conditions. It is shown that the Feynman diagrams with these vertices correctly cover correctly cover the whole moduli space. Then in the second paper, we have the whole moduli space at the tree level. constructed an action based on those vertices in the operator language and proved its gauge invariance. Further we have shown that this theory correctly reproduces After active investigations performed on string field theory, it has been recog the n-point dual amplitudes at the tree level. Here, however, we mailly report the nized that the construction of closed string field theory is much more non- trivial content of the first paper because of the restriction of space. Interested readers than the open string's one. A straightforward extension of Witten's open string should consult with the reference. Ml field theory to closed string was known already in the early stage not to sat First let us consider polyhedra possessing the following properties: isfy the gauge invariance.' ' Kyoto group's closed string field theory satisfied i) They are homeomorphic to two dimensional Bphere S1. (1) the gauge invariance but it contained an unphysical 'string-length' parameter o, the presence of which led to divergent factors at the loop amplitude level. ' The ii) Only three edges join at each vertex. (2) latter difficulty was removed in the so-called covariantized light-cone approach ii't) The perimeter of any face, i.e., sum of the lengths /j by the price of introducing yet other unphysical variables, one bosonic plus two of the edges i of any face F, equals 2ir: fermionic ones, to complete a quartet of zero mode variables together with or. Although this covariantized light-cone string field theory is the unique consistent L'< = 2»- (3) (covarianl) formulation for the closed siring up to now, it is still not sufficiently satisfactory at the point that it uses many unphysical variables and is lacking naturalness. 2ir corresponds to the length of closed string in -241- -242- at a point. In particular, any n-punctured Riemann sphere, corresponding to tree level amplitude of n-closed-string scattering, is represented (in one-to-one correspondence) by an n-faced polyhedron (of above type) for which the n closed strings correspond to the faces and are glued to each other across the edges. The lengths of the edges are variables which play the role of modular parameters. Since the perimeter of each face is constrained to be 2ir by (3) according to our length assignment for the closed string, the number of independent parameters for the polyhedron is c — n, which indeed coincides by (5) with the number of modular parameters 2n — 6 for the n-punctured sphere (genus g — 0 Riemann surface): # modular parameters — e — n — 2n — 6. (6) This implies that, if we use the contact interactions with all the possible n-faced polyhedra, the whole region of the moduli space of n-punctured sphere Ft|. 2. Te'rihedron. is covered and we already get the correct answer for n-point (tree level) ampli Fi|. 4. Ooied path mnoundlni iwo facn on boih itdn of in tude. These contact interactions, therefore, may be useful in operator formalism edge. but they cannot be used directly as string vertices in field theory since the n- and hence 2 modular parameter. The equal perimeter condition (3) says that point amplitude there always has contributions also from the non-contact type the n = 4 polyhedron is such a tetrahedron that the edges opposite to each other Feynman graphs with lower point vertices. have equal length (Fig. 2);' Calling the three different lengths a, b and c, we have a constraint To give correct contact interaction vertices in field theory, we now define a special category of restricted polyhedra: First, we call a set of edges on a a + b + c = 2ir. (8) polyhedron a chtti path C if it makes up a closed loop and surrounds two or On this tetrahedron, there are three different closed paths each of which surrounds more faces. If any closed path C in a polyhedron has a length larger or equal to two faces separated by an edge. If we take a closed path drawn in Fig. 3, for 2ir, i.e., instance, the restricted polyhedron condition (7) leads to D'S*2"' (7) 2(6 + c) > 2ir -• a < *. (9) then we call the polyhedron a restricted polyhedron. The n-string vertex in Hence all the edges are restricted to be less or equal to -243— -2M- precisely fills up the missing region of moduli space of 4-string amplitude pro duced by Feynman diagrams with the Witten type 3-string vertices. It should be noted that this condition (10) that all the edges are shorter or equal to it is implied always for vn > 4 by our restricted polyhedron condition (7); Indeed, for each edge i on the polyhedron, one can always take a closed path C{ surrounding only the two faces sitting on both sides of that edge (See Fig. 4). Then the condition (7) for the closed path d says x + y > 2ir while the perimeter condition (3) for the two faces gives x + /,• = y + U = 2TT, hence implying '• < * (11) foe the length k of any edge. Our restricted polyhedron condition (7), however, Ft|. 5. Two types of n=a reilricted polyhedra obtained by adding I hook line to the prum. implies much more constraints than those, of course, since it should hold for arbitrary closed paths. To gel an image for the shape of higher n restricted polyhedron, the follow ing fact will be helpful: The (n + l)-faced restricted polyhedra can always be obtained by adding a line to the n-faced restricted polyhedra so as to bridge two edges on an arbitrary face in all the possible way. This is because elimination of one arbitrary edge from an (n + l)-faced restricted polyhedron giver H.' n-faced restricted polyhedron provided that the closed path C,- surrounding U.at edge Fig. S. The l«o lypcion n.6 polyh.drt in ii|. 7 are coolint.oi.ily Pig. 7, Contact-type .V.polm Fevnman dlainm. like in Fig. 4 is taken to have the length 2ir. Let us call the added line a hook connected to each other. line. from the positions A to B keeping the length conditions of the restricted poly Adding a hook line to the tetrahedron in Fig. 2, we obtain the n = 5 polyhe hedron, then the resultant n = 6 polyhedron is of type I. This means that the dron, a prism, which is unique since the tetrahedron had only triangular faces. type I polyhedron with the length parameter / (in Fig. 6) approaching to zero The uniqueness of the shape of polyhedra is no longer true if we go to n > 6. For coincides with the type I polyhedron with the parameter V going to zero. Hence n = 6 polyhedra, for instance, we get two different shapes as drawn in Fig. 5 by there is actually no boundary at / = 0 (or i' => 0) In the moduli space covered by adding a hook line to the prism. We will have more and more different shapes our contact interaction which is given by integrating over all possible restricted for higher n polyhedra. polyhedra. Fortunately, however, we need not classify the polyhedra into classes of dif The same remark holds for all cases in which any edge approaches to zero ferent shapes at all for the present purpose. This is because the polyhedra of length. The length / of each edge is positive, / > 0, by definition, and / = 0 may different shapes are eoniinuovtly connected with each other as the length param seem to give a boundary. It is, however, seen easily by the same reasoning as eters are changed as far as they have the same number of faces. This is easily above that it actually gives no boundary since the / = 0 configuration is realized seen by considering the case of n = 6, as an example. Suppose that the hook always in a pair as limiting configuration of two polyhedra (either of the same line on the prism is placed at the position A in Fig. 6, then it corresponds to shapes or of different shapes). the type H n = 6 polyhedron in Fig. 5. Now rotate the hook line continuously We are now ready to give a proof for the central theorem in this paper: Theorem The whole moduli space of the v/V-point closed string amplitude t This possibility that the tetrahedron type interaction can fill up the missing region of the (at the tree level) is just covered by the sum of all the (tree) Feynman diagrams 4-polnt amplitude was first pointed out by Kako. with the string vertices of restricted polyhedra. -245- -246- Vn and 2>„,m. exist on the opposite sides of the common boundary each other is seen if we note that the diagram Fig. 9 with infinitely long propagator should correspond to the "moduli parameter = length of Co™ approaching to 0 and hence ^n,m to the moduli space region "length of Co < 2ir". Conversely, if we are given two arbitrary Ttiiricitd polyhedra Pn+1 Fig.8. CntlesI (rV«rt+m).riced polyhedron P, in which the and Pm+i, we obtain an (n + m)- lenilh oTCn becomes 2it faced polyhedron P„+m by choosing two faces, one from each of P +i and Proof As was shown by Saadi and Zwiebach, the whole moduli space MN n Pm+l arbitrarily, and gluing them to of the W-point tree amplitude (or of TV-punctured Riemann sphere) is covered gether (allowing the twist). Then the already by a contact type Feynman graph (Fig. 7) alone if use is made of the siring vertex consisting of all TV-faced polyhedra which are not constrained by the resultant polyhedron restricted polyhedron condition (7). Since we are now using the TV-string vertex of Pn+m IS a re restricted polyhedra alone, the contact type diagram in Fig. 7 covers only a partial stricted polyhedron with critical con domain Vn, in the moduli space MN. The boundary of the domain VN, that figuration. Being critical is because the perimeter of the glued faces be- -.„,„,. , . . „ . . appears owing to the imposition of the constraint (10) on the integration region J , , ,°. „ fl, r»lD.TwocloiedpiihsC,indC..DbtiinedrromCby«ddin| over the length parameters of the edges (i.e., moduli parameters), is therefore Comes a Closed path (Call Go) On the boundary seiment on C shorter than ft. specified by the surfaces at each of which one of the closed paths realizes the Pn+m with length 2ir. To show that length of boundary value 2ir: Pn+m is restricted, we clearly have only to consider closed paths on Pn+m which cross the boundary Co of P„+i and P +i. Such an arbitrary closed path C sep £ U = 2*. (12) m arates the boundary path Co into two pieces, one of which is shorter or equal to IT since Ctj has length 2rr. Then adding the shorter piece of Co to each of the Pn+1- and Pm+i- side pieces of C, we can draw two closed paths C and Cm as The polyhedron in such .a case we call critical. Consider a boundary surface n in Fig. 10. Since C„ and C are closed paths on the restricted polyhedra P +i 0T>K\C, corresponding to a critical TV-faced polyhedron Pjv where a closed path m n and Pm+i, respectively and hence are larger or equal to 2ir, we can conclude that Co on Pfi realizes the length 2ir (See Fig, 8). Let the number of faces on both C is larger or equal to 2ir. Thus P„+m is a restricted polyhedron. sides of the closed path Co be n and m = N-n, respectively, and suppose that we This shows that the boundary surface (9X>n,m lo realized by the Feynman cut the polyhedron PN along the closed path Co. Then we obtain two polyhedra, diagram Fig. 9 when the propagator collapses in fact just coincides with the (n + l)-faced one P i and (m + l)-faced one P +\. The important point here is n+ m boundary 9Z>jv|c„ of the contact interaction graph Fig. 7 without any excess. Of that they are both restricted polyhedra. This is because the new face given by the course, the one-propagator diagram Fig. 9 itself has other boundary surfaces than cross section satisfies the equal perimeter condition (3) by (12), and any closed SV„, \a as well, which appear when the polyhedron P„+i or P +i in the vertices paths on each of polyhedra P„+i and P +\ are regarded as those on Pn and hence m m m becomes critical. It is now clear, however, that any one of them coincides with satisfy the restricted polyhedron condition (7). Therefore both polyhedra P„+i the boundary realized by a two-propagator Feynman diagram which is obtained and P +i are contained in our string vertices, (n + l)-point one and (m + l)-point m from the one-propagator diagram by cutting the critical polyhedron along the one, respectively. This means that we can consider a Feynman diagram as drawn closed path of length 2ir and inserting a second propagator there. The latter in Fig. 9 in which the two string-vertices are given by the polyhedra P„+iand two-propagator diagrams have yet other boundaries, but they are commm to Pm+i- When the propagator collapses In Fig. 9, the diagram gives the same those of three-propagator diagrams, and so on. This finishes the proof of the configuration as the above contact interaction diagram, and hence the boundary Theorem. dVf/\c, coincides with (some part of) the boundary dV ,m of the moduli space n We should remark that the above proof also shows that the present string domain V covered by the Feynman diagram Fig. 0. The fact that the domains n%m field theory is actually non-polynomial. Indeed the gluing of two restricted poly- -247- -248- The gauge invariance of the action (13) under (14) holds if we have the hedra P„+i and Pm+i produced a yet higher restricted polyhedron /V=n+m. This implies that restricted polyhedra PJV exist for arbitrarily high N and the following identities: corresponding vertices have to be prepared for N. We have thus proved that the Feynman diagrams of our non-polynomial + QB[*1 • • • *Jv-l] + E (-)" '[*l • • • QBIH • • • *J»-l] closed string field theory properly cover the whole moduli space for any tree i=l level amplitudes. The measure of the integrations over the length parameters of = - E E t*« *» - - • *i—> "ff [*••• *fa • • • «i— 3]. the edges of the restricted polyhedra, which we have not mentioned up to here, n+"i-2=;N partition! should of course be chosen in such a way that the contribution of the contact (16) interaction diagram to the amplitude coincides with the Folyakov's one for the which is proven in Ref. 2. The left-hand side leads to a total derivative form restricted moduli space region corresponding to the restricted polyhedra. Then J dm jjj-(• ••) with modular parameters m; (t = 1, •••,£ — N) (independent the above theorem implies that the full amplitudes of our field theory reproduce length parameters) of our restricted polyhedra Pp, as was the case in the 4-string the correct amplitudes at least at the tree level. vertex in the Kyoto group's open-string field theory.1 Then only the surface The full covering of moduli space without gaps is closely related to the gauge- terras remain and the relevant configurations in the left-hand side becomes only invariance at the string field theory. They are the two aspects of one thing, as those of critical poiyhedra PN corresponding to the boundary surfaces dVt/\ci, was indeed the case in the known string field theories, Witten's open-string one1 and hence the equality to the right-hand side holds because the right-hand side and Kyoto group's ones for open-string' ' and for closed-string.' ' ' Indeed we corresponds to the configurations of 81>n,m|o. can construct the action for the the present non-polynomial closed-string field It would be very important to find the direct meaning of the full non-linear theory of the form'. action (13). It is an analogue to the scalar curvature R in Einstein gravity and may have such a geometrical meaning. Another important issue that should be investigated is that of loop ampli s = 5*-gB60-*+f;2jV, (is) tudes. It is now clear, unfortunately, that the single covering of the moduli space is noi realized for the loop amplitudes in the present form of our string field which is invariant under the gauge transformation theory. In fact the one-loop tadpole graph is calculated by using only the Witten type vertex and is known not to realize the single covering. ' The one-loop higher point graphs also seem to fail in giving single covering! This situation UK*) = QaiffA + £) y^lfft^A]. (14) is, however, what is expected generally for the non-local field theory. Our field theory action (13) realizes only the gauge-invariance at the tree level and should be regarded as that in classical field theory. In quantum theory, we should keep Here [*i • • • *tf-i] represents a string field obtained by gluing the N - 1 string the BRS invariance of the path-integral (after a gauge fixing). The non-trivial fields $[,...,4jv-t using the iV-string vertex V/y, and $" is a scalar product transformation of path-integral measure dictates modifications of the action at •t • [4» • • • *y»] f°r tne atie *i = *• K is completely symmetric. The field * as quantum level. The modification procedure was recently clarified by Hata! well as the gauge transformation parameter A are assumed to satisfy ' The single covering will be achieved with such a modified quantum action. (L - t)* = (I - Z)A = 0. (15) [This constraint for the field $ was implicitly assumed in the above discussion since we have not worried about where the origin of closed-string -249- -250- ACKNOWLEDGEMENTS The author is greatly indebted to Taichro Kugo and Hiroshi Kunitomo for fruitful collaboration on which this report is based. He would like to thank Hiroyuki Hata, David Lancaster and Mark Mitchard for valuable discussions and helps. REFERENCES I. T. Kugo, H. Kunitomo and K. Suehlro, Phyw. Lett. B33B (1988) 48 3. T. Kngo and K. Suehiro, "Nonpolynomial Clued String Field Theory: Action and Ita Gauge Iivariance", Kyoto preprint KUNS 988 (1989) 3. E. Witten, Nud. Phya. B36S (1986) 253 4. S.B. Qiddinge and E. Martlnec, Nucl. Phys. B27S (1986) 91 5. H. Hata, K. Itoh, T. Kngo, H. Knnitomo and K. Ognwa, Phys. Rev. D3S (1987) 1318 6. H. Hata, K. Itoh, T. Kngo, H. Knnitomo and K. Ogawa, Phyi. Lett. 172B (1986) 195 7. W. Slegel, Phyi. Lett. 142B (1984) 276; A. Neven and P.C. Wat, Phya. Lett. 1S2B (1987) 343j NacL Phyi. B303 (1987) 2S6; S. Uehara, Phyt. Lett. 100B (1987) 76; 198B (1987) 47; W. Siege! and B. Zwiebacn, Nad. Phyi. B282 (1987) 125; Phya. Lett. 184B (1987) 325; Nad. Phyi. B388 (1987) 332; T. Kngo, la Quantum Mechanics of Fundamental Systems 2, ed. by C. Teitelboim and 3. Zanelli, Plenum, 1989; and reference] cited therein 8. M. Saadl and B. Zwlebach, Ann. Phys. 192 (1989) 213 9. K. Strebel, Quadratic DiBlrlnliall, Springer Veilag (1984) 10. S. Carlip, Phyi. Lett. 214B (1988) 187 11. M. Peskin, anpnbliahed; J. Lykken and S. Raby, unpublished 12. M. Kakn, Pliyi. Rev. D38 (1988) 3052; M. Kakn and J. Lykken Phyi. Rev. D38 (1988) 3067 13. H. Hata, K. Itoh, T. Kngo, H. Knnitomo and K, Ogawa, Phyi. Lett. 172B (1986) 186;Phyi. Rev. D34 (1996) 3360. 14. O. Zemba and B. Zwiebaeh, "Tadpole Graph In Covariant Cloaed String Field Theory", MIT preprint, MIT-CTP-1633 15. H. Hata, private communication 16. H. Hata, Phyi. Lett. 217B (1989) 438; 445; "BRS Invariance and Unitarlty In doled String Field Theory", Kyoto preprint KUNS 968, to appear In Mtcf. Ply».; "Construction of the quantum Action for Path-Integral Quantization of String Field Theory", Kyoto preprint KUNS 987 -251- Kormholes have been argued to play important roles'-*'1 in the theory of gravity and matter. Especially it is surprising that the cosmologucal WARMHOLE SOLUTIONS IN THE SKYRME MODEL constant may vanishes due to the effects of wormholes. Vormholes solutions discovered*-,° so far, however, are not rellevant to realistic models. Some of them arise in scalar mudels' (axion models) with a global I'(II A. Iwazaki charge. 8ut the scalar boson is not still observed. The others were found in gauge theories*''". but they carry no conserved charges which are necessary for their stabilities (Instability of the solutions means that Department of Physics their contributions to a functional-integral are negligible). Nishogakusha University In this letter we shall show stable wormhole solutions in the Skyrme Ohi 2590 Shonnn-machi 277 Japan model of hadrons. The model is an effective theory of QCD. The wormholes are shown to carry conserved topological numbers of Skyrmions". Me also show solutions of wormholes with isospins but with no topological numbers. As is well-known, the Skyrme model coupled with gravity are described in terms of SIMM,) field U. L= -R/lBitG + f'Tr iaAl/a;U)g"/4 t h"Tr( 4.U A If* 411 6)1?) g^ g"/2 - hs Tr ,41Iclu'cLU41J> ff'/l (II with the pion decay constant, f and with an appropriate numerical constant, h. where g^are metricos of a Euclidean manifold. Components of U (US Abstract SlHNt)) are made out of observed hadrous ( n. K. n~"l. Nr is the number Ke present wormhole solutions in the Skyrme model coupled with gravity. of flavours. G is the gravitational constant and R is the Ricci scalar. The wormholes have topological numbers which are identical with those of In the model. solitons(Skyrmion) exist, which are characterized by a topological number. D, (SUINi)) = Z. This number is identified with a baryon Skyrmions: they are baryon numbers. He discuss physical implications of the number". The models are known to effectively describe interactions of mesons wormholes. and of baryons. In our investigation of wormhole solutions, we take a simple ansatz for a line element of the manifold (M = R x Sil -253- -25.1- ds* = dt2 t a'dldCJj" (21 we arrive the equation from Einstein equations. (there dQi" is the line element of the unit 3-sphere Si. Accordingly in the 31a2 - 11/ 8TtGa= = - 3f2/2a2 - lZhVa' (7) space of Si. we may have a field configuration with the topological number. .Indeed, wormhole solutions carrying the topological numbers are easily found vhere we assumed the vanishing cosmological constant (a dot denotes a by taking the following ansatz. derivative with respect to t). The equation does not change if we take £ instead of L as U. L' • L!Si) (3) The equation (71 can be solved explicitly. where L(Sal is a SL'IN'.t matrics depending only on the coordinates of Si. a = Hl-Elt' » 32 x G hVll- E ) } '" (B) I.(Si) is taken to be topological^ nontrivial. For simplicity we consider the case of two flavours, Nr = H. LISjl is with E = 4-reGf2. The asymptotic behaviour of the solution is chosen explicitly such that given such that a'-Ml- clt2 as t— ±°°. The size of the wormhole is ( 32 uGhVIl- Ell "'. which is the value of a 101. L = £o * i t • a. P.o' * £* = 1. It is interesting to see that the wormhole collapes if we neglect the and £n= cosrj. with Pnuli matrices o ,. where n . 8 and 4 are coordinates of Si Id Qr • of the size of our "Skyrmion". dn" * sin'ndS* • sina 71 sin" H d •'). This field LISjl possesses the The solutions represent that there are wormholes inside of which topological number. Skyrmions flow with baryon numbers. Because our model is an effective theory of QCD. the existence of such wormhales can be naturally infered B • B/STt'a1 J -'"g dn d fl d * Tr ll'd, i; L'fl, L' I'd, U » 1 . 13) from QCD itself: the baryon number is a symmetry associated with a global Ulll symmetry. But in QCD the number is carried by fermions so that it is If we choose L* instead of L. then B = -I. It is easy to see that this hard to see such wormholes because of no existence of classical fermionic L (or ill satisfies equations of motion for I. fields. On the other hand, in the model of eq. Ill baryons arise as solitons Sow let us derive an equation of motion for the scale factor alt). (Skyrmions). Thus it is not so surprisinij to see that these Skyrmions can i:sing the following identities. pass tlirough the wormholes. If we add a Ness-Ziiminn term" to the model, the field configuration L ,r 0 . I.'JJ" d . U = -3L."g /a1 and g" a ,Ld ,l!d ,L = -1) X 'a2 161 wi^tl B = 1 kyrmionl corresponds'* to a fermion with non-zero spin. Thus. -255- -256- such an object can not exist in the closed space S3. In our model we do not q • 3 aq/a = B (9. a) take into account of the term. Hence our Skyrmion-like configuration may and be bosonic and can exist in the space Sa. 3 [a2- 1)/8TIG = - QVBit'fV (9.b) »e expect that there are wormhole solutions with even number of B. Because their configurations correspond to bosons, sucli objects can pass where the Skyrme term vanishes identically and through the wormholes even if the Wess-Zumino term is added. Now. let us check the consistency of our solutions. The Skyrme model S(t) = iy-5. yi = sin q cos r, is an effective theory of Q.CD and so its applicability is restricted in a (101 region with a length scale, larger than ahout 1 GeV. Furthermore we note y2 = sin q sin r. y3 = cos q. that the radii of our wormholes are the order of G"2. Accordingly the radii are much smaller than that scale. Below the scale we must use QCD with an arbitrary constant r. itself. Thus the wormholes are not physically acceptable. Q are the isospins associated with the symmetry. I' — V U V and Q" is As a rather realisitc possibility we expect that there exist wormhnle given by solutions with giant baryon numbers (e.g. B = 10*°|, which might arise in a similar equation to eq. \'ii but with large coefficients in the terms of the Q2 = i n' f a" q2 (111 right-hand side: it is expected that the terms are proportional to the square of the baryon number as in the cases of other conserved charges The equation (9.a) implies the conservation of the isospin charge. In the discussed in previous papers*' '• Then the radii of such wormholes are not derivation of the equations we take into account of the fact"'''* that the necessarily small (e.g. the radius is expected to be about f"' for B - square of the conserved charge IQ'I should not change it's sign under Wick 10"). Thus the wormholes are physically acceptable. rotations. The effects of these wormholos may be summarized in an effective The equation 19.bl can be solved in a similur way to the one in Ref.4. interaction such as a 0 +5 0. where 0 is a composite operator with the The solutions represent that there are wormholes carrying conserved giant baryon numbers la is an appropriate coupling counstant). The operator isospins which stabilize the wormholes. These wormholes are physically 0 annihilates or creates the giant baryon numbers (e.g. B • 1U'°I in a length acceptable only if their isospin charges are bigger than (1/Gf'l "2 ~ lO": Bcale of the order of f*'. This effect would be only observed in the presence uiiiei- this condition their radii are larger than f"'. of black holes which actually break the conservation of the baryon number. We have shown that in the Skyrmc model there are solutions of wormholes Finally we show that there are other wormhole solutions with isospins with baryon pumbors (B =±11. The Skyrme term is neccessary for avoiding but with no baryon numbers. Taking an ansatz such as L = Sltl ISItl has no the collapse of the wormholes. We have also shown wnrmhole solutions with dependence on coordinates of Si), we obtain the equations of motion for a(t) isospins but with no baryon numbers. Because the minimum radii of and Sltl. -257 — -258- physically acceptable wormholes are about a hadronic scalo. ue need some References suppression mechanism of the wormholes. 1. S. Coleman. Nucl. Phys. B 310 (1986) 643; Note added: He recieve a paper" where K. Lee and S. 11. Smirnakis discussed T. Banks. Xucl. Phys. B 309 (19881 493. solutions of wormholes stabilized by an energy density of fermions. alt) 2. S. Hawking. Plus. Lelt. 134 B 119841 403. dependence of the energy density is the same as it's dependence of our energy density of the Skyrme term. 3. J. Preskill. Nucl. Pliys. B 323 119891 141. 4. S. B. Giddings and A. Strominger. Nucl. Phys. B 306 I19B8) 890. 5. S. Hawking. Phys. Lett. 195 B (1987) 337: H. Fukutake. K. Ghoroku and K. Tanaka. Phys. Lett. 222 B 11933] 131. 6. K. Lee. Phys. Rev. Lett. 61 (1988) 263. 7. S. Coleman and K. Lee. Harvard preprint. HLTP-89/A 022 (19891: L. F. Abbott and U. B. Rise. Caltech preprint. CALT-68-1523 (19891: S. Midorikawa. 135 preprint. I.\'S-Rep. 754 (1989): A. iwazaki, NishD preprint. Nisho-3 11989) to appear in Phys. Lett. 8. A. Hosoya and «. Ogura. Phys. Lett. 225 B 119891 117. 9. B. Grinstein. Nucl. Phys. B 321 (19831 439: 10. A. K. Gupta. J. Hughes and J. Preskill, Caltech preprint CALT-68- 1557 11989) 11. T H. R. Skyrme. Pron. Roy. Soc. (London) A 260 11961) 127. Xucl. Phys. 31 11962) 556. 12. A. P. Dalachandran . V. P. Nair. S. G. Rajeev and A. Stern. Phys. Rev. Lett. 49 11982) 1124. 13. J. (less and B. Zumino. Phys. Lett. 37 B 119711 95. 14. E. \Utten. Nud. Phys. B 223 119831422. 433. 15. K. Lee and S. It. Smirnakis, Harvard preprint. HLTP-89/A024. -259- -260- *J£JSJ.'H Hiroshima University 1. Introduction Let us start with explaining why we are going to investigate such a "low- dimensional" version of Einstein gravity. In a canonical approach to general relativ ity, the spatial metric is regarded as dynamical variable. One of the main differences between the (2+l)-dimensional and the (3+l)-dimensional gravity is the fact that (2+l)-Dimensional Gravity with there is no local mode of the dynamical variable in the former case. (This is the the Cosmological Constant* well-known fact that there is no gravitational wave in (2+l)-dimension.)-However, there remain a finite number of global modes which are directly related to the topology of space. This global or topological mode may play an Important role in studying quantum cosmology or quantum gravity. Indeed, when one considers the phenomenon like the Coleman mechanism, the effect of the spatial topology is important. Moreover, the topology of closed 2-dimensional manifolds is well un YOSHIHISA FUJIWARA AND Jmo SODA derstood mathematically. Therefore, the (2+l)-dimensional gravity is a toy model with which we can investigate the effect of the spatial topology. Research Institute for Theoretical Physics Hiroshima University, Taieiara, Hiroshima 725, Japan This 3-dimensional model is recently investigated by Witten as a Chern-Simons theory [1]. Although his treatment of gravity is elegant, his analysis heavily de pends on the dimensionality. In fact the Chern-Simons term does not exist in (3+1 (-dimension. Because our eventual goal is the (3+l)-dimensional gravity, the method which is applicable to any dimension is necessary. For this reason, we ABSTRACT employ a conventional approach — ADM canonical formulation. In this context, Hosoya and Nakao pointed out a significant fact [2]. The dynamics of pure Ein The (2+l)-dimensional Einstein gravity with a cosmological constant is studied stein gravity follows a geodesic motion in the moduli space in the case that the in the ADM canonical formalism. Adopting the York's tune slice, we completely 2-dimensional space is topologically a torus (genus g=l). One of the authors (J.S.) solve the initial-value problem and the time evolution equations with an initial [4] analyzed the matter effects on the geodesic motion using a linearization method spacelike 2-surface being a closed Riemann surface of genus one. The result is and conclude that only the global mode part of energy-momentum tensor contribute that the Teichmuller parameters for the torus follow a geodesic in the TeichmuUer to bending the trajectories in the conformal superspace. On the other side, Mon- space but its motion asymptotically stops due to the presence of the cosmological crief independently analyzed the initial value problem of this system and proved constant. that there is a unique solution of the constraint equations in any genus cases g > 1 [3], f This talk is based on the following work. More details and another aapect which cannot be referred to can be found in it. In this talk, we will investigate the dynamics of (2+l)-dimensional gravity with Y.Fujiwara and J.Soda, Teicnmuf/er Motion of (2+l)-Dinmniontl CraWly with tie Cosmoiogical Cotulmt" preprint RRK 89-30 (1989). a positive cosmological constant and find a new feature in its geodesic motion in —261 — -262- conformal superspace. It is interesting from the point of quantum cosmology to N h K a 2 ai ^-Qr- = ~ 9" { 2.2 YORK'S METHOD 2. ADM Formalism 2.1 ADM ANALYSIS In a series of papers York has developed an elegant way of treating the initial- value problem of general relativity [6]. In this section, we adopt his method in our We briefly review the main framework of the canonical formulation of gravity interested case — (2+1 {-dimensional gravity with a cosmological constant. known as the ADM formalism. (For details the reader should refer to [5], for example.) The initial-value problem is a construction of metric g . and extrinsic curvature Kah on a specified spacelike manifold so that they satisfy the so-called Hamiltonian We slice the entire spacetime into a one-parameter family of spacelike hypersur- and momentum constraints: faces, each of which is a t—constant "space". Our dynamical variable is the spatial metric gai defined on this hypersurface. The conjugate variable to it is essentially lb i the extrinsic curvature K"b of this hypersurface. As is always the case with any K^K' -K' -R + 2A = 0, (2.3) gauge theory, the canonical variables in gravity are also subject to the constraints (EqB.(2.3) and (2,4)). So, first of all, we must prepare a set of canonical variables i i Vt[K' -g' K) = a. (2.4) b gat, K° that satisfy these constraints on an initial spacelike hypersurface. How to set up such initial-value data will be discussed in the next section. Secondly, we Here R is the spatial scalar curvature and A is the cosmological constant, specify a lapse function N and a shift vector N° to solve the Hamilton equations. These constraints serve to eliminate redundant dynamical degrees of freedom After solving the Hamilton equations, the entire spacetime geometry will emerge from the metric and the extrinsic curvature. As is well known, the true number of as dynamical degrees of freedom is "two" each for the metric and extrinsic curvature 1 2 h h di = -(JVdt) + g^dx" + N'dt)(dx + N dl). (2.1) in the 4-dimensional spacetime and "zero" for each in the 3-dimensional spac This means that there is no gravitational wave mode in the 3-dimensioual case. The Hamilton equations or the time evolution equations are given by However, there remain nonlocal modes or global and topological modes, which we i r % = -2A ^i + V„JVb + V4JV0 (2.2a) would like to investigate in this paper. -263- -264- (1) First we specify such a time slice as r = trK ~ K=comtant on each spatial On the other hand, the Hamiltonian constraint hypersurface. In other words, the trace part of Kaf> represents time. This time slice 2 is called York's time slice. K^ - K - R + 2A = 0 (2.9) (2) Thus r = K is constant on the initial spacelike manifold so that the mo reduces to a quasi-linear differential equation which determines the conformal factor mentum constraints can be cast into the form: -A* + R = Me"* - f |r 2 - 2\J e*, (2.10) ,{K*-\?K): (2.5) 0 where A = J^V,,1?., Af = E .£"' and JZ is the scalar curvature for j . Here the If we define E"» 3 K"h - ^K, eq.(2.5) means V,S°4 = 0. By construction, oJ resulting scalar curvature R is given by E°' is traceless. E0' is therefore transverse and traceless, or TT-parf of extrinsic curvature. 7. = e-*(R-M). (2.11) Now take an arbitrary metric compatible with the spacelike manifold M. It can be shown that we can always construct a transverse and traceless tensor, E°\ on Note that all the derivatives and the coefficients in eq.(2.10) are given by our ai oi al ok this (Af, jol). K can be written as Jf"' = E +|s T0, where T0 is an appropriate starting gab and E . Thus our problem to construct the initial-value data is now initial time. reduced to solving eq.(2.10) for the conformal factor (3) Our starting g . and Kai may not in general satisfy the Hamiltonian con In this way we arrive at the initial-value g . and A that satisfy the constraints. straint. We are going to construct j . and TC from ff and K^ so that they nJ We have to say one word about an important point of York's method. What can satisfy the constraints by taking advantage of conformal transformation. Consider happen, if we were to start with different metric and extrinsic curvatures, j^t, K"° (Af, jT ) whose metric is conformally related to the starting metric j in (M, g ), nJ 2 6 ?„» = <= V ( - ) 5al and H° .This means that we have only to consider a conformally equivalent class of gai in solving the initial-value problem. Define 2° = e~ME0'. Then the momentum constraint V,S°' = 0 (2.7) 3. Solving the Constraints on a Torus In this section we shall apply York's method to the cases that the space manifold automatically holds as can be easily proven. Here V( is the covariant derivative Af is a 2-dimensional compact manifold without boundary. It is a well-known fact with respect to s„ - Note that k that such manifolds are classified into its topological equivalence classes. We can classify them by its gemu g. Main attention is focused on the case — s=l or a 20 (2.8) torus. In fact this is the one of the two cases for which the initial-value problem can be explicitly solved. (The other case is g = 0 or a sphere.) -265- -266- First of all, we borrow some known facts in mathematicians' works [7]. The Letting 0 = 2m/ JT$ — 4A, the initial-value data on a torus are space of second rank TT-tensor E . or "holomorphic quadratic diiFerential'' is lo s 6 -6 cally isomorphic to R for g=\ and R ' for g > 2. They further observe that (3.5) »--'(! !)• for E^ should be constant everywhere on a torus (ff=l) while for Eot is simply zero for a sphere (ff=0). For a higher genus case, Eai must have ig — 4 zero-points 2 And from M = 2m in eq.(3.2) and using the traceless condition of Eol, somewhere on M. E„ = mcosfl, E„ = —mcosS Now it is easy to solve the constraints. We construct a torus by identifying the (3.6) two pairs of opposite sides of a square whose coordinates (z, y) are as shown in {: Fig.l. The starting metric can be taken as where 9 is a newly introduced parameter (0 < 0 < 2ir). We can see that the two parameters m and 9 represent a point in R2 which is locally isomorphic to the s°>=(! I) o,b = z,y (3.1) space of TT-tensor E . on a torus. In higher genus cases g > 2, we cannot easily solve eq.(2.10) explicitly, for E , on a torus. necessarily changes its value on M so that the conformal factor cannot be constant. The scalar curvature R for this metric is 0 and 4. Time Evolution of Torus M = E E»» ol (3.2) = 2(EL + E2,). From now on, it is convenient to write the time evolution equations for ai (gai, Eo6), not for (sa4, K ), because we have adopted York's method. Indeed Because E . is constant on a torus, write M = 2m2 where a parameter m is a Eoi, the TT-part of extrinsic curvature plays an essential role in our analysis. We constant. Eq.(2.10) now becomes can rearrange the time evolution eqs.(2.2a) and (2.2b) after some calculation as: J A^ + 2mV* - Qr0 - 2A) t* = 0. (3.3) %* = - Brit - MTC., + VaNt + VtN. (4.1a) A trivial solution of tbJB equation is a constant ^, -(^-^V^W-r^E,,. (416) i 2m (3.4) rs '-4A V 0 The relation between the York's time r and the parameter ( is deduced from eqs.(4.1a) and (4.1b) as: It can be shown In general that if eq.(2.10) has a solution, it is unique. Re cently Moncrief [3] has given a rigoroua proof of it using the properties of elliptical £ = -AN + (£aSE" + lr' -2AJN + N'Var. (4.2) differential equation. The above trivial one is therefore the unique solution. Note that r2 - 4A must be positive. Because we take r = constant on each spacelike hypersurface, the last term drops. -267- -268- First we choose the lapse function and the shift vector so that we can definitely Let us focus on the particular case 0 = 0, for it has an essential feature of the describe the time evolution of a torus. We set the shift vector equal to zero. This more general case 0^0. The resulting 3-geometry is choice implies that our initial coordinate frame is the comoving frame. Secondly the lapse function is set equal to one' . Note that all the information on the geometry (4.7) of the torus is contained in the 2-metric only, because we fix the boundary (see «--*+>[{£$ "+{3$ « Fig.l). An interesting feature of this result is the ratio of the lengths of the two cycles Because of our lapse function and shift vector choice, the time evolution equa (each along x and y), tions can be cast into simple forms, sinh at , cosh at tanhar . . sinh aig' cosh a<0 tanh at ' (4.3a) di ' Recall that t goes from t„ < 0 to 0. In the limit t —> 0, the above ratio vanishes. ^ = -2J«'S„SU (4.36) Namely, the x-cycle collapses. On the other hand, we can reverse the direction of time ( so that t goes from tQ < 0 to —oo. As t —* —oo, the above ratio asymp with totically tends to a constant — l/tanhaca, while the overall scale factor increases J exponentially. £ = r -4A. (4.4) Integrating eq.(4.4), we obtain 3. Summary „ cosh2a* 1 , r-2a r = -2a . . . . or ( = — In—-r-, 4.5 smh2at 4a r + 2a What could be found about the global or topological mode in the (2+1)- dimenBional gravity, mentioned in the introduction? What is the topological mode where A is henceforth supposed to be positive and a = vK. As r runs from T„ to of torus? Let us explain it by using the concept of conforms! superspace. oo, t goes from („ < 0 (corresponding to r ) to 0. Notice that the range of t is g The concept of conforms] superspace stems from the usual superspace. The t<0. canonical approach to the (2+l)-dimensional gravity can be understood as a de Then the solution of eqs.(4.3) that satisfy the initial condition (2.19) and (2.20) formation theory of 2-dimensionat Riemannian manifold M. Let Riem(M) be the turns out to be space of Riemannian metrics on M and Diff(M) the group of diffeomorphism on M, As is well known, a formal definition of superspace is then given by 9,a t •'(::)(T;)(::) «••> _ fl»«"(Af) where ah = sinh at/ sinh atQ, eh 3 cosh at/ cosh a<0, J = sia J and c = cos { The deformation process may be viewed as a trajectory in the superspace whose f The choice N = 1 is compatible only with a sphere and a torus. For we can see from eq.(3.2) point is spatial 2-geometry. As is well known in geometrodynamics, however, the that N must be a function of spatial position, not a constant, since E«i necessarily changes its value on A/ in a higher genus case (g > 2). 2-geometry is also a carrier of information about time. This extra variable can be -269- -270- identified with conformal factor. Then what we would like to know is a deformation y of conformally equivalent Riemannian manifold. This is the so called "conformal superspace" which is defined as superspace modulo conformal mapping Conf(M), i » S ) = . *"?M {M Di//(M)-Con/(Af)' a G g^t) u In section 2.2 we have seen that taking York's time slice is equivalent to consider ing "conformal superspace" — superspace modulo conformal mapping. Diff(M), however, should be limited to Diff0{M) which is isotopic to the identity since _ > I- the spatial coordinate is (at most) continuously deformed in time in our canonical o " l approach. Then the remaining conformal superspace is the so-called Teichmuller space. Thus we would like to know the motion in Teichmuller space or Teichmuller Fig.l The torus is made by the identification of the two pairs of the opposite sides deformation. This is the global or topological mode in the (2+l)-dimensional grav of a square, 0 Mathematicians know that locally ' a point (i; = 0) REFERENCES Teichmuller space ~ I R2 1,9 = 1) (5.1) R6'"8 (g > 2). 1. E. Witten, Nucf. Pfiys. B311 (1968) 46. a denotes "homeomoiphic to". So the mode is directly related to the spatial 2. A. Hosoya and K. Nakao, "(2+l)-Dimensioaal Pure Gravity for an Arbitrary topology. The ratio between the two cycle lengths o{ the torus is the one, as Closed Initial Surfaces" preprint RRK 89-11 (1989). explained in the previous section, and the other \a the parameter that represents a 3. V. Moncrief, Ann. Phys. (N.Y.) 16T (1986) 118. "twist" of the torus. 4. J. Soda, included in this proceedings. In [2], it v '1 shown that without the cosmological constant the trajectory in 5. C. Misner, K. Thome and J. Wheeler, "Gravitation", Chapter 21 (Freeman, the Teichmulle. t ace necessarily goes into a singularity — the torus must collapBe ; San Francisco, 1973). or explodes, The presence of the cosmological constant causes a new effect as was 6. See, for example, J.W.York Jr., J. Math. Phys. 14 (1973) 456. shown in the previous section. That is, we have explicitly shown in torus case that the cosmological constant causes an asymptotic convergence effect of the geodesic 7. H. M. Farltas and I. Kra, "ffiemann Surfaces" (Springer-Verlag, New York- motion. That is, the Teichmuller deformation stops asymptotically in time. This Heidelbergg-Berlin, 1980). frozen mechanism of the dynamical degrees of freedom is remarkable. This is similar to "no-hair theorem". -271- -272- Linearized Analysis of (2+l)-dimensioii!U Einstein-Maxwell Theory respectively. The stroke indicates the covarianl derivative defined by the spatial metric />... Here we discard the surface term, because we shall concentrate on Jiro SODA t (2+l)-dimensional space-time M = R x E where S is a compact orientable two Research Institute for Theoretical Physics manifold. The canonical conjugate momentum ir,J to n.. is given by ir" = ,/h[K» - h'>K). (4) A consistent quantisation of gravity is one of the big problems in physics. When The ADM action for Einstein's equations takes the form1'1 we meet a difficult problem, usually we attempt to reduce the problem to simpler one. Although we would like to quantize (3+i)-dimensional gravity, following the S = f^xi^h.. -NH- N.H*} (5) above principle we shall study (2+l)-dimensional gravity. (2+I)-dimensional grav ity has a special property that the local degrees of freedom do not exist. Oweing to this speciality, (2+l)-dimensional gravity can be reformulated as a Chern-Simons B* = -2»|'t' (?) theory which is an exactly solvable system. From this point of view, however, it is not yet clear how to interpret the result physically. So we follow the conventional with TT = Trj = —-^/fiK*. Note that the Hamiltonian constraint can be rewritten as method. ADM method, to get some implications about (3+l)-dimensional gravity. Given a Einstcin-Hilbert action H = G yiP**' - JhRP\ (8) G«« = 5*"*(*«*J' + Vi»-aW (9) s = yV-<7;W*, (i) This tensor, Cr. ,,,, is the so called supermetric on superspace. This metric has there is a standard prescription for obtaining a Hamiltonian formulation. The signature ( — ,+,+) in 3-dimensional metric space Riem(E). This is different from canonical theory begins with the following decomposition of the metric tensor superspace whose naieve dimension is one. The evolution equations obtained from (2.5) are di' = gmAz*dz* (2) = -;V!di3 + h (dx> + N'di)(dz> + N'dt) dh.. 2N where n,v range over 0,1,2 and i,j range over 1,2, Using this (2+1) decomposition 5 NW v of the metric, we obtain iir = ^'''A, -* > - ^'M - *^)+^ - ''"'i*> i t m H , + (AfV>)|t-ir .Vr ;-ir*'yV|V (11) S = J Ny/h{KijK'-K + ll ) (3) where A'. = rv(h... - N. - N,.) is the extrinsic curvature and A' = A' W> is Together with the constraint equations its trace. Rm and ft'" denote the three and two dimensional scalar curvatures, H = 0 (12) -273- -274- W = 0 (13) ^-fi*- into the action eq.(15) in the phase space. Due to the special gauge .'V = -V((J, I lie final form of the action becomes we obtain Einstein's equations Apv = 0. Equations (12) and (13) are the Gauss- Coddazzi equations giving necessary and sufficient conditions for the embedding of hypersurface with second fundamental form K-- in a space-time satisfying Einstein equations. The momentum constraints (13) have clear meaning that it is a generator where of space diffeomorphism, Dif[(T). The Wheeler's superspace can be defined by J (i'J) Riem[Z)/Diff(E) (14) N = N/2v. The complexity of the Hamiltonian constraint (12) is the origin of difficulty for Here reducing the phase space to the physical space. 2 ) ik j(«)(« =fd x^h^f a) in (2+1 (-dimensions implies that the space-time is locally flat. Globally, however, K* = I>(a/ "72* (16) (») there are "topological degrees of freedom" to consider. with v = / tPxy/h. The deformation of h., is represented as Let us start to analyze the pure linearized gravity. For the moment, we simply assume that the deviation, «,,„, of the actual space-time metric ^E^UV*"- (17> (°) (0) This equation defines the Teichmuler parameters p and the corresponding Bel from the flat metric x\ILI, is "small". We mean by "linearized gravity" that approxi trami differentials /iL.,. Substitute the expansion eq.(16) and eq.(17) for K'> and mation to general relativity which is obtained by substituting equation (21) for jf„„ -275- -276- in the Einstein-Hilbeit action and retaining only the terms quadratic in h^,,. The ihe well'known action result is given by 2 3 5 = f **{*&& - h7*k + i{Aj?,)]} (29) s = /d *(rrr2„-rjT;A), (22) In the topologically trivial space nothing happens. As we are considering the torus where ?CIU, = 1/2(A,„,„ + hSVJ, - h^,). This action is also rewritten as case, fif? and TT^T are spatially constants. So we get S = jfJKyK* - K> + riitrt.,. - r^. + n/2(Au, - kik.k)l (23) S= lit jSx^h^-Tjf^ 30 where K{. = Toj., n = A00 and N{ = Aw. To cast this action into first order form, * ( » let us define the momentum n'> = Kif> - K*'. (24) Note that the Weil-Petersson metric g"$ does not depend q" in contrast to the full gravity. This fact is understood as follows. As we regard the deviation from Then, we get the background metric as small, we are on the tangent space at some point in the conformal superspace. As a consequence the geodesic motion is straight. It is i , s./j -{-«Aw-[^- +rwrwrrtttriJ(+2Ar^+n/2(*UiJ(-A(M)]}. natural, because any gendesic motion is locally straight. At this point we would (25) like to emphasize that the clear-cut result for the case of g = 1 heavily depends r Our next task is to solve the constraint equations on the constancy of Aj . We can not expect the geodesic motion in the conformal superspace in the case of g > 2. n'i = 0 (26) We are now in a position to discuss the effects of matter fields on geodesic ft.. .. -A.. ., =0. motion. Our action to consider is For this purpose we use the following decomposition S = J'iPxJgR-i Jd'xJgg^'F^. (31) ir(> = *SL + (jH')i' + l/2i/«ir w (27) To analyze this system, we assume that the Maxwell fields are sufficiently small so A.. = A^ + (£H')y + l/2r/y/., that quartic term can be negligible. Then full action reduces to J k Md T where (LV)'> = 8'V + 0*V' - t)'idkV . Hew Ajf rr represent transverse- tiaceless parts of A., and IT1'. The general solution of constraint equations a;» f ir' = i« -W(T/A) + r,«T j (32) (28) A = A?T - l/2r, A + ^.^ + t^e-* T 2 + T + A r fy y + »W - J + "/ (V* - \w* oo> 2 .; .;> - [iPxl**. , + where ^ is an arbitrary function. Inserting these eq.(28) to the action, we obtain 4~5(' »i + 5''%) W] -277- -278- The small perturbation term can be written as surface with genus g — 2, pinching some cycle produces a degenerate Riemanii surface which is the boundary of the superspace. Although we can not go through h.T^hTTTrr + hffl + hlT? (33) this point classically, ...re may be a chance to surmount this pinching point as a quantum mechanical tunneling process. If this is possible, topology changing Solving the constraint equations, we can obtain the longitudinal and the trace parts effects can be formulated on the universal moduli space which has previously been which are represented by matter fields. These are, however, higher order effects studied in superstring theory. At this point matter fields must play an important which we disregard in our approximation scheme. Finally retaining the relevant role. Of course we should study (2-t-l)-dimensional gravity in the case of g > 2 part only, we obtain before challenging this big problem. Application of our method to (3-H)-dimensional gravity is interesting and W^T\ S = fdt /W-W - ^4r + tractable in the case of simple topology. We are also planning to study the Einstein- (34) f g°> Maxwell theory in the case of special initial data. That will be helpful to understand our results. ! where Fa = J d xuj*T... This eq.(34) is our main result. The geodesic motion in the conformal superspace is deviated by the transverse-traceless parts of the energy REFERENCES momentum tensur.i.j.its global mode part. Note that the final formula need not 1. R.Arnowitt, S.Deser and C.W.Misner, in "GRAVITATION" ed. by L.Witten, assume specific matter fields. The reason why we concentrate on the Maxwell Wiley, New York-London (1962). fields is the existence of global modes of Maxwell fields on torus. The interaction between the global mode part of gravity and that of Maxwell fields are manifest in 2. A.Hosoya and K.Nakao, RRK 89-11; RRK 89-16. our formula eq.(34). 3. R.Ainowitt and S.Deser, Phys. Rev. 113 (1959) 745. We have succeeded to get quantitative understanding for (2+1) -dimensional 4. A.Hosoya and J.Soda, RRK 89-13 (1989). gravity in the case of linearized theory. Although analysis for full gravity is too dif 5. R.P.Geroch, J. Math. Phys. 8 (1967) 782. ficult to make any quantitative statement, we expect that the qualitative features 6. D.Friedan and S.Shenker, Phys. Lett. B175 (1986) 287. remain true for this case. Furthermore we might speculate that the global defor mation of space is dictated by the global part of energy-momentum tensor in any dimensions. This is the motivation of our previous work1'1 in which we analyzed the Abelian gauge theory in topological^ non-trivial space. In this case, however, the global dynamics may not be the geodesic motion in the conformal superspace .due to the potential term. Now we understand the qualitative behavior of (2+l)-dimensional gravity cou pled with the matter fields in the case of g = 1. Next we would like to explain how to relate our analysis to the topology changing phenomena. Imagine the Riemann -279- -280- *RZnTP Hiroshima University December 1989 BRS Current and Related Anomalies in Two-dimensional Gravity and String Theories HIROSHI SUZUKI Research Institute for Theoretical Physics Hiroshima University, lakehara, Hiroshima 725, Japan The subject of this talk is covered by the following references: [1] K. Fujikawa, T. Inagaki and H. Suzuki, Phys. Lett. 213B (1988) 279. [2] K. Fujikawa, N. Nakazawa and H. Suzuki, Phys. Lett. 221B (1989) 289. [3] K. Fujikawa, T. Inagaki and H. Suzuki, Hiroshima report, RRK 89-17 (to appear in Nucl. Phys. B). [4] H. Suzuki, Mod. Phys. Lett. A4 (1989) 2085. -281- The BRST quantization of the CSGT discussed so far [4,11,12] is based on the conven tional Faddeev-Popov gauge-fixing for the Yang-Mills gauge symmetry. We would like to BRST Quantization Btress here that the present approach makes it possible to reveal the structure of full local symmetry involved in the theory. In our previous paper [13], the generalized Hamilton!an of formalism was applied to obtain gauge-fixed actions for the TYMT [2] and the TSM [3] Chem-Simons Gauge Theory from the pure Pontrjagin action and the "Riemann-surface fl-term". In quantizing the CSGT, we encounter a new feature that the theory involves the second-class constraints as well as the first-class ones. For BRST quantization the presence of the second-class Hiroshi Imai constraints needs a new device. Ordinarily, these are handled by working with the Dirac Department of Physics, Niigata University, brackets defined in terms of the canonical measure on a corresponding hypersurface. But Niigata 950-21, Japan this procedure prevents the use of the canonical commutation relations in the extended phase space [8] where the Lagrange multiplies and the ghostB are included as ynamical variables. To overcome the difficulty, Batalin and Fradkin (BF) has proposed a further Abstract extension [9,10] of the extended phase space within the spirit of the generalized Hamil- tonian approach; the corresponding new dynamical variables should be introduced for We quantize non-abellan gauge theory with only a Chern-Simons term in three second-class constraints to become effectively first-class. We shall intensively use thifl dimensions by using the generalized Hamlllonian formalism of Batalin and Eradkin idea to quantize the CSGT. Introduction of such dynamical variables leads to a new fea for irreducible first- and second- class constrained systems, and derive a covariant ture: The CSGT possesses not only the Yang-Mills gauge symmetry but also another action (or the theory which is invariant under the off-shell nilpotent BRST trans formation. symmetry, which can be used to define supersymmetry similar to the one in TYMT [2] in four dimensions. The analysis in this paper follows the Ref. [14]. 1 Introduction 2 Second Class Constraints and the BRST Charge Recently a new land at applications of quantum field theory has been suggested by Atiyah [1] in the study of three- and four-dimensional manifolds. Witten has pushed forward Let us begin by considering the pure Chern-Simons action in a three dimensional Euclidean this idea by constructing actions of topological field theories, the topological Yang-Mills space, theory (TYMT) [2] in four dimensions and the topological sigma models (TSM) [3] in S, = ±J -283- -284- As usual, the primary constraints (3) lead to the secondary constraints, the ghosts (C„ P.) and their conjugates B\ [V, C'). Now we look lor the operator Qm'" n and Q™ t wluch satisfy the relations = D,n{ + ^-E" d,A, = 0, (5) {Qf.Qf } = »• (2°) where the classical Hamiltonian is given by They are found to be 2 Qmi„ 2 Ho = Jd x[-A0 • {Prf + £>M,)]. (6) = /d x(C, •<)>• +P,B'), (21) On the other hand, the constraints (4) lead no secondary constraints. Qf = l It is worth noting thai we do not set Aa to zero as the case of the TYMT. It is the covariance here that requires for the constraints (8) to be treated on the same footing as = -jfi + Y*^, (24) (7) especially (9). Because of the same reason we have added the last term in (7). The algebraic structure for the constraints is whose Poisson brackets are , 1 1 (25) {*"W,<>'(y)} = rVWi (x-y)1 (ii) {fn-7} = «*„ (mAy))=i', (is) W(x), fiM(y)} = /"V (x)f J(x - y), (13) = f<„, (26) W'(*>. #*(y» = /" V(x)i3(x - y), (14) {#"(x),*M(y)}=0, (15) we can define the BRST charge for ^' as {r(x),#*(y)) = -^"V'iJ(x-y), (16) , {ir(x),tf„} = 0, (17) = /d'x[C,-(tt' + ^« '*))+P,.fl']. (27) {ir°(x),//„} = (], (18) (ir'(x),ffo} = 0. (19) It is easy to verify that the second-class constraints -285- -286- For the first-class part of the algebra for the constraints, eqs. (11-13), the BRST We choose the gauge-fixing functional to be charge is * = l(?x[Cx + C°Xa + C X, + ^ • W - P° • (JV„ - i»,*') 2 0 Q7 = fd z[C- Q + BV + B V0\, (30) X = d,A' - d where No and B° are the Lagrange multiplier-momentum pair for the constraints tf>°. Xo = (£>' -*'x)(JV, - DiA0) + lN-A0)xN0 + ^Ba, Since the theory at hand contains both first- and second-class constraints, the full 1 BRST charge is given as the sum Ql + Q7 supplemented by the terms / tPxl—P • (Ci x X. = D'{F„-D^,) + {N-Aa)».Ni + ^Bi, (34) C) + kf%it€i3^i • {$j xC)], where a and 0 are constants with appropriate mass dimensions. Since we have defined new an 3 set of first-class constraints T, 4>° d T', we have to look for the classical Hanoi! tonian Q = Qi + Q + |rf x[-'P'(C xC) + A ^ . $ ] 3 l e i ( jXC) p according to the argument In the Ref. [9,10]. As a result, the classical Hamiltonian Es given by 3 = J d x [C-( rsM+i-e"^*^, (32) 1 i a t I = Jd x[x-Ai + T -Aa + B-N + B'-Nlt+V-C + C-i> we obtain the following algebra for constraints, +P» • 4 + £" • P„ + ij* • 6] - JW (37) {r(x),T1(y)} = /'fcT'(x) S1(x-y), Redefining the gauge and vector fields as .4,, = {—N,Ai — $,;), $,, s (V4Q,<&,) and elim 1 inating TT", P, V, T^ P" and rj' by using the equations of motion, we find for instance W">W,*w(y)} = 0, NQ = V^i Ni = T^ + 7?o*tt where V„ and F„¥ denotes the co variant derivative and {r(x),*w(y)} = /"V'(x),S'(x-y), the field strength with respect to the redefining gauge field A». {T"(x),7*(y)} = /r,iT°(x)SJ(x - y), As a result, one obtains a covariantly gauge-fixed action ir-(x),0w(y)} = O, S =/ o?* [^<'4. • (M, + \A. x A,) {r"(x),7*'(y)} = 01 (33) M which shows the constraints T, ijP and V are ot first-class. -B„ • {(D„ + *„x)(.F"" - V"'i") + |B } •M?» • {(P, + *„x)(JP"' * C - (Z>"$») x C + VC) 3 Covariantly Gauge-Fixed Action -(C„ - (B„ + >t„x)C) x (*»• - Z>" -287- -288- Here 5 is anti-commuting with Grassmatin odd variables. One can easily verify that the The action (41) possesses still local gauge symmetry. (This situation is similar the one transformation (39) satisfies ofT-shell nilpatency. It is worth noting that in terms of the of the two step gauge-fixing case in the TYMT.) This symmetry can be fixed by adding gauge field defined by A,, S -4,,+*^, the BRST transformation for the gauge fieldbecome s s &Ap = CM — DpC, which is the same as the one for the gauge field in the TYMT in four S, = J = ^ * k p f M» = -V,C, 6 C = \C x C, This is invariant under supersymmetric transformation [8] E.S. Fradkin and G.A. Vilkovlsky, Phys. Lett. 55B, (1975) 224; I.A. Batalin and G.A. Vilkovisky, Phys. Lett. G9B, (1977) 309. b'A* = 0, [9] I.A. Batalin and E.S. Fradkin, Nud. Phys. B279, (1987) 514. *'*,.= C,„ t'C, = 0, (42) S'C, = -fl„, i'B, = 0. [10] I.A. Batalin and E.S. Fradkin, Phys. Lett. 180B, (1986) 157. -289- -290- [11] E. Guadagnini, M. Martellini and M. Mintchev, Phys. Lett. 227B, (1989) 111. [12] D. Birmingham, M. Rakowski and G. Thompson, ICTP preprint IC/88/387; D. Birmingham and M. Rakowski, ICTP preprint IC/89/38. [13] Y. Igarashi, H. Imai, S. Kitakado and H. So, Phys. Lett. 227B, (1989) 239. [14] Y. Igarashi, H. Imai, S. Kitakado, J. Kubo and H. So, Niigata Univ. and Kanazawa Univ. preprint NIIG-DP-89-3, HPICK 136/89. [IS] L. Alvarez-Gaume, J.M.F. Labastida and A.V. Ramallo, CEH.N preprint, CERN-TH.5480/89. [16] S. Ouvry, R. Stora and P. van Baal, Phys. Lett. 220B (1989) 159. [17] F. Delduc, F. Gieres and S.P. Sorella, Phys. Lett. 225B (1989) 367. -291- 1. Introduction The Chern-Simons term in three-dimensional gauge theories has very interest Super Wess-Zumino-Witten Models ing topological properties and possesses rich implications in diverse phenomena of from particle physics and condenced matter physics. In particular, Witten discovered Super Chern-Simons Theories a close connection [1] of the Chern-Simons theries with the Wess-Zumino- Witten (WZW) models in two dimensions. The connection of these theories arises in two ways. When the Chern-Simons theories are quantized on a three-dimensional manifold E x R with E being a compact two-dimensional manifold without boundaries, one obtains the space of conformal blocks of the WZW models as NORISUKE SAKAI the physical Hilbert space. On the other hand, when £ has boundaries, one Department of Physio, Tokyo Institute of Technology obtains the WZW actions as effective actions on the boundaries. The physical Oh-okayama, Megaro, Toijo 152, Japan Hilbert sapce is an infinite dimensional representation of the current algebras. Since then many authors studied the Chern-Simons theories, especially in and connection with conformal field tl.ec.ies [2]. Conformal field theories are very important to study vacuum configurations of string theories as well as critical phenomena of two-dimensional statistical physics. We can hope that the Chern- YosHiAKr TANII Simons theories may provide a new powerful setup to study superstring theories. Physics Department, Saiiama University Therefore we are led to consider supersyrnmetric Chern-Simons theories and their Urawa, Saitama 338, Japan relation to supersymmetric conformal field theories in two dimensions. In the following we will discuss a connection of the super Chern-Simons the ories with the super WZW models. For more details of the discussions, see the original paper [3]. We will consider the super Chern-Simons theories on a disk x Abstract R, which is a typical example of three-manifolds £ x R with E being manifolds with boundaries. (In ref.[3| we also studied the case of a torus x [0,1].) We will The two-dimensional super Wess-Zumino-Witten models are obtained fiom obtain the action of the super WZW models on a two-dimensional boundary. By the super Chern-Simons theories Tor a three-dimensional manifold with a topol exploiting the freedom of adding total derivative terms in the three-dimensional ogy of a disk x R, By appropriately choosing total derivative terms in the three- action, we can always maintain the N = \ supersymmetry in the two-dimensional dimensional action, the N = \ superaymmetry can be maintained on the bound surface. For particular values of parameters it possesses also the /V = 1 super- ary. The N = 1 supersymmetry on the boundary can be realized by choosing symmetry. particular values for parameters of the total derivative terms. -293- -294- 2. Bosonic Chern-Simons theory With this boundary condition we obtain the partition function VA Before discussing the supersymmetric case let us first review the bosonic / —^expfiSBlA]) theories [2]. The bosonic Chern-Simons theory has the action 2 3 rvl u r <-> 3 = J -^ «(F_3) exp(-.| try d i£'M,Mi), where Vg is the volume of the gauge transformations satisfying 17 = 1 on the 5B = -| tr fd'x ^'{ApduA, + 1A„A„A,), (2.1) boundary and the field strength F-2 — 0 is the constraint. It is convenient to make a change of the variables where Ap is a Lie algebra valued gauge potential. Under the gauge transformation 3 l Ap —' A n = g~(d + A^g, the action changes as l x A- = U~d-U, Ai=Ai + U- diU, (2.4) 3 i under which the functional measure and the delta function change as SB[Ai] - SB[A,} = -| tr Jd x e"'"[-\g-dliga-'eilgg-'dfg - B^gg'^A,)]. (2.2) VAi = VU VAi det-CL, 6{F.2) = i{A,) (det0_)-'. (2.5) For three-dimensional manifolds without boundaries, the second term on the right hand side vanishes while the first term is proportional to a winding number Here D- is a covariant derivative in the adjoint representation. The determinant of a homotopy class of the gauge transformation. When 2nfi is an integer the factors in eq.(2.5) cancel each other in eq.(2.3). By integrating over Ai we obtain latter has no effects in exp(ti5fa) either. On the other hand, for manifolds with boundaries, these two terms do not vanish and therefore the action is invariant J V° (2.6) only under gauge transformations satisfying g = 1 on the boundaries. Another = yn<2>Crexp(iSwzw[tf]), aspect of the action (2.1) is that it is independent of metric and is invariant under the general coordinate transformation. where V^U denotes the integration over the boundary values of U and Let us consider the quantization of this action on a manifold with a tolpology l l x l Swzwltf] = - £ tr /d'z {U~a+U U~d.U + 2A-d+UU~ ) of a disk x R. We use the coordinates x° and x (or the light-cone coordinates \ (2.7) 1 2 i* = 4r(z° ± 2 )) for the directions on the boundaries and z for the direction 1 i l l -|tr I d xU-d,U[U-'a+U, U-d.U] perpendicular to them. We shall regard x+ as "lime" and apply the canonical quantization for constrained systems. We can easily see that the field .-1+ plays is the WZW action with a Rat Minkowski metric. Since the integrand depends the role of the Lagrange multiplier. The variation of the action with respect to only on the field U on the boudary, the integral of U inside the three-manifold has A+ gives a constraint on other fields Aj (i = -,2). To avoid singular boundary cancelled Va. The partition function (2.6) has a form of that of a two-dimensional contributions in the constraint equation, we impose a boundary condition on the field theory. Thus we have obtained the WZW action as an effective action on multiplier: A+ = 0. the boundary starting from the Chern-Simons theory. -295- -296- 3. Super Chern-Simons theory of component fields which are obtained by a nonlinear field redefinition The supersymmetric Chern-Simons theory in three dimensions is described K = A*- 2*7(1*, by a Lie algebra vaiued spinor superfield [4] 3 fl X'a = Xa + (7"S(.A)a + [A„, (7"A)„] + [A, X„) - |[A7„A. (/A).], ( - > ^a = \a + ABa + All{i"S)a + \xJ», (3-1) 6^ = 8^ + 1^,*], °Y„ = [x'0,*]. where A^ and x denote a gauge vector potential and its spinor superpartner, and A and A denote an auxiliary scalar and its spuior superpartner. Here we consider For three-dimensional manifolds without boundaries, the supersymmetric a flat metric in three dimensions. The N = 1 supersymmetry transformation is Chern-Simons action has been worked out [4] realized by differential operators Qa acting on the superfield 3 2 3, 3) S„„ . = J tr f d xd 0 {9"D^D^h + §{*", ¥}D^ «*„ = £"^ *a, QL = JJ - (7'9),d„, (3.2) b 0 ? where £ is a constant Grassmann parameter and superscript (3) denotes three dimensions. 3 = & tr | d 1 [ -2£"'"(J4;fl„J4'(, + |^U'p) " |* Y The gauge transformation parameter is given by a Lie algebra valued scalar + total derivative terms]. superfield $ with scalar components 0 and F, and a spinor component ^ This action is invariant under the supertransformation (3.2) and the infinitesimal * = 0 + (ty + HJSF. (3.3) gauge transformation (3.5) for manifolds without boundaries. We can assume not Introducing covarinant derivatives which anticommute with Q),', the gauge that x' is scalar under the general coordinate transformation but transforms transformation of the spinor superfield is given by as a density with an appropriate weight. Then the action is invariant under the general coordinate transformation without introducing metric. t-e-'lrf' + ty, D<,3) = ^ + (l"S)ad„. (3.4) For manifolds with boundaries, the action (3.7) is not really invariant under The infinitesimal form of the gauge transformation reads these transformations but is transformed into a total derivative. At this point, we note that the total derivative terms in the action affect the dynamics if there S9a = D^i + [*„,*], are boundaries. Therefore we have to retain all possible total derivative terms, including those not appearing in eq.(3.7), consistent with symmetry requirements SA=F + [A, 6A„ = drf + [A„, Sxa = -[-l"9^)a - [A,, (7"0)(,1 + [A., f | + Ix., 4] - [A, „]• two generators Q\' and Q\' corresponding to parameters £j and Ei respectively. We can simplify the gauge transformation property by introducing another set Since repeated applications of two supertransformations involve translations in all -297- -298- 2 three directions, translation in x (the direction perpendicular to the boundaries) on the *a superfield. The supersymmetry we have kept here may be called (0,1) cannot be avoided: supersymmetry. Similarly, one can keep another supertransformation along e\ instead, which may be called (1,0) supersymmetry. ( (3) {Qa>, <3 '} = 2 (7°5o +-r% + 7%) J. (3.8) We would like to keep all terms, which are not total derivative, of the N = 1 supersymmetric Chern-Simons action (3.7) for manifolds without bound Therefore we cannot demand invariant under both two supertransformations aries. The (0,1) supersymmetric action which coincides with these terms can be represented by Qf' and Q$ , if there are boundaries. On the other hand, if uniquely determined apart from total derivative terms. It can be given as a (0,1) we keep only one of these supertransformations, say Q\' with a parameter £:, superspace integral we observe that the algebra is isomorphic to the TV = 1 supersymmetry in two 1 l 3 2) dimensions Scs ' = - £ 'j d i^(-2V^i*u5+i3S «:i - 2V^*ufl+*12 / S) + 2v 2*!3fl_*3l + 2*J2H( *1! - 2(*2J02*U - *tl5!*2j) 3 J :) 2 (0< ») = ^S- = (Qi ) , Q? = i-^ + V262d.. (3.9) + 2[*„, *,a]*j3 - -i{9», tfst}^*,, + -i{¥2i,(?2*!i}4>1i In fact, we find a simple relation between the two-dimensional supertransforma- - \/2i{*u, *H}5+*21 + 2i[Hf 'tfn, *2l]*JJ + >'{*J1> *H }^l2'*U tion Q[ and the three-dimensional one Q\ ': , + 2i[*ii,*ij]*2I*2i + 2i*u*n[*Jt, *!j]). (3.13) gt'lgiMaB, = e«,«,8,g(J) (j 10) where the (0,1) covariant derivative is defined by D^ = i^--s/2B,d.. (3.14) This relation suggests a reduction of the N = 1 superfield *„ into two superfields *„i and #„i which transform irreducibly under the N = \ superiransformalion We require the (0,1) supersymmetry and two-dimensional Lorentz invarince on in the Ej direction the boundaries for total derivative terms. We find that there are only four pos sible total derivative terms which can contribute to the effective action on two- dimensional surface 3 ,, »u = A, - \/2-4-«j, S1=-£tr|d *^aJ(i*210i *21), *u=(/i-Mj) + iSj(xi-fl2Ai), 3 (3.12) S2=-ftr/d X^?-a,(*21*12), (3 15) *n = Aj + (i4-^j)tfj, ,; t A9 - / 53 = tr d3x a,( u ) *u = v 2/l+ + iflj(,va-fl]Aj). "4 / 2 * *" ' S, = - &tryd3x^ls (i*„* *2i). Let us note that the t\ transformation is realized as a differential operator Q\,<» > 3 si when acting on the *„i and *„] superfields but is realized as Q[ when acting Thus the most general (0,1) supersymmetric action is given with four real pa- -299- -300- rameters 6j, • • •, 64 as After substituting the solution, we find that the action becomes total deriva 4 tive. Therefore we obtain an effective two-dimensional action as a local field theory: ! 2 n n n 4. Effective two-dimensional action S = - ^tr{|d ^(-2v^e-">£|( »e 'e- 'fl+e ' - 2ianiD 3 n n n n n x K. As in sect.2 we shall regard the light-cone coordinate i+ as "time" and + |d *^2V2V 'e2e '[e- '£ * = *„-2ie*|,. (4.1) n t In deriving this action we have imposed an additional boundary condition 32e ' = 0 to discard component fields with noncanonical dimensions. The component This *L containing an arbitrary parameter c is the most general Lagrange mul form of the action is given by tiplier consistent with the symmetry and the dimensional analysis. By varying the multiplier, we obtain a constraint 2 l l s = - £tr{ fd x[2 r/-fl+r/ tr a_r/ - VSi^a+Xi D^tu + V^fl-*n - 202*11 + |«u, *»] + ••[0i2'*u. *2i] + a(-)iiXid.X + (A-A,)1) -•'[*21,*ll*ll]=0. i l (4 6) To obtain the constraint, we have used a boundary condition of vanishing multi + {/}- 4a)(i«(A - A,){Ai, A,} - +=i\,\,U-d-U - jA, A,A,A,) ] ' 3 i 1 l plier: *L = 0. - [d x2U-d,U[U- 6+U, f/-fl_f/]}. The constraint equation (4.2) can be solved by introducing Lie algebra valued (0,1) scalar and spinor superfields ill and fli, which can be combined into an In order to have a positive definite kinetic term for the scalar field If, we need jV = 1 superfield: to require the Chern-Simons parameter /J to be positive. If p happens to be en_e«i#jS,(ei*,n,en,) (4.3) negative, we can instead take x~ to be time in quantizing the Chern-Simons action and consider (1,0) supersymmetry. The result is the left-right reflection The solution of eq.(4.2) is given by of the above WZW action except an overall sign. n ( *„=e- 'D lV', The action (4.5) has two parameters a and 8. By adjusting the normalization *„ =e-n'fl,en', (4.4) of flj we can rescale the parameter a to be +1, 0, or —1 if it is positive, zero, or n 5, n n n #1, = -^- '(DJ n,)e ' +2e- 'e,e ', negative respectively. Furthermore, by a field redefinition flj — e"n'fljeni, the which has a form of pure gauge except for VJI. action with /? = 4o is equivalent to the action with 0 = 0. —301 — -302- For any value of these parameters the action (4.5) is manifestly invariant References under the (0,1) supertransformation generated by Q\ . Actually it has the right field content for the JV = 1 supermultiplet and is invariant under the [1] E. Wilten, Commun. Math. Phys. 121 (1989) 351 two-dimensional N — 1 supertransformation when the parameters are chosen [2] G.V. Dunne, R. Jackiw and C.A. Trugenberger, Ann. Phys. 194 (1989) as a ^ 0 and 0 = 0 or 4a. In fact, we can rewrite the action (4.5) for or ^ 0 and 197; G. Moore and N. Seiberg, Phys. Lett. 220B (1989) 422; S. Elitzur, /3 = 0as G. Moore, A. Schwimmer and N. Seiberg, Nucl. Phys. B326 (1989) 108; H. Murayama, Univ. of Tokyo preprint UT-542 (March, 1989); M. Bos N=l i i 1 a i 5 = -£atr{/d xd SG- D GG-' DaG and V.P. Nair, Phys. Lett. 223B (1989) 61; J.M.F. Labastida and A.V. (4.7) 3 , 1 I l Ramallo, Phys. Lett. 227B (1989) 92; W. Ogura, Phys. Lett. 229B (1989) + /"d xd 92G- e2C?<3- 0\ i)>G-DfG}, 61 where we have introduced an N = 1 superfield and the covariant derivative [3| N. Sakai and Y. Tanii, Tokyo Inst, of Tech. and Saitama preprint TIT/HEP -146, STUPP-89-107 (Revised, November, 1989) n n !) l G(i, «!,«,)= e"' 'e ', 0< = -i£- + \/2a-eld+. (4.8) [4] S.J. Gates, Jr., M.T. Grisaru, M. Rocek and W. Siegel, Superspace or 00 \ one thousand and one lessons in supersymmetry (Benjamin/Cummings, The action (4.7) is invariant under the N = 1 supertransformations generated by Reading, 1983); R. Brooks, Nucl. Phys. B320 (1989) 440 both Ci2' in eq.(3.9) and [5] E. Abdalla and M.C.B. Abdalla, Phys. Lett. 152B (1985) 59; P. di Vecchia, V.G. Knizhnik, J.L. Petersen and P. Rossi, Nucl. Phys. B253 (1985) 701; Q^ = -i^-s/2a-%d . (4.9) + T.L. Curtright and O.K. Zachos, Phys. Rev. Lett. 53 (1984) 1799 Note that Q(," is not related to Q^' in eq.(3.2) by the relation (3.10). The N = 1 supresymmmetry algebra in two dimensions is different from the three- dimensional one (3.8). For a = +1 eq.(4.7) is the action of the usual N = 1 supersymmmetric WZW model in ref.[5]. For a = -1 the action is invariant under Q\ in eq.(3.9) and <2j!) in eq.(4.9). We can relate this theory to the usual /V = 1 theory by interchanging x° with ~xl. -303- -304- but also extend to the nonabclian case [8] by making use of a variational method based on the properties of a TQFT. Actually, the variational method has already proved to be useful for a direct derivation of link invariants in three dimensions without relying on the two dimensional Topological Quantum Field Theories In conformal field theory [9],[10]. By this direct proof, we could have cleat Higher Dimensions understanding of the equivalence between the three dimensional Chern- Simons action and link polynomials. Therefore we summarise the vari ational method of Ref. [9],[10] briefly and make some comments on the Ichiro Oda difference with our variational method. Their method uses the funda mental identity of the Chern-Simons action Department of Physics, Chiba University, 1-33 Yayoi-cho, Chiba 260, JAPAN November 9, 1989 and the well-known property of the Wilson line operator under an in finitesimal variation of the path in x produces the F^v insertion, namely U(*i.»i)—> u<*i,*i r" Frl^> uu,*i> where U (Xi i Xi )=J? Q is the Wilson line operator associated to the Topological quantum field theories (TQFT's) have recently received path connecting two arbitrary points x, and xx . Our variational method much attention. It is now believed that the language of TQFT's provides shown below is slightly different from one of Rcf. [9],[10], since even if a beautiful framework for both theoretical physics and mathematics. we use a similar identity, we never think of deformation of the Wilson First, Schwarz has constructed a TQFT and identified the Hay-Singer line. Instead, we use the variational method for the direct evaluation of analytic torsion with a partition function about ten years ago [1]. Then, 2-point function as we will see later. VVitlen [2] has constructed a new class of TQFT's in three and four di We begin by rewriting the mathematical definition of the linking num mensions which are closely related to the Jones polynomials of knot the ber concisely [11]. Let M be a compact, oriented n dimensional manifold ory [3], Floer cohomology groups of three dimensional manifolds [4] and without boundary. Let U and V be disjoint, compact and oriented sub- Donaldson polynomials of four dimensional manifolds [5]. manifolds of M with dimension p and n-p-1, respectively. We assume that Recently, HorowiU ->nd Srcdnicki have shown that the 2-point function U and V are boundaries of surfaces of one higher dimension, namely, U= of the TQFT of Schwara produces the linking number of two surfaces of i) Y and V=3 W. Mathematically, such manifolds are called homologically dimension p and n-p-1 in an n dimensional manifold [6], Their derivation trivial. We will see later that this condition is essential for the derivation method of the linking number is heavily based on the Hodge theorem and of the linking number. The linking number of U to V, which we write some results of the eigenfarm of the Laplacian. Also, oilier authors deal L(U,V), is defined as follows; When V and V intersect in a finite number with the same topic [7]. of points pj , the weighted sum over pj with +1 or -1 according to In this talk, I not only rodcrivc their results [6],[7] in the abelian case a suitable orientation convention is the linking number L(U,V). Roughly -305- -306- speaking, one can interpret the linking number as the intersection num v'+dv", w'—»w'+dw" etc. Of course, this system has not an infinite but ber. One can also define the linking number of V to U, which we write a finitenumbe r of reducibility once we fix the dimension n of manifold M. L(V,U). This is weighted sum over intersection points between V and Y. Horowitz and Srednicki have solved these both problems by making use Then, L(U,V) is bilinear with respect to U and V, and has then an im of the Hodge theorem and some results of the cigenform of the Laplacian portant non-commutative relation [6]. We will use a variational method [8]. As a firststep , we do the gauge LCu,V)= C-)r"M>+,LCvMi) u> fixing of the action (2) in a covariant gauge. The gauge fixed and BRS invariant action becomes which implies that for instance the linking number L(S ,S ) in S is van ishing. Examples may be useful to help to illustrate the linking number &•/* =f[BA -307- -308- where [ eQ/S ] denotes the functional measure jQB jQC oD N - — ichi = t*--»- c^C^ „<*> pother reducible ghosts, anlighosts and NL fields) and(x)dcnotes tensor product. Using the identity (5a) except the last lerm^Q,do( }> which we will specifically take account later < B«) Ktj)> ' »-*•< --•/Ni-f-l N _CM-0.' ^'f"(fj L -^*^) = c_^r»-t'tN.-^jkiflv_ff|...+wu, To derive the last equation, we have used the part integral which is the only assumption for the functional measure. It is worth mentioning that the partition function in the denominator is automatically cancelled by the partition functio.' appeared in the numerator, therefore, there is no where Q is the BUS charge, and we have used the fact that the BUS trans problem with respect to the metric dependency on the functional mea formation of anlighosts becomes Nakanislii-Lautrnp fields in the present sure as mentioned before. After using the Stokes theorem twice and the formalism. homological triviality or V, namely, V= ~j)\X, a short calculation shows As a third step, we use one of the variational identities (5a), the part the following result in a flat metric integral and the Stokes theorem as follows, Consider the equation < J, 6wj^c'3>> = i L CU,V^ CP < Btt> <*Ct^>j> u where (ut-4) IT* p; (Vp-i)/ ju r ,Sf> ®^c,f .^Mr > ••••A>re -309- -310- which is an integral formula for the linking number of two surfaces of di (which means that these surfaces are everywhere smoothly connected) as mension p and n-p-1 in S . Incidentally, one can apply another identity well as homologically trivial. Although the variational method in this (5b) for^"dB(x)C(y) y and obtain the same result as (7) and (8) in a case is also useful and the path of discussions proceeds the same as the similar way. It is interesting to realize that the homological triviality of above abelian case, some complicated features will occur. The gauge U and V is essentially and naturally needed in our derivation method. fixed and BRS invariant action is equal to the nonabelian version of (4) The problem to be remained in order to complete our derivation is whose exterior derivative d is replaced by the covariant derivative D. The that one must show variational identity for B field becomes B,x1 iSt } n < L [jQ'* ») > = ° <1 £ '"" VtfH Crrtl.....,,h ex) In fact, this proves to be true because of the BRS invarianceof the vacuum and f B. = ______}Q,i/r 5 •"••*•%) Ut) Next, we turn our consideration to the nonabelian TQFT [1] SB/*, ...^p txi And we use the useful property of 2-poinl function (11) CJ = f Tr [ E> A DC ] 0") fiA ]} where D denotes a covariant derivative with flat connection one form for < TrUw iMle e(3i 2 e Ml> g a semi-simple Lie group, thus D =0. D is defined in a graded mannar, A ,} for example, Ddi.^ =d of,^ +[A, 0(1(1 ], but Dt(1Mi =d 0fij*.| + a = < Tr f Bc« 2 e^ T>ci;» J?e ' ] > i> {A,g(u4l) 1 "d 'hen B and C arc p and n-p-1 forms transforming in a 0 representation under G. The field equations are DB=0 and DC=0, and the gauge transformation is B-^ B+dv and C->C+Dw owing to D =0. where d» denotes the exterior derivative with respect to the variable y. Note that this nonabclian theory has an unsatisfactory point, which is to If one substitutes (12) into (13) and use the part integral, and integrate introduce a fat connection field A as a background field by hand in order with respect to x over U and y over W, and then use the Stokes theorem, to have gauge invariance. We will sec that we can improve this point by one obtains the following result, establishing other extended theory [13]. Let us consider the following gauge invariant 2-point function Btx) l|} , (ft) < Tr t J [ •£ e ctp 2 e J > < where L(U,V, h,ttl1x)s f f L_yWJft_.il where two Wilson line operators connect a point x in U with a point y J in V. When proving the gauge invariancc of (11), we have to use the fact •AJOO ywiji p/«n—pi C that the exterior derivative is well defined at points x and y. To achieve • 1n...,,,, iX•'*-•• Ain't ® i^rHIA . ... A J-|/- this fact, we restrict surfaces U,V,Y and W to be holonomically trivial • Tr \ H* _ einA p.* e**A ] fir) -311- -312- where R is generator matrices of a semi-simple Lie group G, and f~. and "Y • In this talk, I have referred to topological quantum field theories (TQFT's) are closed curves having the bascpoint x=y. in higher dimensions. Mathematically, we have derived the conventional To visualize the meaning of the generalized linking number (15), let linking numbers by using the abelian TQFT in higher dimensions, and us take a simple case G=U(1). In this case, eq.(15) becomes then derived the generalized linking number, which is mathematically new, from the nonabelian TQFT in higher dimensions. i n •1* .A L cu,v,A,?.>•*!} = i: y>y(fi"\ e * ' tit) where II sign(p • ) denotes the conventional linking number which has been obtained in the abelian theory. Some examples are giver, in Fig.2(a) and Fig.2(b). These figures show two circles U and V in R which have I would like to thank Shigcaki Yahikozawa for collaboration works the conventional linking number zero and one, respectively. Let us con on TQFT's. This talk is based on works with him. I am grateful to sider the specific situation where two parallel dolled lines arc removed Akio Sugamoto for valuable discussions and continuous encouragement. and put on a U(l) connection with phase factor c' circulating around Finally, 1 would like to thank Professor Abudus Salam and E. Setzin for two lines according to the orientation shown in the figure. The general hospitality at 1CTP during summer in 1989 where this study was started. ized linking numbers are thus REFERENCES for Fig.2(a) and f ClJ A.S.Schwarz, Lett. Math. Phys. 2 (1978) 247; Comm. Math. Phys. 67 L(u,V/ A,ir ,*o » e * c<*> (1979) 1; A.S.Schwarz and Yu. S. Tyiipkin, Nucl. Phys. B2I2 (1984) 136 for Fig.2(b), whore we notice that (17) is equal to the conventional link [a! E.Witten, Comm. Math. Phys. 117 (1988) 353; "Quantum Field The ing number and (18) becomes equal to it taking the limit o( —>0. In ory and the Jones Polynomial",IAS preprint IAS-IIEP-88/33 to appear these examples, M has a boundary while we have previously assumed that in Comm. Math. Phys. *J M=0, so that we should modify these examples to become 3 M=0, First cut these figures by a family of horizontal planes, where every plane has [3 J V.F.R.Jones, Bull. AMS 12 (1985) 103; Ann. Math. 126 (1987) 335 the topology of Rl except two points. To remove these singular points, we connect the boundaries of the two points by a handle. If we add a Ctyl A.Floer, "An inslantion invariant for three manifolds", Courant Institute point at infinity to this topology, the resulting surface Is equal to a torus T preprint (1987); Bull. AMS 16 (1987) 279; Comm. Math. Pliys. 118 . And then identify the top surfaces with the bottom one, we eventually (1988)215 obtain T . Therefore, the generalize.! linking number of these examples corresponds to the linking number in T3 which is mathematically new [& J S.Donaldson, J. Din". Gcom. 18 (1983) 269; ibid. 26 (1987) 397 linking number. LG J G.T.IIorowitz and M.Srcdnicki, "A Quantum Field Theoretic Description —313— -314- of Linking Numbers and Their Generalization", preprint UCSB-TH-89- FIGURE CAPTION 14 Fig.l. Examples of ordinary linking number are ihown. Both U and V are V are circlea in R . 3 M.Blau and G.Thompson, "Topological Gaugr Theories of Antisymmet ric Tensor Fields", preprint SISSA 39 March 1989 Both Y and W are disc, therefore the homologies] triviality, that is, U=»Y and V=»W hold. Ordinary linking number of each each figure (a),(b),(c) and (d) ia lero, one, one and J I.Oda and S.Yahikozawa, "Linking Numbers And Variational Me'^od", two. One can see that L(U,V) =L(V,U) holds from fig.(b) and (c), for instance, ICTP preprint IC/89/287 1 Cotta-Ramusino, E.Guadagnini, M.Marlellini and M.Mintchev, "Quan tum Field Theory and Link Invariants", preprint CERN-TH 5277/89; Fig.2 E.Guadagnini, M.Martellini and M.Mintchev, "Pcrturbative Aspects of the Chern-Simons Field Theory", preprint CERN-TII-523 3} I.Oda and S.Yahikozawa, "Topologically Massive Nonabclian Gauge The ories In Higher Dimensions", ICTP preprint IC/89/288, to appear in Phys. Lett. B — 315 — -316- 5 o 6 D) •i—i M—I CNJ i—i 2 6) FIG.2(b) -321- Action Principle for Chiral Bosons 1. Motivation Let us consider the chiral Schwinger model [1], described by the following Lagrangian, C = -±F,.F" + C„ (1) Koji Harada L, =^y(ifl,+«v7it,(l-7«))*. (2) Institut fur Theoretlsche Physik Since the right-handed fermlon i/fc = ^(1 + 75)^ has no interaction vertices, we der Universit&t Heidelberg can expect, at least perturbatively, that it only gives a field-independent constant in the Fhilosophenweg IS, D-0900 Heidelberg following vacuum functional: West Germany Z[A]= [ dVu\Jexp{i / i'xCr}. (3) On the other hand, it is known that the path-integral in (3) can be performed exactly, Z[A] = «,{^|i'xA,[an" - {B> -r»^(6" +*•)]*.}, (4) where a(> 1) is a constant which represents a regulariiation ambiguity and fl* = t"d,. Abstract The non-local expression (4) can be written in a local form by introducing an auxiliary scalar field at, We present actionB for chiral bosons Interactiong Abelian and non-Abellan gauge fields. Z[A] = / dj,cxp{i j d'xCa} (5) The "minimal" chiral Schwinger model is solved by using chiral boaoniiation. We prove with that the correct effective actions for chiral QSDi and chiral QCDi are obtained from the b> = \{8,+? + •(!?" - <")dtA. + \. (6) bosonized actions. f One may regard £g as a boaonized action of Lr. Since the right-handed fermion is expected to give a trivial contribution to Z[A), we would like to extract the counterpart in its boBonic representation (5). In other words, — 323— -324- we are interested in the bosonic representation (bosonization) of the ''minimal" chiral Let us consider the Hamiltonian system described by HB subject to the chiral con Schwinger model described by the lagrangian straint (10). The vacuum functional is obtained, by noting that dei{Q}£}} is a field- independent constant, as £"""-SS-IFJI,P"'+ £;'", (7) s 7 Zth[A)= J"rf«Wir^(n(*))|ilri{Q1Q}|" arji{i jd x(r^-7iB)} = J'dfrxpii Jd*xCtk) (12) where ifu, ~ 5(1 - 73)^- with In this way, we ate lead to the concept of chiral bosonization, i.e., equivalence between C = fa? - tf'}3 + 2e*'(X - A ) - j«3(A> - *,)* + lfaA,A*. (13) Weyl fermions and chiral bosons. th 0 X This Lagrangian is a gauged version of the Lagrangian by Floreanini and Jackiw [3], CJPJ as ij»ft' - (^')2- Since they have given a direct connection of their free <£-field to 2. Chiral constraint in phase space a free Weyl fermion a la Man dels tarn, we may regard C^ as a bosonic description of ££in(8). Actually, it is easily verified that Z [A] is equal to Z[A], Z [A) = Z[A] up to a At the first sight, it seems to be difficult to throw away thr redundant part in CB- ch tll t 3 constant. Therefore Z ),[A] is an alternative local expression of the generating functional Even for a free boson, the Lagrangian Co as £(d^) cannot be devjded into two pieces so e that each piece rep risen ts a chiral boson, i.e., a left-mover or a right-mover. However, it Z[A}(4). can be done if we work in phase space. Note that our procedure introduces interactions to the FloreaniniOaciciw Lagrangian We begin with the Hamiltonian obtained from CB, which is not manifestly covariant. »B = \[r> - We impose here the following " chiral constraint", Now we have a bosonic Lagrangian £,« which corresponds to CJ". The solution of n(*)s*,r>)-lt>)«0. (10) the model described by the Lagrangian, £3" = -\Ff,F>* + £,n is the following: It turns out that the chiral constraint fl(*) is second-class [2], l A, = iB^ + (a-l)Bf^-a8,h}. (14) {n(x)(n(V)} = -25'(*' -y). (11) -325- -326- This is the same as the one obtained by Jackivv and Rajaraman, except for the fact that We impose here the following "chiral constraint", the h field i* now a self-dual ftcld,(do-&i)h = 0. (In their case, A is a harmonic excitation.) Q" = ir\* hPT + -J-lrX'di hh~ 'aO. (19) 4r 4. Nan-Abelian generalization One can easily see that this is second-class and rf constraint surface. The non-Abelian generalization of the Floreanini-Jackiw Lagrangian has been given It turns out that the Hamiltonian H% on the constraint surface is obtained from the by Sonnenshein[4]. following action: l , /„[*]= __L [ fcirh-^do - dl)hh'8lh+ -L ! fxeuUrHiHjHi, (15) f.»[M] =M»] + -^Jj'^rh-alh(A0 - /,) 4x J Mr JD where A is a field which takeB values in the symmetry group and ff, = h~ld,h. Now the + ±Ji'*r{Ao-Aij'-±Jd>X1rAl,A>. (20) problem is how to introduce interactions correctly. TKis is the desired result. It is surely a non-Abelian generalization of (13) and iB also a We can proceed as in the Abelian case. Let us start with the usual bosonized action, gauged version of the Sonnenschein's action (15). XB[KA\ - - i- fd'xirH.H" + T^- / d^^irHiHjHk OX J 1** Jjj 5. Effictive action of chiral QCD^ + -i- Ii'xirRrW -c")A, - £- [d'x1rArA'. (18) 4x J ox J From this action, one can get the following Hamiltonian, Finally let us prove that the action (20) gives rise to the correct effective action of l T T T chiral QCDi. In the following, since the a-dependent term does not play any important HB = Jds[-2rirP hP h + irP h{A0 - Ax) - — irh'^^iAo - A,) role, we disregard this ambiguity. l l 3 - ^irh-dlhh-dlh-'~1r(A0 - Ax) + ~.rA„A"]. (17) Let us first summarise well-known facts. Folyakov and Wiegmann[S] obtained the Here P is a "velocity" variable (not a momentum conjugate to h) and may be written as following expression for the fermion determinant of QCD2; P.j^IM-'+Mo-AOA"1],,, (18) W[A] = - ilojrfe<{iy (S„ + A,)) r and T stands for transpose, (P ),; = P,,. = -r(ou-') (21) — 327- -328- We have two bosonized actions which give rise to the correct effective action of chiral r(j) = ~ lPxirQ,G> + ~ I t'^irQ&jGu (22) QCD,;(16) and (20). Actually, under plausible requirements, we can get a one-parameter ox J 1/r JQ family of actions which give the correct effective action of chiral QCD%. It contains anions with G, = g~xd,g. Fields u and v are introduced by the following relations, (16) and (20) as special cases. If we further require the Poincace invariance, it turns out A+ stf-^+u, that only the actions (16) and (20) survive. This fact reveals a particular status of the ^_ =u-'a_u. (23) action (20). Since, in chiral QCDj, only the >1- component couples to the fermion, the effective 6. Epilogue action of chiral QCD7 may be obtained in the following way; WW[A] = W[A+ = 0, A.] = -rtu"1). (24) This talk is based on the following papers. K. Harada, Heidelberg preprints HD-THEP-89-18 and-22 What we want to prove is the following relation: K. Harada and K. D. Rathe, Heidelberg preprint HD-THEP-89-30 l j ihaf{iI.i[h,A]} = exr{iWt\A\). (25) The following list includes only a few references. More (but far from) complete lists of references may be found in the above papers. The proof is not J.»[M] = U[Au-'.0]-r(ir'). (26) meeting. This work was spported by the Alexander von Humboldt Foundation. which is proved by a direct computation. References [1] R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 5s (1985) 1219, 2060(E). J Jhap{il,k [h,A]) =«p{-tf(u-')} J ihuf{iU [hu-', 0]} [2] P. A. M. Dirac, Lecturtt on Quantum Mtchamci (Yeihiva Univ. Press, New York, =«,{-iT(u- >))Jj{h -') xpiil [hu-l,o)} (27) u C lk 1964). 1 =Mni1,eip{-ir(u" )}, [3] R. Floreanlni and R. Jackiw, Phys. Rev. Lett. 69 (1987) 1873. [A] J. Sonnenschein, Nucl. Phys. B309 (1988) 782. where we UBed eq.(26) and invariance of the Haar measure. -329- -330- Path-Integral Quantization of a Particle Coupled with 1 Introduction Chern-Simons Gauge Field Recently Polyakov [1] has claimed that the gauge field transmutes a charged scalar particle into a spinning particle. He has evaluated the dressed propagator for a scalar particle with the gauge field which corresponds to a free spinning particle in the language of the functional integral for a relativistic particle. The effective action for the charged particle consists of a length of the particle's path and its torsion term induced by the Chern-Simons gauge field. Since the torsion Chigak Itoi term contains higher derivatives of the particle's position, the particle may acquire Department of Physics, College of Science and Technology a new degree of freedom[2], which is a unit tangent vector on the path. This ffihan University should be identified with spin degrees of freedom because of the commutation Kanda Surugadai, Chiyoda-ku, Tokyo 101, Japan relation determined by the torsion term. Thus, the particle belongs to the spin J representation of SO(3) if the coefficients of the Chern-Simons term is i5 = 37: Sc.s. = -^jd?x^A^Ay. (1.1) Abstract Polyakov suggests that the propagator for a charged scalar particle should effec The transmutation of a charged scalar particle into a spinning particle is tively become that for a Dirac particle at J = 1/2. In Polyakov paper however, studied. We evaluate the partition function of the particles in three dimensions many problems still remain. He does not comment on an ambiguity in the reg coupled with Chern-Simons gauge field. The regularization scheme is discussed ularization of the self-energy and does not clearly define the measures in the in detail. It is shown that the Polyakov claim is justified. functional integral over random paths. The purpose of this report is to clarify the problems elucidated. We define the This work is in collaboration with S. Iso and H, Mukaida. regularization of the dressed propagator and refer to an ambiguity which exists in -331- -332- that regularization. We show that the propagation of the dressed particle can be [3]. X"(s) and Y"(i) denote position vectors Pi and P? respectively and are effectively described by functional integral over random paths with torsion. We parametrized by s and t which denote the length of he loops: explain that torsion can be rewritten into a spin factor which is more convenient (£)'=(£)'=!. 0 2 Loop Splitting Regularization The limit e -* 0 can be taken in eq.(2.3) by means of this regularization. Then one obtains Here we study a charged scalar field coupled with Chern-Simons gauge field. /(Pj - Pi) = iJT, r= fdsX-[nx n = cosor ni + sina n2, (2.7) where P are closed paths, L(P) is their length and one can rewrite the total twist T into K{P) = [VA t'f^'^r1": (2.2) T-/(±.[n,KA,] + d,). (2.8) The functional averaging over Af is performed as follows: The principal normal nt and the binormal n2 = X X ni can be chosen as the frame except a singular case that the path is locally straight line. In this framing, Since the right-hand side in eq.(2.3) is a line integral along the two coincident X • [ni x hi] is equal to the torsion C. We note an ambiguity of an integer N in loops, it should be defined by a certain regularization. Here, we define the reg- this regularization. iJT il i,c 1 11 ulaiization by a coincidence limit Pi -* /\ of two separated loops P\ and P3 K(P) = t = t 5 + " '. (2.9) -333- -334- Though we cannot determine the regularization without the ambiguity at this where a — s, u and eq.(3.2), is identical to eq.(3.1) at ti = 1. In order that stage, we discuss it in section 4. the equations (3.2) have solutions on Dj, the following integrability condition is necessary 3 Relation between Total Twist and Spin Factor 1 B.C.-a.C, = B\Bl-Ei tBl. (3.3) In this section we rewrite the total twist into a more convenient form. It From this equation (3.3) we obtain is necessary to study a role of the total twist eq.(2.9) in the functional integral / Cds-i Cds = / dudse • [8,e x fl.el. (3.4) eq.(2.1) over random paths. Let e be an unit tangent vector along the path. A JPl JP) JDi The fields can be extended to Oi as Pi is a circle with a certain winding number principal normal ni and a binormal n3 = e x ni satisfy M. In this case, the second term in left-hand side of eq.(3.4) vanishes. Next, 3,e = flm.a.n] = -Be + Cn d,m = -Cm, (3.1) 3l we define e(s,u) on Di. The following "M-meron" configuration satisfies the where B and C are the curvature and the torsion respectively. Here we define requirements for e at u = 0 and u = 1/2 fields e(a, u) and n,(s, u) (i = 1,2) which are extended fields of the fields e(s) and e = (—atnwu stn—-—, smiru cos—-—, COJTTU). (3.5) rii(j) on A to a two dimensional space D with a boundary Pi(u = 1). e(s, u) and L L Since the integral is n,(s,u) satisfy the following boundary condition e(s,l) = e(a),e(s,0) = const. We divide D into two parts J3i(l/2 < u < 1) and Z)j(0 < u < 1/2). D| has two / duds e • [d.e x d„e] = 2irM, (3.6) JD, boundaries Pi(u =1) and Pj(u = 1/2), and Di has one boundary Pj(u = 1/2). the integral of torsion over Pi is rewritten in terms of e(s, u) For instance, D\ and Dj are a cylinder and a hemisphere respectively. First we define el'. u) and n,(j,u) on D\, The extended equations on D\ are / Cds = *[e] - 2JTM, (3.7) 3„e(s,u) = Eflini, where •*i (3.2) djn(s,u) = -flie+ECo^rij, *[e] = J dudse • [d.e x d.e]. (3.8) -335- -336- Thus eq.(2.1) is rewritten into this local gauge transformation. Since e„ is a unit vector,X>e is integrated over solid angle 4JT at each r. Note the constraints h = vX? and e» = h'^X", which Z = Y e-ra«i')+iJ*|«l-l.iJ(Af-W)] (3 g) P are obtained by integrating out the multiplier field k. If one integrates out k,e It is known in the functional integral method for spin that the action is given by and h, a theory described by only X is obtained. This corresponds to a theory J $[e]. In the section 4, we show that e' * is a spin factor for spin J and eq.(3.9) eq.(3.9) which contains higher derivatives of X. In ref [6] the propagator F has describes spinning particles. been calculated as follows: F = J°° dle-mL J dpe**'-*') J Vee'ti irLp-°+'J*['\ (4.3) 4 Functional Integral for Relativistic Particles in Three where m = m — (y/ire)~l and p denotes the zero mode of k. The formula Dimensions 0 L < t,\Texp{£ds3 • S(a))|ei >= J•Dee-Jn')+I, M-)SC) (4.4) In this section we study the particles whose partition function is given by eq.(3.9). Since the definition of functional integral is not clear in the expression enables us to rewrite the propagator in the following form if J ^ 0 (3.9), we define the functional integral rigorously. Let us study the following ,F(p) = (-iJ-'p . J + m)-\ (4.5) propagator which is invariant under local coordinate transformation in one di where J are generators of SU(2) and S(s) is a c-number source. Note that J must mension. t VXVkVeVh be an Integer or a half integer. States |e, > and |e/ > are elements of the spin J ^(X/eylXiC-) = / —J, « s, (4-1) representation. Eq.(4.S) indicates that the charged scalar particle is transmuted where into a spinning particle. Eq.(4.4) can be proved if we introduce coherent states S = /' dr{moo+ <*„(*" - he.")} + W*[a], (4.2) Jo for SU(2) group[4]: and X" is a position vector of the particle with a fixed boundary condition X(0) = |c >= eiHe„x«M|e.««|-'|o >t X,,X(1) = X/. The action is invariant under local coordinate transformation where eo = (0,0,1) and |0 > denotes the highest weight vector in this represen T — /('•)./(") = °. /(I) = 1 'f h(r) — //i(/(r)). V,„„ denotes the volume of tation. Eq(4.4) is proved due to properties of the coherent state as follows: -337- -338- Here we will discuss the ambiguity of the regularization in eq.(3.9). Let us A charged scalar particle is transmuted into a spin J particle. It has been consider the model of eq.(3.9) as first quantized model in operator formalism. shown by means of the functional integral for relativistic particles. It is noted J The Foisson bracket between e*s are determined only by the term J$[e] in the must be an integer or half integer. The result is consistent with the transmutation action as follows: of statistics. Thus, the Polyakov claim is justified. 1 {ei,ei} = i€i,kei,J~ . (4.6) Note that this is independent of the regularization procedure. To obtain a con References sistent quantum theory in operator formalism based on eq.(4.6) we should take the representation of SU(2) for spin J. The coherent state method enables us [1] A. M. Polyakov, Mod. Phys. £etf.,A3(1988)325. to rewrite the operator formalism for e into the functional integral representa [2] C. Itoi, Phys. Lett. 211 B(1988)146. tion eq.(4.4). The result shows that M - N must vanish in eq.(3.9). Thus, we conclude that the regularization should be determined as a consistent quantum [3] C. H. Tze, Int. Jour. Mod. /V-,A3(1988)1959; theory can be obtained. In the regularization determined above the topological [4] A. Perelomov, Generalized Coherent States and Their Applications, (Springer- relation between the two splitted loops may be fixed. Verlag, Berlin, 1986); [5] This model eq.(3.9) is studied also in operator formalism: 5 Summary N. Imai, K. Ishikawa and I. Tanaka, "Quantum Theory of Relativistic Particles We have investigated the partition function for charged relativistic particles with Torsion", Hokkaido University preprint EPHOU-89-003 coupled with Chern-Simons U(l) gauge field and summarize our result. [6] S. Iso, C.Itoi and H. Mukaida, TIT/HEP-152, NUP-A-89-12; to appear in There is an ambiguity in the regularization procedure for self-energy of the Phys. Lett. B. charged particle. Although the ambiguity exists in the regularization, it should be determined with no ambiguity in order to obtain a consistent quantum theory. -339- -340- 1 Introduction Effective Actions of ID and 2D Heisenberg Several years ago, Haldane suggested that one-dimensional quantum Heisen berg antiferromagnets have finite gap for integer-spin, and are gapless for half- 1 Antiferromagnets in CP Representation odd-integer spin [1]. He derived the 0(3) cr-model with the 9-term as an effective continuum field theory for Heisenberg antiferromagnetic spin chains and found that integer- and half-odd-integer spin chains are topological^ dis tinguishable by 6 = 2vr5, 5 being their spin. This remarkable conjecture is supported by Bethe's exact solution for spin-1/2 case [2] and by several recent Hisamitsu Mukaida studies[3]. Bethe ansatz indicates the existence of gapless excitations. Since Department of Physics, Tokyo Institute of Technology the 0(3) (T-model in (l+l)-dimensions at 9 = 0 does not have any gapless Oh-okayama, Meguro-kv, Tokyo 152, Japan excitations, these can appear obviously due to the 0-term at 8 = ir. In (2+l)-dimensions, Dzyaloshinsldi, Polyakov and Wiegmann stressed that the Chern-Simons term plays an important role in the CP1 model similar to the 0-term in (l+l)-dimensions [4]. The Chern-Simons term changes the statistics of z-boson to that of fermion or to fractional statistics and may af Abstract fect the energy levels of the excitations. On the other hand, it is pointed out that the Chern-Simons term does not arise from 2D Heisenberg model from We derive the low-energy continuum field theories of quantum Heisenberg the microscopic viewpoint, since the lattice spacing regularization does not antiferromagnets on one- and two- dimensional square lattice. In (1+1)- 1 viokte parity invariance and further parity may not be spontaneously broken dimensions, we obtain the CP model with the fl-term. On the other hand in 1 [5]. But this naive argument do not always forbid the appearance of the Hopf (2+l)-dimensions, we find the CP model without the Chern-Simons nor the term in (i !) dimensional case. In fact, the Hopf term may not break the Hopf term. r parity invariance as the case of the fl-term in (1+1) dimensions. Because it This work is in collaboration with C. Itoi. -341- -342- can contribute to the action as ir times integer. Therefore it is worth while Namely, the physical states are in the spin N/2 at each site. With the above deriving the CP1 model explicitly in order to clarify the effective lagrangian constraints, we obtain the partition function Z as in (2+l)-dimensions. Z = J DaDaDXe-s, (2.4) In this report, we derive CP1 model as an effective theory of one-and two- dimensional quantum Heisenberg antiferromagnets. The effective action is where obtained by integrating out the small fluctuation modes in the partition func s = f dr{Y. Si* + J- E a.-** • B&J + E ^< (kf - tf)} tion. We find an extra terra which contains first order time derivatives of •/0 i * 1 the Chern-Simons term nor the Hopf term in (2+l)-dimensio;ial CP model. ,8i admits the gauge transformations a; —• Oje , so that the measure DaDa contains redundant variables. As we shall see later, the redundant variables 2 Path-Integral Quantization for Spin can easily removed. Let us start with the hamiltonian of the quantum Heisenberg antiferro Now we change the variables on a sublattice B (We shall call each sublattice magnets with spin N/2 on a square lattice. A and B.) H = J £§<•§„ (2.1) which leads to where < i,j > denotes a summation over all nearest neighbors. In the path- S, - -S;. (2.7) integral approach, it is convenient to describe spin operators in terms of bosonic Then the lagrangian in (2.5) becomes oscillators: £ = iat^, a,= h). (2.2) £ - E ** - E ** - i E W + E * (kf - *0- (2-8) We put the following physical state condition (aja, - N)\phys >= 0. (2.3) -343- -344- 3 One-dimensional Lattice Further we define 26d]Q2, H fl|fl = f z+z£ and f||; = £z—z£, the action does not contain f||/. (It corresponds «Ji = —Jrf^+I + a2i). &i• = ^(a2i+l -a i). (3.1) 3 to the fact that (3.3) are first-class constraints and f||/ is a gauge freedom.) It is reasonable to assume Therefore we should remove D£p from the measure D£D£ in the partition **~0(1), &,-~0(6), (3.2) function. After integrating out fj., fj_ and §|R, we get the desired effective action. where b is a lattice spacing parameter. 2 Sm = rfr a a a The constraints |a« j — N = 0 can be written " 7 jj" / «** {jt^jl V0*| +JVy6| V1*| +(flb*fli*-ft***)+A(|*| -1)}, 3 3 (3.8) 1 = ta,f + ^l&l ~ l**| , 0 = faiai + fjifa . (3.3) where we have taken the continuum limit by 26 £ —> / <£c. Note that the Our aim is to obtain the low-energy effective action for z by integrating out gaussian integral over fj. and fx cannot be performed without redefining the the small fluctuations £. Here we calculate the effective action up to order variables by eq.(2.6). This is the action for the CP1 model with the 9-term at 0(3„3) derivative. The following terms in lagrangian (2.8) contribute to this 9 = TTN. Although the S-term changes its sign under parity transformation, order the partition function is parity invariant because of e" = e~". This result is J I = E{ 2\/tf (£r - zf) + 2NJ&SPZ - 6v^V(eV,z + V^) - JV|V,*| ) equivalent to that obtained by Haldane and Affleck[l], +A(|Jf-l)-2M„(£z + zf)}, (3.4) where P denotes the following projector j Pofl = i„0 - *0i/|. (3.5) Vi is a covariant derivative as defined by Vi = 9i-i8]Z. (3.6) -345- -346- 2 4 Two-dimensional Lattice order O(d0d, ). a 2 Now we apply the similar method to two dimensional case. Let us define L = 4 £{ VN[Z£ + li) + rjifti + ifciji + ATj{2rftPj7, + 4fPf + JV6|V,-ar| new variables: -b^/N^ViZ + V^r?,) - -^[{(zjftfri + ff%rb) + A.c.} + (1 ~ 2)] J vNZil = -( OH + Oli+fa + a22+M+jii + a22+ili)t +&[{[-eViia + fj(2an» + %M] + A-*} + (1« 2)]} + A(|z| -1) +iMfjiZ + Urn) + iAa(-/N[&+ zQ + Tftrj, + ^j. (4.4) Via = 7(-ajjr + aw+jii + "22+h+ii, — aM+fl,)i V211 = T(-0M - a2!+h + 02i+|!| +£, + a j ji,), Note a permutation symmetry 1 «-» 2. We see the lagrangian (4.4) does not 4 2 + contain IJ,||/, f||/. Therefore we can remove these variables from the measure fa* s -( aii-a i /!+a2t+ii,+ii, — ai3+ii,)- (4-1) 2 + l a PI DrijDr)iD£D£. After integrating out the remaining small fluctuations in i-l and assume (4.4), we obtain the effective action zj* ~0(1), n„i ~0(4), s<" = N ildTSdx{wiW^a2? + ^T{Viz{2 + A(|z|' ~1] 1i M ~0(4), ft* ~0(i3). (4.2) +^[d2{dizd0z - dozdiz) + di{d7zdaz - dazfyz)]} (4.5) Constraints are described as follows The last two terms are derivative of the 8 term which were conjectured by l = |**|* + £(lwaM&#|')~l««l' Haldane, Dombre and Read [6]. These terms should be vanished if z-fields are smooth. Therefore we confirm that 2-D Hesenberg model in the long-wave limit is equivalent to the CP1 model. 3 0 — z2*f jjr + f 2*zai + -T=(fiijHM + 'Tsu'fta*) (4- ) 5 Discussions We wish to check whether the Chern-Simons or the Hopf term may appear We have derived a CP1 model explicitly from ID and 2D quantum Heisen- in the effective action or not, we calculate the effective action up to order berg antiferromagnets. We show that a first order time derivative term should 0(&Via) (i=l or 2). The following terms in lagrangian (2.8) contribute to the -347- -348- be added to the CP1 model lagrangian (1+1)- dimensions.but no such terms I. Affleck, Nucl. Phys. B257 (1985) 397. are contained in (2+1) dimensions. Here we comment on the possibility that the Chern-Simons or the Hopf [2] H. A. Bethe, Z. Phys. 71 205 (1931). term arise from 2D models of strongly correlated electrons. If the parity is spontaneously broken, the Chern-Simons term can be induced. For instance, [3] I. Affleck, Kennedy, Lieb and H.Tasaki, Phys. Rev. Lett, 59 (1988) 799; Wen, Wilczek, and Zee suggested that a chital ordered state is a local minimum Comm. Math. Phys., 115(1988) 477; I Affleck and E. H. Lieb, Lett. Math. in a antiferromagnetic model with interactions between first and second nearest Phys.,12 (1986) 57. neighbor sightsp]. In this case, the Chem-Simons term is induced if the chiral ordered state is a true ground state. If the Chern-Simons term is regarded as a fermion determinant in a Lorentz invariant field theory, it cannot be induced [4] I. E. Dzyaloshinskii, A. M. Polyakov, and P. B. Wiegmann, Phys. Lett.,217 without breaking parity invariance. On the other hand, the Hopf term as well A. (1988) 112; P. B. Wiegmann, Phys. Rev. Lett, 60 (1988) 821 ; A. M. as the 8 term in (l+l)-dimensions can keep parity invariance. Since the Hopf Polyakov, Mod. Phys. Lett, A3 (1988) 325. term is topologically invariant, the corresponding partition function is parity invariant if the coefficient is quantized. Therefore we cannot reject that a [5] X. G. Wen and A. Zee, Phys. Rev. Lett.61 (1988) 1025; see also E. Fradkin CP1 model with Hopf term is derived from some other models of strongly and M. Stone, Phys. Rev. B38 (1988) 7215. correlated electrons even if parity is unbroken. It is important to find out any 2D models described by the CP1 models with the Hopf term. [6] F. D. M. Haldane, Phys. Rev. Lett. 61 (1988) 1029; T. Dombre and N. References Read, Phys. Rev. B38 (1988) 7181, [7] X. G. Wen, F. Wilczek and A. Zee, Phys. Rev. B39 (1989) 11413. [I] F. D. M. Haldane, Phys. £e«.,93 A (1983) 464; Phys. Rev. Lett.,50 (1983) 1153; J. Appi Phys. 57 (1985) 33S9; Phys. Rev. Lett.,61 (1988) 517; -349- -350- Nonlinear Sigma Model and 1. Introduction Quantum Antiferromagnets* Since the discovery of the copper-oxide superconductors, many people have come to consider that the magnetism in the two-dimensional CuO, layers may be responsible for the high-T,. superconductivity in the lamellar materials such as £a, ,M CuOk(M = Sr,Ba). Experimentally, recent neutron scattering mea surements in La,CuOA revealed novel two-dimensional antiferromagnetic spin fluctuations [1] . HlSASHI YAMAMOTO Theorists have therefore concentrated on the magnetic properties of the sim Institute for Nuchar Study, University of Tokyo, plest models to take into account the strong electron correlations in a nearly Tanashi, Tokyo, 1S8 Japan hall-filled band such as the Hubbard model, or the two-dimensional antiferro magnetic Heisenberg model as the strong coupling limit at half-filling. Abstract Some recent studies have shown that these lattice models are represented by It is broadly viewed that the magnetism may play an important role in the t/(l) gauge invariant CPl nonlinear a model or that coupled with hole fermions, high-Tc superconductivity in the lamellar CitO, materials. In this paper, based as low energy effective field theories [2,3]. The long-range behavior of the origi L on the one-loop analysis of the CP (or 5") nonlinear However, it is broadly viewd that in order to understand high-T£ supercon T^Neel temperature). TN decreases as the ralio(=or) of interlayer coupling over that within Cu — 0 layers becomes small. For the realistic case (o 2; ID"5) ductivity it will be necessary to elucidate fully the rich diagram of phases as a the phase diagram and the behavior of magnetization and the spin correlation function of temperature, doping, etc. Among them there is a three-dimensional length as functions of temperature are investigated in detail. The results show antiferromagenetically (Neel) ordered phase near the superconductive phase. Ex that our anisotropic field theory model could give a reasonably well description periments on La1_lSr(CuOi show that this phase has a remarkable feature that the Neel temperature TN below which long range Neel order (LRNO) is present, of the magnetic properties indicated by some experiments on La,CuOr upon doping, drops quickly to zero when 6 is changed by 0.01 ~ 0.03. Theo retically, to the Heisenberg model in space-dimensions d < 2 Mermin-Wagner's • Talk presented at KEK workshop (Tsukuba, November 1989). The work baaed on the collaboration with G. Tatara. 1. tchinose (Univ. of Tokyo, Komaba) and T. Matanl (Freie theorem [5] applies, and 50 the spontaneous break down or the appearance of long Univ. Berlin) range order cannot occur at any finite temperature. It is therefore believed that — 351 — -352- the antiferromagnetic ordering at finite temperature observed in experiments are ,vhere drived by a weak three-dimensional interplaner coupling between Cu02 layers. 0 = l/hBT, n = (»,(»), n,(*)), D„n = (d„ + iA„)n, A„ = ^T^n, In this paper, based on the CPl (or S") nonlinear a model, we give the field (2.2) theoretical analysis of some magnetic properties of quantum antiferroniagnets with the constraint |np = |n |! + |nj|2 = 1. Here the fields are periodic functions such as La CuO,. In Sect.2, using the auxiliary field method, we calculate in of the imaginary time T. d-space dimensions the one-loop effective potential of finite temperature CPl 2 The model is transformed through the relation: y> = n 2 summary and discussions. Z(0) = j D?D and the saddle-point approximation where the replacement Ccp< = -} \D.nW (2.1) ature Green's function we get an effective potential for ip3 (antiferromagnetic -353- -354- is regularized by the momentum cut-off A related with the lattice spacing(= a) long range Neel order) and trc (spin mass gap): of the original crystal model as Aa = constant. The cut-off dependence can be absorbed into the coupling constant renormalization:jV-IflK „/3<7,.( 2 d 2 A/ ,93 = T = 0) = —M ~ /ifJ{ (Af:a renormalization point, /„:a dimen- (2 6) -J'-^-^+jS/iS^+F^)- - sionless renormalized coupling) when d < 3. 3.1. d<2 (MERMIN-WAGNER-COLEMAN'S THEOREM) From this, the stationary-phase conditions for ac and ^3 follow easily On the other hand, if we put finite temperature, any solution with ac = 0 is absent in (2.10). This is the manifestation of Mermin-Wagner-Coleman's theorem [5], [7] . 0 = -r^ = <™, . (2.8) "V dip. Let us more explicitly see the d = 2 case. After integration and renormaliza tion eq.(2.10) becomes for d = 2 3. Long Range Order '-te-H-J-'Z-M1-^*)' (32) In this section we study the condition under which the nonzero (p)t (LRNO) exists as the solution of eqs. (2.7) and (2.8). From (2.8) we immediately see where for simplicity we have taken the infinite cut-off limit. In the absence that the presence of LRNO corresponds to that of observation is that in the third term on the right-hand side (r.h.s.) of (2.7), plays the role of infrared cut-off in the one-loop momentum integral. Using the !R = «"- Va, = "t = 0i (iii)for /„ < r, <4t = f (^ - $), standard trick, this term can be rewritten as the sum of contributions from zero ordered phase). Here fR = ir, or /(= /A) = /e = ir is the critical coupling. and finite temperature parts: LRNO exists in the weak coupling region (iii). d d+1 d However once we incorpolate the finite temperature effect, the last term of 1 ,=. f d V 1 f d k 1 f d k 1 3 u eq.(3.2) with at = 0 diverges logarithmically and gives no solution with -355- -356- thermal phase transition occurs exactly at T — 0. The value of yHr^the inverse where we have set /, = a-1/ (0 < a < 1). Evaluating the integrals gives of spin correlation length) at finite temperature is obtained universally by 0 2 =4(vi, ~h)+ A[A >1(«,7) + BK.A,or,7)] " _, "; * (3.5) + [g^+ where To conclude the d < 2 case, our effective theory with a mean-field type approximation shows no LRNO (no spontaneous breakdown of SU(N)) when A(a, 7) = 7' (v/o7 + 1 - JZPj) + or"' log (y/af + 1 + v^), d < 1, or d < 2 at any nonzero temperature. The result is consistent with the B(0, A, a, 7) = C(0, j9,00, o, 7) = D. Mermin-Wagner's rigorous theorem about the absence of long range order in the Here, we have introduced the two-dimensionally symmetric cut-off A in the fcj, k, Heisenberg spin model [5]. integral and A3 in the i3 integral. 3.2. d = 3 AND THE ANISOTROPIC CASE In the following we will restrict ourselves to the ire = 0 phase where LRNO could appear. In this case the term flfo-,., A, a, 7) vanishes, and we will neglect Now we are naturally led to suppose that the third space-dimension may be the term C(0, /J, A, a, 7) appearing in the finite-temperature part of the one-loop important for the real appearance of LRNO observed experimentally in some integral, which vanishes if we let A go to infinity. This approximation is valid undoped materials. We are then urged to extend our analyses to the d = 3 unless 0hcai A3 is very small, where c is the spin wave velocity and we have set case where the infrared divergence is absent in the one-loop integral (3.1). But he = 1 so far. Then renormalizing the remaining quadratic cut-off dependence before proceeding to the concrete analysis, we should recall that in real metarials by the condition: showing the superconductivity such as £a2Cu04 the interlayer coupling (= /,"') dV,, 1 between x - y planes (Cu — 0 layers) is known from the experiments to be -^r =-JT< (3J) very weak compared with that(= /"') within planes. Taking this fact into 4 0"c U,=v„=T=0 /fl consideration we extend our model to a following general anisotropic one: where fR is a dimensional renormalized coupling in x — y plane, we can write (3.5) as £ rio-L-BC* (3.8) = 5 ( L (fy>)' + f(M' + «tf - jf)) (3-3) Vt l " 2fR 6/J' ' For this equation to be meaningful fR needs to be positive. This means In this model the stationary-phase equation (2.10) is replaced by - • 'ir3 / < /. H — -, (3.9) n hi l,jf [A 1 ,, .> ^(0,7) 1 with / = /A . Note that fc has a finite limit if or goes to zero. When the above -357- -358- condition is satisfied, we find from (3.8) that LRNO exists if and only if 4. Application to Quantum Antiferromagnets 4. 1. HUBBARD-SlGMA MODEL T Below TN LRNO, in other words, the staggered magnetization is given by nearly half-filled band described by the following Hamiltonian: rF (3 11) ^ = ^y\/^ - - *-- £ '^i±(i|.C„ + f/£nx,»lt-/i£n„ (4.1) x,±ii,a x x,o To make our discussion simple, we have neglected the existence of C term in where Cj„, Ct„ are the creation and the annihilation operators of one-body eq.(3.5). Hence we note that the eqs.(3.10) and (3.11) are approximately good electron state localized at x, and a represents the third component of electron only when a is not very small. For the range a = 10"6 ~ 10~4 where the com spins s, taking up (f) or down (J). nzs s Cj„C„ is the number operator of parison with experiments is p issible, the C term in (3.4) may not be negligible. electrons which takes the values of zero or one. In (4.1) the first term on the r.h.s. More quantitative arguments near ar = 0 needs therefore more careful treatment is the hopping term of electrons, and the hopping amplitude i^ (/i = x, y, r to the anisotropy in the lattice spacing or = ay = a < a,. In this paper we will take o = 3.88A, the shortest Cu - Cu distance in the Cu — 0 planes and a. = 6.68A, the distance between the networks. The second term expresses the on-site Coulomb repulsion (£/ > 0) between two electrons on the same site. In the third term the chemical potential ;J is introduced to enforce the relation: < uit + ntl >= 1 - 6, 0 < 6 < 1. (4.2) -359- -360- where 6 is the doping parameter or the concentration of holes and measures the two-dimensional lattice constant a by Aa = \/5jr) is an "effective" Then the effective field theory describing the low energy spin dynamics of from the lattice action (4.1), we cannot expect that the relations between macro the above model at large If is derived" as the following continuum action: scopic and microscopic parameters are exactly exact. Instead we will think that the coupling constant of the CP1 model is a free parameter, and should be deter mined by the experimental data. After all there are three free parameters in our h 20-a3Ay J (43) theory, that is, the anisotropic parameter or, the antiferromagnetic Heisenberg exchange coupling J or the spin wave velocity he and the sigma model coupling /. They shall be chosen by the comparison with experiments. 2 2 with the constraint |»p = InJ + |n,| = 1. Here J„ = 4lj/(7 > 0 (Jx = Jy = Note that the model given by eq.(4.5) is essentialy equivalent to that consid J < J., antiferromagnetic Hcisenberg exchange coupling) and the factor \{S) ered in 3.2 if transformed to the form of 0(3) nonlinear 2 where A = («3/a) and o(= J./J) are the anisotropy parameters for the lattice spacing and the Heisenberg coupling. Ac = n(l)\/2 + aJa is called the spin wave •-•s^-'-v (4-7) velocity, as seen from the structure of derivatives. Rescaling the imaginary time r by c, the action (4.4) becomes the familiar As before, the third term on the right hand side of (4.6)can be rewritten as form of CPV model with anisotropic coupling. the sum of contributions from the zero (= /j) and the finite temperature (= I}) Sic parts 3 jtAcp, = j J dr Jd z[\D;nf + \D,jn\> + ^0^ + \DTn\>], (4.5) o where / = /A"3 (A is a two-dimensional momentum cut-oft" and is related to the (4.9) * For the detailed dlscussin of the derivation, aee refs (?].[3|, [10] '3 = /|0M^-i)l- — 361 — -362- ! where u = i; + k\ + aXk^ + ac. 1^ is evaluated analytically since the ^-field is normalized as ^3 set to zero. The region T > Ty is the disordered phase, where ipz = 0 + (A- + eq.(4.6)by setting ip} = 0. In this way, from the stationary-phase conditions where As = (oA)> A,. For /,, performing the two-dimensional integral explicitly we can calculate easily the magnetic quantities such as the Neel temperature, we obtain the correlation length and the magnetization of pure and doped La2CuOt. The correlation length was previously calculated in [4] by using a different approach in the framework of the two-dimensional ir-model, and we will see if our results 1 /, = -, f dx Pog (1 - e-y/t'MW) - log (l - e' )]. are consistent with theirs. (4.11) In this paper we will take a = 3.8SA, the shortest Cu - Cu distance in The remaining x-integral will be evaluated numerically to investigate the mag the Cu - O planes and a3 = 6.68A, the distance between the networks.We netic properties of the model based on (4.6)and (4.7).It is the subject of the next verified that the physical quantities like TN, £/a and the magnetization are not subsection. sensitive to the cutoff parameters, the lattice spacings. The three parameters in our model; /, Ac and a are determined by comparing our numerical results with 4.2. APPLICATION TO LaCuO AND ITS NUMERICAL SOLUTION l < the experimental data on pure La2CuOt. The neutron scattering experiment has been done on La^CuO^y [12] . Their best sample has the antiferromagnetic In this subsection, we will study the solutions of the stationary-phase condi- long-range order up to T„ = I95K. Based on these experimental data on the tions(eqs.(4.6)anJ (4.7)) with the help of numerical calculations, and investigate correlation length Chakravarty et. al. [4] have estimated the coupling constant the magnetic properties of LajCuOA in detail. of the 1 the vj-field so that the correlation length £ of the spins are given as 4 = v^o" - and determine a as a function of Ac by requiring Tv = 195K. Finally the spin wave velocity Ac is chosen so as to reproduce the experimental data on the Therefore, there is the long range Ne'el order in the phase -363- -364- Let us show them. 2.0 x 10"" for Tv = 195K) gives much smaller {"'(dotted line). The value of or, Phase Diagram We first observe how the Neel temperature depends on we found, is also in good agreement with the result of the previous analysis [14]. Let us check that our calculation contains the results of two-dimensional analysis two parameters Ac and a. The dependence of Tv on a is shown in Figs. 1 and 2. When a is zero, the Neel temperature is zero, as expected from the Mermin- in [4] by setting or = 0. Chakravarty et. al. obtained the correlation length by Wagner theorem. As a increases, the effect of the third dimension becomes the renormalization group approach, and we directly calculated the mass of the larger and T„ increases. As easily seen in these figures, or must be very small sa-field which is dynamically generated. It is easy to see that essentially these two 7 5 results are same. The correlation length with or = 0 (dashed line) differs from the (s= 10" ~ 10- ) in order to obtain small TN =: 195A'. The dashed line in Fig. 1 5 is an approximate expression for T„, which was derived by letting the cutoffs in result with a = 1.7 x 10" for T smaller than 300K. For the range of temperature the i-integral in eq.(4.1i) go to the infinities as done in ref.[6]. This is legitimate from 300K to 500K, these two are quite close, and both are consistent with the when or is not small, because the cutoff A, is proportional to a'. There is, experimental result. However, contrary to the naive expectation, there appears however, small but finite difference between this approximate expression and the difference between these two results at temperature higher than 500K. numerical result even near or ~ 1. Therefore, the numerical calculation is needed It should be also pointed out that smallness of a means very short correla especially for small values of a we are interested in. tion length in the third direction, as seen in the following way. The spin-spin correlation function is written as On the other hand, TN increases with Ac, because the temperature and Ac l appear in equation (4.6) in a form of /9Ac. Fig. 3 is the J~-TN phase diagram for some small values of a. Remenber that for small a, J is proportional to he. < „,(„„,(«,, > * Ej^H + H+ ^ +l4 + r, The lower region in this diagram is the Neel ordered phase. It can be seen that at the zero temperature, there is the Neel order for any value of coupling J of / d-kdk- =- , the antiferromagnetic Heisenberg model, and it is consistent with the results by ( J k\ + k\ + k\ + u,\ + length £. Fig. S shows our fits to the results of experiments [12] on La7CuOt. < Vi{!)Vi(0) >- e-v^W The Neel temperature of this sample is 195K. The best fit (solid line) is obtained for fie = 0.39eVA, a = 1.7 x lO-5. This value of Ac is the same with that obtained as |i| — oo. That is, the inter-layer correlation length ^ is Ac cannot reproduce the experimental result. For example, Ac = 0.6eVA(or is shows that <$3 is about 2A while £ a 200A at room temperature. -365- -366- Magnetization The calculated results of the magnitization M as a function there is no long-range Neel order for 6 > 0.16. However, in the experiments, the of temperature is shown in Fig. 6. It is zero at T and behaves as M oc Neel order disappears at about 6 = 0.03. JT^ — T near the Nee! temperature. This temperature dependence is consistent This fact suggests that the hole hopping gives important effects to the spin with the result of the molecular field theory. If the infinite cutoff limit is taken dynamics. In order to estimate these effects we have to treat the interactions of - in evaluating eq.(4.9), M is written analyticaly as M) *^^/{jT-, - T- , hole fermion and spin variables appearing in the full effective theory of Hubbard which is proportional to JT -T for T =: T . M takes its maximum value N v model more elaborately. at T = 0, and its maximum value is obtained from equation (4.6)as A/(T = 0) = Jl - 2/^/7/irn2 + 0[-Jo). Thus it is independent of a( and T ) for very N 5. Summary and Discussions small a. This expression also shows that the increase in / makes the system disordered. For / = 0.685/,., M(T = 0) = 0.57, which is close to 0.60 predicted Using the effective field theory approach we have studied the magnetic prop by the spin wave theory. The sample we refered to in Fig. 5 has magnetization erties of quantum antiferromagnet. The low energy spin dynamics of d- 0.35 at T = 0, which gives / = 0.89/c. However, the correlation length cannot dimensional Hubbard or the antiferromagnetic Heisenberg model is expected to be fit by using this / for any value of tic. This is because for large /, £~l rises be described by the (d + l)-dimensional CPl{^ 52) model (coupled to hole linearly as shown in dotted-dashed line in Fig. 5. Smaller / is also rejected by fermions). the same reason so that / must be about 0.69/ . Since the absolute value of the c We have first studied in S~ model under what condition the long range an magnetization is very sensitive to the defects and impurities of the sample, we tiferromagnetic (Neel) order (LRNO) appears, which is experimentally observed consider the value 0.57 is good as a theoretical prediction. But it is also possible in quantum antiferromagnets such as La,CaOt. We have adopted the one-loop that our approximation about the shape of the £o Cu0 crystal may contribute 3 4 stationary-phase method to study the issue. It is found in a weak ( -367- -368- related with the lattice constant in La^CuO^ we investigated the various mag References netic properties such as phase diagrames, spin correlation, magnetization. The [1] G. Shirane, Y. Endoh, R. J. Birgeneau, M. A. Kastner, Y. Hidaka, M. Oda, results are in a quantitatively good agreement with the experimental data on M. Suzuki andT. Murakami, Phys. Rev. Lett. 59 (1987) 1613 pure La^CuO, if the parameters are chosen to be [2] X. G. Wen and A. Zee, Phys. Rev. Lett. 61 (1988) 1025; F. D. M. Haldane, a — 1.7 x 10~5 (anisotropy) Phys. Rev. Lett. 61 (1988) 1029; E. Fradkin and M. Stone, Phys. Rev. he — 0.39eVA (spin wave velocity) B38 (1988) 7215. I / = 0.685/. (ir-model coupling) [3] I. Ichinose and T. Matsui, Tokyo preprints, UT-Komaba 89-8 and 89-17. [i] S. Chakravarty, B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. 60 To conclude, our (3+l)-dimensional anisotropic nonlinear ir-model seems to be (1988) 1057; Phys. Rev. B30 (1989) 2344. able to give a good description of the magnetic properties of both the Neel ordered [5] N. D. Mermin, Phys. Rev. 176 (1968) 250; N. D. Mermin and H. Wagner, (below Tv) and disordered (above TN) phases in the range of approximation adopted in this paper. Phys. Rev. Lett. 17 (1966) 1133. -1 Our result of tic is smaller than that estimated by various experiments. In our [6] I. Ichinose and H. Yamamoto, "Finite Temparature CP" Model and Held theoretical approach it would also be possible to taken systematically the Long Range Neel Order INS preprint, INS-Rep.-768/UT-Komaba 89-19. higher-loop corrections into our analysis. They would change the above values [7] S. Coleman, Commun. Math. Phys. 31 (1973) 259. for parameters to fit the results to the experiments. [8] I. Ya. Arefeva, Anna]. Phys. 117 (1979) 393. In the doped case the T„ — 8 diagram obtained by the simplest approxi [9] I. Ya. Arefeva and S. I. Azakov, Nucl. Phys. 162 (1980) 298. mation to take the effect of hole-dynamics to spin dynamics, indicates thai cur treatment of hole-doping enVt js still incomplete to explain the experimental [10] H. Yamamoto, G, Tatara, I. Ichinose and T. Matsui "Magnetic Proper results. However the tesult definitely shows that the presence of holes works ties of Three-Dimensional Hubbard-Sigma Model" INS preprint INS-Rep.- towards disordering the spin system, and that it is necessary to consider the hole 788/UT-Komaba89-24. hopping to explain the experiments. This observation may give us an interesting [11] R. Shanker, Phys. Rev. Lett. 63 (1989) 203, and preprint NSF-ITP-89- suggestion for a mechanism of high-T,. superconductivity. 116. [12] Y. Endoh, K. Yamada, R. J. Birgeneau, D. R. Gabbe, H. P. Jensen, M. a. Kastner, C. J. Peters, P. J. Picone, T. R. Thurston, J. M. Tranquada, G. Shirane, Y. Hidaka, M. Oda, Y. Enomoto, M. Suzuki, and T. Murakami, Phys. Rev. B 37 (1988), 7443 [13] E. Manousakis and R. Salvador, Phys. Rev. Lett. 60 (1988), 840 -369- -370- [14] T. Thio, T. R. Thurston, N. W. Preyer, M. A. Kastner, H. P. Jenssen, D. Fig. 1 R. Gabbe , C.Y. Chen, R. J. Birgeneau, and A. Aharony, Phys. Rev. B38 (1988), 905 FIGURE CAPTIONS 1. T"„-or diagram with fixed J. Solid lines are numerical results for J = 850A' and J = 1300A'. The dotted line is the result with J = 850A' of an approximate equation valid for not small a. f is 0.685/c 2. Ty-a diagram in Fig. 1 in the small a region. l s 7 3. J~-TN diagram for a = 1.7 x 10" and 2.0 x 10" . The lower and the upper region is Neel ordered and disordered region, respectively. •1. The relation between Ac and a for fixed TN; Ty = 195K and 245K l 5. The inverse correlation length (~ versus temperature. Using / = 0.685/,. Fig. 2 with TN fixed at 19SK, the best fitfsolid line) is obtained for Ac = 0.39eVA. The result with Ac = 0.6eVA is shown by a dotted line. Dashed line is the result with a — 0 (Ac = 0.39eVA) using the same /. Dotted-dashed line is the best fit using / = 0.89/c with TN fixed at 195K (Ac is 0.64eVA). The experimental data are taken from [12]. J=1300K 6. Antiferro magnetization versus temperature for two values of /; / = 0.685/c and / = 0.89/,.. Ac is 0.39 and 0.64, respectivelly. TN is 195K. 7. TN-S phase diagram. / = 0.685/c, Ac = 0.39eVA. The lower region is the Neel ordered region. —r- 10" 10" 10' 10- 4 -371- -372- 1 ^v*-- I * - o o *i»-2H IO h' bO, • t-i PK o o CM \ • i i co CM o o o o © o o o o o I 7 1 o . o CM .1 u sS^T*^ // 0. 6 / / II / / / w. / E-i *D / o • / hO r-o< •f! fa fa O. S / " 1 / i 1 in o % "String Amplitudes": What Can We Do about the Divergent Integrals? been assumed all along in the process which reduced the amplitude to the form (2.1). The inadequacy of (2.1) as an oftshell amplitude shows in the fact that it Kaoru Amano is modular invariant only if Department of Physics, Tokyo Institute of Technology Meguro-ku, Tokyo, JAPAN J2 kkj = 0. (2.2) t The rules for perturbative construction of scattering amplitudes ' constitute the The modular invariance is a consequence of the reparametrization invariance of most well-established part of string theory. They lead to neat integral expressions the string world sheet and the amplitude should possess this invariance if the that represent the amplitude as the sum over the geometry of the string world derivation has been consitent. The condition (2.2) is satisfied in any onshell kinematics. sheet. However, quite unfortunately, if we really try to evaluate the expression we have to go a long way from there, for the integral does not represent the The integral (2.1) almost always diverges. To be precise we have amplitude in any direct way. This is the problem I'd like to discuss in this note. I will first show that the traditional integrals that purport to be string amplitudes Theorem. For real |resp. complex] kj • kj (1 < t < ;' < 4) satisfying' (2.2), the are divergent, and then consider a remedy. ' integral (2.1) converges [resp. absolutely] if and only if 2 h -kj = 0 [resp. Re ki- kj = 0 ] for all i < ;'. (2.3) Here is an example of the convetional expression for a string amplitude. Note how stringent the restriction (2.3) is. Onshell kj satisfy this conditition only when they are pararell to each other. I'm not going to prove the theorem /(i^yrfrr /*] n «*•* <"> here. Instead I give a plausibilty argument. We choose the following fundamental region FQ tor T: This integral appears in the massiess four-point amplitude for type-II super string in the one-loop order. The indices i and ;' distinguish external particle states. Fa = jreC |lmr>0, |Re r\ < |, |T| > l}. (2.4) State j carries an incoming momentum of kj attached on the loop at the po sition VJ. The string loop forms torus T(r) which is identified here with the Then represent the Xii'k'tj in ">e fonn parallerogram on complex plane with corners at 0,1, r and r+1, where the com Xiji^i - e*V*itawU-f»>{l _ 1*.. .t £ _ + , .) + ...}, (2.5) plex Teichmiiller parameter r runs over T, a fundamental region of the modular k f + 2ui c c group. The Xij are the exponentiated corelations on the torus, and they take on 2 real value. Another point to note, a very important one here, is that (2.1) is an where p = e ''" and V = VJ- i/j or v; - Vj depending on whether Imi>;- > Imi/; or onshell expression. All the momenta kj must satisfy kj = 0 because this has not. We use this to expand the integrand in (2.1) and carry out the integration -377- -378- term by term over one of i/j, say, v\ and over the real part of r and of the 3 rest of UJ. Then the integral looks like a Feynman integaral in the Euclidean Obviously divergent amplitude is not acceptable in a case like the one-loop four- version, i.e. Wick-roted one, or the sum of such integrals. And the sum includes point amplitude. Although the origin of the divergences seems to relate to phys the integrals that would be constructed with massless particle propagators, as it ical singularities, the integral fails to give imaginary part required for unitarity should since the string spectrum contains massiess states. Then let us recall once again that the external lines are on the mass shell. The kinematics is generally but only gives infinities which we don't want. Now what can we do about it? over the threshold for physical intermediate states and therefore the Euclidean Respecting the observation that the integral representation corresponds to integral only gives meaningless divergences. (The argument is not affected by the the Euclidean Feyiiman integral, we may modify the integral in the following "ultraviolet cutoff" that (2.4) dictates.) In sum the integral diverges because it way: is basically a Euclidean version Feynman integral and contains massless internal 1. Postulate an offshell integral for the Euclidean region E = {k | Reft; + states, while the external momenta are kept on-shell. %)2 SO /<"" a" • < i }• The integral should be chosen such that it defines Thus we expect divergence from very general observations. Clearly the con an analytic function l(k) on a region D C E. ventional integral representations for string amplitudes converge only in excep 2. Then continue Z(fc) analytically from D to the region F = {k \ lm{k{ + tional cases, like tree-level tachyon amplitudes in bosonic strings, or the massless 1 kj) < 0 for all i < j }. loop amplitude in superstrings with no more than three external states (for which 3. And define the physical amplitude by the limit value of l(k). the integrand vanishes identically). The firststep takes guesswork but it must be easier and safer than to directly Noii. The integrals for the massless amplitudes are generally divergent guess at the physical amplitude. The integral should take on a form very much even at tree order. Does this not contradict the fact that finite expres like (2.1), and, if we guessed right, steps 2 and 3, which are the Wick rotation sions do exist in the literature for such amplitudes? The authors may in reverse, would take us to the correct Minkowskian expression. To give a few have a different idea, but I imagine the expressions are obtained by di examples, first consider (2.1) as an offshell expression for step one. The expression viding up the integration region into pieces, evaluating the integration depends on the choice of T and so let us name the integral 2>. Or we may avoid over each piece in different kinematic regions where they converge, and the ^dependence by modifying the integrand in such a way that makes the ,!ik then putting together the results (actually their analytic continuations). integral modular invariant. One way is to insert a factor of (ImT)~i< j/*) and we obtain an integral £„„. The integrals %„„ and Z> define different analytical functions. However, they all lead to the same physical amplitude as far as I know — that is, at least below the poles for massive intermediate states, beyond which I did not perform analytic continuation, and if we take T to be F0 or its image under a modular transformation. Note that although we have used offshell extensions of the same on-shell integrand it is far from obvious whether -379- -380- they define a unique amplitude when continued to the shell after integration. amplitude. In fact, if we take T = Fa and choose a negative To promote the above prescription to a proper derivation, we need to clarify References what class of offshell extensions are acceptable for the use in the Euclidean region and to provide some stringy reason for that. For the time being it may be better to 1. J.H.Schwarz, Phys. Reports 89 (1982) 223; look at the previous treatment as an interpretation of the analytical continuation E.D'Hoker and D.H.Phong, Rev. Mod. Phys. vol. 60 (1988) 917 often talked about in a more casual way in the texts on string amplitude. The 2. S.Weinberg, Phys. Lett. 156B (1985) 309 moral we obtained here is then that such continuation is potentially ambiguous since it involves oflshell extension. I would like to stress that the divergence we 3. N.Marcus, Phys. Lett. 219B (1989) 265 discussed here and hence ambiguities it entails are a universal phenomenon in 4. K.Amano, Nucl. Phys. B328 (1989) 510 (to appear) string in the conventional formulation. Without controlling them in a consistent 5. M.Green and J.H.Schwarz, Phys. Lett. 109B (1982) 444 manner, we cannot really talk about string theory. Note. It is possible to give a geometric interpretation to offshell exten sions linu and If. The idea is that we can define an ofishell extension retaining the traditional sum-over-the-world-sheet point of view by in serting vertex functions of the usual form but with oflshell momenta. The only trouble is that part of the Weyl anomalies from the vertices remain uncancelled and therefore the extension depends on the choice of the representative world sheet (metric) we associate to each point in the moduli space. In the simplest cases, such anomalies will only come from e1*''* in the vertices, so the use of the metric e^iviv' will mean that the factor e"'*i"">'r> is to be inserted in (2.1) for each external line. Thus the extension T? corresponds to the choice g = iviv' and Zfni/ to j = r^f iviv'. (This has been my interpretation of a suggestion by P. Nelson.) We should also note however that the class of offshell extensions thus obtained is not exclusive enough for the definition of the — 38] — -382- 1 Introduction Necessity of Finite Size Term in WZW model Given a commuatarion relation and a Hamilton!an,w* can find its corresponding path integral —coherent-state path integral approach— by repeatedly inserting some partition of unity (i.e. a complete set of states) into a transition amplitude. The states of the complete set must be parametrized by a continuous parameter Satoshi Iso unless we cannot operate differentiation or integration to the Lagrangian. Coherent states Department of Physics, University of Tokyo, provide such a complete set. The method of coherent state path integpal can be applied to any Bunkyo-ku, Tokyo 113, Japan group and has many physical applications [1][2]. For example spin coherent states, coherent states far SU(2) group, were elegantly used to prove the equivalence of spin s antiferromagnetic Chigak [to! Heisenberg spin chain and (1+1) dimensional 0(3) tr model with & term(fl = 2JT«)[3J. It was Department oj Physics, College of Science and Technology also used to rewrite a path integral of a bosonic particle with a tortion term to that of a Nihon University spinning particle[4]. Kanda Surugadai, CMyoiia-fcu, Tokyo 101, Japan Recently Stone derived the coherent state path integral for the current algebra of a system and of chiral fermions (i.e. level 1 Kac-Moody algebra)[5]. He explicitly used the Fock space of the fermions. In this papacr we generalize his work to any representation of level k Kac-Moody Hlsamitsu Mukalda algebra without using the language of fermlons. Derpertment of Physics, Tokyo Institute of Technology, In section 2 we briefly review coherent state path integral of a compact Lie algebra and Oh-okayama Sfeguroku, Tokyo 1SS, Japan show how to derive commutation relations far Lie algebra generators. In section 3 we apply Abstract the coherent state path integral approach to the Kac-Moody algebra. By using a coherent state path inetgral method, we construct an action which reproduces the Kac-Moody algebra of level k. We show that it Is the light cone WZW action with a finite 2 Compact Lie Groups size correction term. The current operators also contain a finite size correction term and are In this section we briefly review the coherent state path integral for a compact Lie group G. For shown to satisfy the original commutation relations. The finite term is necessary to derive details see, for example, Porelomov[6] or Stone[7,5]. Let T be an irreducible representation and the exact phase space and also to understand the constraint 0 < 1J < k (for SU(2)) in terms |0 > be the highest weight state. We now define a coherent state \g > by \g >= g\Q > where of the path integral language. g S G. (Precicely g is a representation matrix.) This set of coherent states is overcomplete -384- but has the partition of unity tations of the Lie algebra generators 71, are defined by \=Jdg\gXg\ (2.1) t r(g)= transition amplitude we obtain its path integral representation: ^r"(9) = i[H{g),r(g)] (2.6) < 9/\c~iliT\9i >= jDg exp{ijdt < 0\ig-lg~g-lSg\0 >). (2.2) where H{g) —< g\H\g >. The left hand side can be calculated directly by using the equation of motion. The ambiguity disappears and Since the action is first order in time derivatives, it is a phase space path integral. The symplectic structure is given by the first term of the action. If the path i € [0, T] -» g(t) 6 G ^r{g) = i k [T'(g),rig)] = ifiihT lg). (2.8) i I < 0|r'4/l0 >= ~i [ < Q\(g-ldg)7\Q >= / w. (2.3) JC=BS JS JS The action is invariant under transformations g{t) — g(t)h[t) where h(t) leaves |0 > fixed up 3 Loop Groups to phase. (We call a set of these elements H the isotropy group of |0 >.) Therefore the exact In this section we proceed the same calculation for Kae-Moody algebra. Stone[5] constructed phase space is reduced to GfH, H is usually the maximal torus. (For special representations coherent state path integral for a singlet representation of level 1 Kac-Moody algebra by using it is larger.) the description of the representation in terms of chiral ferrnions. We will generalise his result This action reproduces the original commutation relations. If the path integral were to any representation of any level Kac-Moody algebra. We show that coherent state path denned on G/ff, the symplectic structure w had an inverse and we could calculate the integral reproduces the well-known light-cone WZW action with a finite sine correction term. Poisson brackets. But in this case the action has gauge in variance and the above method In this case coherent stales are labelled by elements {9(1)) of (he loop group LG [8]: cannot be applied. Then we use another method, following Stone[5]. Equation of motion of the action (2.2) is not unique and has an ambiguous factor A: ?:i6S'« g{x) = e"("> 6 G. (3.1) iff"1?- g-'Hg = A(0 (2.4) Coherent states are defined for each configuration {g{x)} by where \(l) is any element of the Lie algebra of isotropy group H of {0 >, Bosonic represen* |(s(i)}>=eJ"")|0> (3.2) -385- -386- where |D > is the highest weight state for Kac-Moody algebra and The second term looks unfamiliar but after integration by parts it can be written as 1 , i k 1 XM = i/iirfJ(iM4 (3-3) Swzw = ^{Jfcti trg- a,gg-d,S + \Jdzdtdst' trg-'digg- dsgg-'aig}. (3.10) b oA (The normalization of trace is trT*T = M5 . ) By repeatedly insertig the partition of unity 11 is just the WZW action in the lightcone cordinate at the fixed point [10], What is really a transition amplitude can be written in terms of path integral as in the case of a compact unfamiliar is the first term. For J representation of SU(2) Kac-Moody algebra, I.e. Lie group: < o| j(»)|o >=< o| 52 jjy"""-r\o >= J-T; (3.11) <{si)\>-""\M>=jD{g}e's, (3-4) 5 = Jdt« {j(x))|^|Mx)} > - < {g[x))\H\{g[z)) >) = Si + S„. (3.5) the first term (a finite size correction term) is where d{g] is the Haar measure on LG. The first term 5D gives the symplectic structure on (&)/.„... = jJdzdHg->Bl9y. (3.12) LG. Calucuiation of So is straightforward, From the Kac-Moody algebra It seems negligible when L —• oo but this term plays an important iole to derive the exact k k [•/'(*), J'M] = if' J (*M* - if) - ±kPS'{x - »), (3.6) phase spase, Without the Bnite size correction term (J = 0) the action SQ is invariant under the commutation relations for X[u) are given by global gauge transformations g(x,t)->g{X,t)h(t) (3.13) [X[A), X(B)] = X([A, A]) - j^7 / trMB. (3.7) where h(t) is an x-inde pen dent element of G. This invariance originates from the isotropy Mow by using the above algebra we obtain group of the highest weight state |0 > for J=0 representation. The phase apace is LG/G. <{g)Hg + sg)> = < o|e-Jfejr+Jjr|o > However if J ^ 0 the transformation leaves (Sb)/m.i» invariant only when h(i) is an element = < 0|1 + /'di(6X + s[SX,X] + '-[[6X,X),X] + •• -)|0 > +0(6X') JO 2! of the isotropy group H of |0 > for J $ 0 representation. Therefore the phase space is now re | = l + ^<0|/d*tr./(x)j- 15«|0> duced to LG/H. (For J £ 0 representation of SU(2) Kac-Moody algebra it is LSU{2)/U(\),) SNJ'1' l! d' ir3''SSd-i!''9,g). (3.8) Next we calculate the boson representations for currents. It is easy to show that it also contains a finite size correction term: Here we extend g{x,t) 109(1,1,1), 3 S [0,1] such that 9(1, t, D) = 1 and J(I, (, 1) = 3(1,1). Therefore So can be written explicitly as [9] ./'(*) = , So 2 [So)f,n,u + SVzw = j < 0|(r(!./(I)ir'r)|0 > -jL(r(fl,M- 7"). (3.14) = jjJdxdt +-^~ Jdzdtditrg-'B,gB,(S-'B,g). (3.9) size correction term while the second is the familiar term. The algebra for these bosonic -387- -388- representations of currents can be calculated with the same method described in the case of If we set A(y) = 5(y — x)T* the Hamiltonian becomes J'[x) and we reproduce the original a compact Lie group. We set commutation relations for currents (3.G). B = ~fdxtr{J{x)A{x)) (3.15) Finally we want to make a comment on the finite size correction term. According to the representation theory there are restrictions for representation J and level A.(for SU(2) and denote a sum of (S0)fi„iu and (Sff)/„,, d,J[x) + i[A{x), J{x)] = ~dtA. (3.17) (S0);inile = j$ JdxdtdsiT'lg-'d^g-^g]). (3.22) Next if we demand the equation of motion for only the finite size correction term, that is, M By changing the variable g(x,t,a) — g{x,t,a)h(x) where h(x) = exp(i^-xT )t the action SSt,n,1e = 0 we have the same equation except the range of the ambiguity: changes by g~ld,g + ig~*Ag = A(i,() € Lie algebra of H (3.18) SS° = IZN /^^(T-lg-'a^s-'B.g]). (3.23) From eq.(3.16) and (3.18) if we restrict the ambiguity A to an x-independent element of the (Note that the transformation is singular at s=0.) Since the above transformation shifts J by Lie algebra of the isotropy group H, g(xft) satisfies the equation of motion for the whole k/2, two theories of J and J + \% are equivalent and we can restrict 2J between 0 and k. action. Note that we can solve the equation of motion because the bosonic representation of the currents also contain the finite size correction tprm. 4 Conclusion Now we apply the same method used in the case of a compact Lie algebra to calculate the commutation relations for the currents. The Hamiltonian equation of motion for the currents We constructed a coherent state path integral for any representation of any le .'el Kac Moody are algebra and showed that the action is the light cone WZW mode) with a finite size corection £-/'(»)-W^W (3.19) term. The current operators also contain a finite size correction term and were shown to While the left hand aide can be calculated directly by using the above equation of motion. satisfy the original Kac-Moody algebra. These finite size correction terms are necesarry to derive the exact phase space. We also showed how the restriction of 0 < 27 < it can be ^J'(V) = •$ -389- -390- One of us(5.I) thanks H.Murayama and M.Fukuma for fruitful discussions and their en couragement. C.I. is grateful to C.Iso, N.Sakai and the Tokyo Instilete of Technologyfor the kind hospitality extended to him. We would like to thank T.Fujita for reading the manuscript. References [1] J.R-KIauder and B.S.Skagerstam, Coherent States, Applications in physics and mathe matical physics (World SienliHc, Singapore, 1985) [2] P.B.Wiegman, Nuc.Phys. B323(1989) 311 [3] E.Fradkin and M.Stone, ILL-TH-88-12(I988) [4] A.M.Polyakov, Mod.Phy.Lett„A3(1988)325, S.Iso.C.Itoi and H.Mukaida, TIT/HEP- 152.NUP-A-89-12 (to be published in Phys.Lett.B) [5] M.Stone, Nuc.Phys. B327(1989) 399 [6] A.Perclomov, Com.Math.Phys. 26(1972)222, Sov.Phys.Usp. 20(9),sept 1977,703 [7] M.Stone, Nuc.Phy.B314(1989) 557 [8] A.Preseley and G.Segal, Loop Groups (Clarendon Press.Oxford 19B6) [9] Alekscev and Shatashvill also derived the same action by using the coadjoint orbit method. A.Alekseev and S.Shatashvili, Nuc.Phys. B323(1989)719 [10] E.Witten, Coin.Malh.Phy».92( 1954)455 [11] See for example, P.Ginsbarg HUtp-88/AD54 Applied Conformal Field Theory Quantization of k can be also derived from the associativity of Kac-Moody algebra. It.Murayama, Zelt.f.Phy.C 42(1989)397 -391-