Volume 15 1B, number 1 PHYSICS LETTERS 31 January 1985
SUPERCONFORMAL INVARIANCE IN TWO DIMENSIONS AND THE TRICRITICAL ISING MODEL
Daniel FRIEDAN, Zongan QIU and Stephen SHENKER Enrico Fermi and James Franck Institutes and Department of Physics, University of Chicago, Chicago, IL 60637, USA
Received 25 October 1984
We discuss the realization of superconformal invariance in two dimensional quantum field theory. The Hilbert space of a superconformal theory splits into two sectors; one a representation of the Neveu-Schwarz algebra, the other of the Ramond algebra. We introduce the spin fields which intertwine the two sectors and correspond to the irreducible representations of the Ramond algebra. We give the determinant formula for the Ramond algebra and the discrete list of possible unitary repre- sentations. We have previously noted that the Z2 even sector of the tricritical Ising model is a representation of the Neveu- Schwarz algebra. Here we complete the picture by showing that the Z2 odd sector forms a representation of the Ramond algebra. This system is the first experimentally realizable supersymmetric field theory.
Superconformal field theories are the supersymmet- sentations of the Neveu-Schwarz algebra and thus the ric generalizations of conformally invariant field theo- possible dimensions of superfields in two dimensions. ries. Every supersymmetric local field theory is supercon- We pointed out that the simplest nontrivial examples formally invariant at short distances, so the realizations in the discrete list corresponded to the Z 2 invariant sec- of superconformal invariance completely determine tor of the tricritical Ising model, which describes generic the possible supersymmetric field theories. In statistical tricritical phenomena in two dimensions [5]. The super- mechanics superconformal field theories describe spe- symmetry of the corresponding conformal field theory cial critical points at which the macroscopic physics is was also noted by Zamolodchikov [6]. supersymmetric. In this letter we point out that a superconformal In two dimensions the superconformal algebra has field theory has a richer structure than given by its infinitely many generators. The structure of the algebra superfields alone. The Hilbert space of the theory also is rigid enough to place significant restrictions on the contains irreducible representations of the Ramond al- field theories in which it is realized, extending the re- gebra, corresponding to conformai fields distinct from strictions imposed by the ordinary infinite dimensional the superfields and in fact nonlocal (i.e., double-valued) conformal algebra, the Virasoro algebra [1,2]. with respect to the fermionic parts of the superfields. The first examples of supersymmetry appeared in We call these the spin fields. The Z 2 odd operators of string theory [3,4]. In fact, they were examples of two the tricritical Ising model are spin fields. dimensional superconformal invariance - theories of The fermion parity (-1) F is multiplicatively con- free massless superfields on the world-surface of the served so we can project on the sector of even fermion string. The two superconformal algebras, the Ramond number. This selects the bosonic parts of the superfields [3] and Neveu-Schwarz [4] algebras, act as gauge sym- and a subset of the spin fields. The result is a new local metries of the string. bosonic field theory. We call this the spin model corre- In ordinary conformal field theory the scaling dimen- sponding to the original superconformal theory. The sions of fields are determined by the unitary representa- construction of the spin model from a superconformal tions of the Virasoro algebra [2]. In superconformal field theory is a direct generalization of the construction field theory the superfields correspond to irreducible of the superstring from the free fields of the Ramond- representations of the Neveu-Schwarz algebra. In ref. Neveu-Schwarz model [3,4,7], which is based on the [2] we gave the discrete list of possible unitary repre- construction of the critical Ising model from the mass- less Majorana fermion. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. 37 (North-Holland Physics Publishing Division) Volume 151B, number 1 PHYSICS LETTERS 31 January 1985
The infinite superconformal algebra is generated by Virasoro algebra is c = 3d/2. A single free super field has the super stress-energy tensor T(z) = T F (z) + OTB (z) = 1. It consists of a free scalar field with c = 1 and a where z = (z, 0) is a complex super coordinate, T B (z) is Majorana fermion with c = 1/2. the ordinary stress energy tensor (spin = dimension = 2), A superfield ¢(z) = ¢0(z) + 0¢l(Z ) is a conformal and T F (z) is its fermionic superpartner (spin = dimen- superfield if it obeys the operator product expansion sion = 3/2). We omit discussion of the ~ dependence -2 1 since it is equivalent to the z dependence [1,2]. The T(Zl)¢(z2)~hO12Zl2 ~(z2)+ 2 z]-21 D2~b+ 012z]-21+ ~2~b. correlation functions of T F (z) can be double-valued (4) because its spin is half-integral. Thus the Hilbert space This is equivalent to the commutation relations of the theory divides into two subspaces: the Neveu- [To, q~(z)] : 08¢ +4 (Dv) D4~ + hOv) ¢, (5) Schwarz subspace on which T F (exp (2hi)z) = T F (z), and the Ramond subspace on which T F (exp (27ri)z) = where - rF(Z). The natural operator representation of superconfor- T o =2~,fdz dOu(z) T(z) (6) mal field theory comes from the radial quantization. is the generator of the infinitesimal superconformal We write z = exp (r + ia) and take 7- to be euclidean transformation 6 0 = Do/2, 6z = u - 06 0. The complete time, a to be periodic space. The complex super coor- algebra of superfields is generated by products of the dinate (log z, z-l/20) describes a cylindrical superspace. super stress-energy tensor with the conformal super- The super stress-energy tensor can be expanded in fields. operator Fourier coefficients In radial quantization the dilation operatorL 0 acts as hamiltonian. The L n and Gn are lowering operators TF(Z ) + OTB(Z ) = ~ z-n-3/2TF, n + Oz-n-ZTB, n. n when n > 0 and raising operators when n < 0. A state (1) Ih) of energy h is called aground state if it is killed by all the lowering operators. The raising operators acting We will use the more conventional notation Gn = 2 TF,n, on a ground state generate an irreducible representation L n = TB, n. In the Ramond sector the indices of the Gn of the algebra. The eigenspace L 0 = h + n is called leveln. are integers; in the Neveu-Schwarz sector they are The vacuum 10) is the ground state of lowest energy half-integers. The Fourier coefficients satisfy h = 0. It belongs to the Neveu-Schwarz sector. It is invari- L ?n = L_n, G?n = G_n, ant under the global superconformal group OSP (211), i.e. it is annihilated by the five generators L_ 1, L 0, [Lm,Lnl =(m -n)L m +n + ~1 C^ (m 3-m)~m+ n, L1, G1/2 and G1/2 . The conformal superfields of the theory are in one to one correspondence with the [Gm, Gn]+ = 2Lm+n +21 d(m2 -'~)6m+n,x ground states of the Neveu-Schwarz algebra, the super- field ¢(z, 0) of dimension h being associated with the [Lm, Gn ] = (4 m -n) Gm+ n . (2) ground state [h) = q~(0,0) 10) of energy h. The relation between ground states of the Ramond The hermiticity conditions follow from reality of the algebra and conformal fields is more complicated. G O super stress-energy tensor. The (anti-)commutation re- commutes with L0, so it acts on the ground states, lations are equivalent to the operator product expansion which therefore come in orthogonal pairs [h +) and T(z 1)T(z2) -- ¼ d zi 3 I h- ) = G O I h +). By the general principle that ground states for the Virasoro algebra are associated with ordi- +(2O12z~-] +21-1 z12 D 2 + 012z~ 1 02) T(z2) , (3) nary conformal fields [ 1,2], the Ramond ground states I h ± ) are created from the vacuum by spin fields 0 2 (z) where z12 =z 1 -z 2 -0102, 012 = 01-02, and which are ordinary conformal fields of dimension h: D=O 0 + 0~) z. I h ±) = O 2 (0) 10). From the action of T F (z) on the The coefficient 0 is a number which characterizes the Ramond ground states we find the operator product operator implementation of superconformal invariance in a particular theory. The usual central charge of the TF(Z)O±(w) 2' (z - w)- 3/2 as O* (w), (7)
38 Volume 151B, number 1 PHYSICS LETTERS 31 January 1985 wherea+ = 1,a =h - ~/16. Thus TF(Z ) is double- stress-energy tensor is T = DXu D2XU/2, and the oper- valued with respect to the spin fields. In fact, all of ator product (3), with 0 = D, can be verified using the the fermionic parts of the superfields are double-valued two point function (XU(zl) Xv(z2)) = 6 u In z12. For with respect to the spin fields, and all of the bosonic a single free superfield, the Majorana fermion ~b is equiv- parts are single-valued. Ramond boundary conditions alent to the Ising model at its critical point. The ground can be regarded as due to a cut connecting spin fields state energy with Ramond boundary conditions is 1/16. at z = 0 and oo. The operator product (7) is equivalent The spin fields are the dimension i/16 order and disor- to the relations der operators of the Ising model. For arbitrary D the spin fields are products ofD Ising order or disorder K?_n - 1/20+(1) +- iO-+(1) K n = a+O~(1), (8) operators and have dimension h = D~ 16 = ~/16. They where form the fermionic vertex constructed by Corrigan, Goddard and Olive [9], who wrote the relations (8) 1 f dzzn+1/2(l_z)l/22TF(Z) for the degenerate case h = 0/16. The connection be- Kn - 27ri Izl 39 Volume 151B, number 1 PHYSICS LETTERS 31 January 1985 tured in two by two block form. The superfields of The Neveu-Schwarz representations are given by the the theory are block diagonal, taking the Neveu- hp, q with p-q even [2] ; the Ramond representations Schwarz sector to the Neveu-Schwarz sector and the by the hp, q with p-q odd. Ramond sector to the Ramond sector. The spin fields Each unitary field theory with d < 1 must be built are block off-diagonal. The dimensions of the super- from the possible representations on the list for some fields are given by the ground state energies in the fixed rh. The striking feature of the list (11) is that the Neveu-Schwarz sector, since they make Neveu- possible representations of the two superconformal al- Schwarz states from the vacuum. The dimensions of gebras occur at the same values of 0. This strongly in- the spin fields are given by the ground state energies dicates the existence of a discrete series of supercon- in the Ramond sector, since the spin fields make formal models in the sense described above. Ramond states from the Neveu-Schwarz vacuum. The For th odd, ~/16 is not an allowed ground state ener- complete superconformal field theory is not local, gy, so the supersymmetry of the periodic model on the since the fermionic operators are double-valued with cylinder is broken. When rh is even, hrh/2 ' rh/2+l = ~/16, respect to the spin fields. There are two ways to pro- so the supersymmetry on the cylinder is unbroken. ject onto a local field theory. The first is to restrict to The constraint of unitarity applies level by level, the Neveu-Schwarz sector, giving the usual algebra of since the eigenspaces ofL 0 are orthogonal The nth superfields. The second is to restrict to the I" = 1 sec- level of an irreducible representation is spanned by the tor, giving the spin model. The complete superconfor- vectors mal theory can be reconstructed from the superfields by representing a pair of spin fields as endpoints of a G_mlG_m2... L_nlL_n2 ... Ih), cut in the plane across which the fermionic fields 0 40 Volume 151B, number 1 PHYSICS LETTERS 31 January 1985 with the tricritical Ising model and KadanoIt |lb I con- tkPNs(k) = H (l+tk-1/2)/(1-tk). (15) firmed the identification using correlation functions k=0 k=l calculated by himself and Nienhuis [16]. The tricritical We arrived at the determinant formula for the Ramond Ising model [5] is an Ising model with annealed vacan- algebra by numerical calculations: cies. It has a tricritical point at a certain temperature and vacancy chemical potential. The model is invariant det(M~) = 1, det(M~) = h - ~/16, under the Z 2 symmetry which flips both the order operators, the Ising spins, and the disorder operators det (M'~n) = det (Mn ) = (h - ~/16) PR (n)/2 [17]. We pointed out in ref. [2] that the Z 2 even sec- tor is described by representations of the Neveu- X H[h h (~)] PR(n-pq/2) forn>0, (16) - p,q Schwarz algebra. Here we complete the picture by where the product runs over positive integers p, q with pointing out that the Z 2 odd sector is described by pq/2 <~ n and p-q odd. PR (k) is half the dimension of representations of the Ramond algebra. Thus the tri- the kth level: critical Ising model is a superconformal field theory. The ground state energies allowed by unitarity and tkPR (k) : H (1 + tk)/(1 - tk). (17) ordinary conformal invariance are [2] h = 0, 3/80, k =0 k=l 1/10, 7/16, 3/5, and 3/2. The ground state energies al- Both determinant formulas (14), (16) have now been lowed by unitarity and superconformal invadance in proved by Meurman and Rocha-Caridi [12] using the the Neveu-Schwarz sector are hl, 1 = 0 and h2, 2 = 1/10. technique that Feigin and Fuchs [13] applied to prove They decompose into irreducible representations of the the Kac determinant formula for the Virasoro algebra. Virasoro algebra: Curtwright and Thorn [14] have independently proved these determinant formulas. (0)N S = (0)VIR • (3/2)VI R , As with the Virasoro algebra, M n becomes diagonal (1/10)N S = (1/10)VLR $ (6/10)VIR. (18) and positive ash ~ 0% and det(Mn) has no zeros in the quadrant d > l, h > 0. Thus there are no negative met- These are the dimensions of the Z 2 even operators of ric states ford/> 0, h/> 0. On the other hand, ford< 1 the tricritical Ising model [ 18]. The allowed representa- and any h, there is always some n for which tions of the Ramond algebra are hl, 2 = 3/80 and h2,1 det(M n (d, h)) ~< 0. A negative determinant indicates = 7/16, each consisting of exactly one irreducible rep- at least one negative metric state, so every unitary resentation of the Virasoro algebra *1. These are the di- representation with d < 1 must lie on a vanishing curve mensions of the Z 2 odd operators of the tricritical h = hp, q (d) or, in the Ramond case, h = ~/16. The rest Ising model, i.e., the leading and subleading magnetic of the argument which gives the list (I l) is essentially spin operators [ 19]. These operators take Z 2 even states the same as used in ref. [2] for the Virasoro algebra. to Z 2 odd and vice versa and so intertwine the Ramond The list (11) consists of the so-called first intersections and Neveu-Schwarz sectors of the theory. The disorder [2] of the vanishing curves h = hp, q (d). It can be shown operators are the P = -1 spin fields. The F = 1 projec- that there is a negative metric state between first inter- tion of the superconformal model is a maximal mutual- sections on each vanishing curve, which leaves only the ly local algebra of operators. This picture of the tricriti- first intersections as possible unitary representations. cal Ising model is supported by the operator product For the Ramond algebra there is a slight variation in relations of ordinary conformal field theory [ 1 ]. the argument, because the collection of curves (hp p+l) All of the Z 2 even operators can be constructed makes first intersections with the single curve h = c/16 from the super stress-energy tensor and a single con- while for k :~ 1 the collection (hp, p+k) makes first in- formal superfield. To do this we use the complex con- tersections with the collection (hp, p+2-k). jugate coordinates (~, 0) _and the (h, tt) notation which The first nontrivial model has rh = 3, ~ = 7/15. c = gives the dimension (h + h) and the spin (h - h) of a 7/10. This is the only value ofc which occurs in the dis- crete lists for both the superconformal algebras and the 4:1 A superconformal representation does not typically decom- Virasoro algebra [2]. In ref. [2] we identified c = 7/10 pose into only a f'mite number of Virasoro representations. 41 Volume 151B, number 1 PHYSICS LETTERS 31 January 1985 conformal field [1,2]. The Z 2 invariant operators con- are represented by exponentials of a free superfield. sists [18] of the energy operator e with (h, h) = (1/10, An extra supercharge is placed at infinity and the net 1/10), the vacancy operator t with (6/10, 6/10), and an charge is neutralized by superconformally invariant irrelevant operator with (3•2, 3/2). There are also Z 2 contour integrals ofh = 1/2 exponentials of the free even fermionic operators ~ with (6/10, 1/10) and TF superfield. To obtain nonvanishing correlation functions with (3/2, 0). The ordinary stress-energy tensor is of bosonic fields there must be an even number of neu- combined with T F to form the super stress-energy tralizing supercharges. By this method it is possible to tensor, and the (3/2, 3/2) operator is interpreted as describe correlation functions of all conformal super- TF TF. The remaining even oper_atorsare combined to fields corresponding to degenerate representations of form a superfield ~b = e + 0 ~0 + 0ff + OOt with (h,/70 the Neveu-Schwarz algebra (as well as certain others). = (1/10, 1/10). All of the correlation functions of the Bershadsky, Knizhnik, and Teitelman [23] have inde- Z 2 even operators can be derived from the correlations pendently constructed this integral representation and of the superfield d;, since correlation functions of the used it to study the operator product structure of con- super stress-energy tensor are completely determined formal superfields. by the commutation relations (2), as in ordinary con- The correlation functions of the tricritical Ising mod- formal field theory [1]. el can also be calculated using ordinary conformal prop- We have calculated correlation functions of the su- erties. The results of Dotsenko and Fateev [22] show perfield of the tricritical Ising model by a supersym- that there is a unique conformally invariant theory with metric version of the method of null vectors and linear the appropriate field content (at least in the Z 2 even differential equations [ 1,20]. Here we present the re- sector), so it must be the supersymmetric theory. Rela- sult for the four point function. Derivations and addi- tions like eq. (7) can be checked by the ordinary con- tional results will be presented elsewhere [21]. The null formal techniques. We are currently constructing ex- vector [G_ 3/2 - (5/3) L_IG_I/2] I1/10) gives rise to plicitly supersymmetric representations of combined a linear super differential equation on correlation func- spin operator and superfield correlation functions. tions of d~ which, along with the appropriate singular- We should note that Rocha-Caridi [24] calculated ity and monodromy conditions, determines the four the partition functions of the representations with point function: c = 7/10. We should also mention interesting recent work by Goddard and Olive [25]. Using Kac-Moody (qb(1) d/,(2) d/,(3) qb(4)} = [z12z23z34z411-1/5 algebra techniques they have found a fermionic oscilla- tor realization for the c = 7/10 representations, among × (If(r/, ~')12 +hlg(r/, ~')12), (19) others, thus proving their unitarity. where r/=z12z34/z13z24 , ~ = 1 - rl-z14z23/z13z24 There are several applications of superconformal and field theory in superstring theory. The Faddeev-Popov ghosts of the fermionic string [26] form a superconfor- f= (1 + ~'~ d/dr/) ([~(1 - r/)]-l/l°Fl( 1, --g,-g,r~))22 mal system. The ghost spin fields complete the covariant construction of the fermionic vertex [27]. Ising model +~s ~[r/(1 -r/)] 9/l°Fl(~, ~, 1~ ,r/), techniques can be used to calculate multi-fermion am- plitudes [28]. Nontrivial superconformal field theories g = (l+~rld/d~)([rl(l_rl)]l/2Fl(s,4_g, "g,8 rl)) with d = 6 give possible compactifications of superstring +-g1 ~-[~(1 -~)] -1/2F1 (_ 6 3 2 theories from ten to four dimensions [29]. In particular, ~-,--e,--e, ~), the N = 1 supersymmetric nonlinear sigma models with Ricci-flat internal spaces are conjectured to have zero A -_4 ~ U(4/5) 1-' (2/5) 3/F(1/5) F(3/5) 3 . (20) beta function [30], which implies superconformal in- It is also possible to construct correlation functions variance. Such models can be defined on the world-sur- for conformal superfields by a straightforward super- face of the open type I superstring if they are extended symmetric extension of the Feigin-Fuchs Coulomb gas by boundary terms which depend on a Yang-Mills integral representation [22] *2. Conformal superfields gauge field on the internal space. The Ramond represen- tation of the superconformal sigma model determines .2 For related works, see refs. [18,19]. the fermionic spectrum of the compactified superstring, 42 Volume 151B, number 1 PHYSICS LETTERS 31 January 1985 and the index tr(P) gives the chiral fermion content. [8] E. Witten, Nucl. Phys. B202 (1982) 253. Another set of superconformal models which can pro- [9] E. Corrigan and D.I. Olive, Nuovo Cimento llA (1972) 749 vide compactifications of the string are theN = 1 super- E. Cortigan and P. Goddard, Nucl. Phys. B68 (1974) 117. symmetric generalizations of the nonlinear sigma mod- [10] Y. Aharonov, A. Casher and L. Susskind, Phys. Rev. D5 (1971) 988. els with Wess-Zumino term, at zeroes of their beta [11] V.G. Kac, in: Proc. Intern. Congress of mathematicians, functions [31 ]. The super Kac-Moody current algebras (Helsinki, 1978); in: Group theoretical methods in phy- of these models can be used to show the absence of sics, eds. W. Beiglbock and A. Bohm, Lecture Notes in massless fermions in the closed string sector, and thus Physics, Vol. 94 (Springer, Berlin, 1979), p. 441. the breaking of supersymmetry in the compactified [12] A. Meurman and A. Rocha-Carida, in preparation. [131 B.L. Feigin and D.B. Fuchs, Funkts. Anal. Prilozhen. 16 string. (1982) 47 [Funct. Anal. Appl. 16 (1982) 1141. Finally, we remark that the tricritical Ising model [14] C.B. Thorn, private communication; can be realized experimentally (for example by adsorb- see also T. Curtwright and C.B. Thorn, University of ing helium-4 on krypton plated graphite [32]) and so Florida preprint (1984). provides the first instance of a supersymmetric field [15] L.P. Kadanoff, unpublished. 116] B. Nienhuis and L.P. Kadanoff, unpublished. theory in nature. [17] L.P. Kadanoff and H. Ceva, Phys. Rev. B3 (1971) 3918. [181 B. Nienhuis, J. Phys. A15 (1982) 199; One of us (D.F.) is grateful to the Landau Institute D. Huse, J. Phys. A16 (1983) 4357. and Nordita for the invitation to present this work at [19] M.P.M. den Nijs, Phys. Rev. B27 (1983) 1674. [20] V1.S. Dotsenko, Stat Phys-15 (Edinburgh, 1983) [J. Stat. the Landau-Nordita Conference on Field Theory and Phys. 34 (1984) 781];Nucl. Phys. B241 (1984) 333. 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