-1-
TOPICS IN TWO DIMENSIONAL CONFORMAL FIELD THEORY
by
Gavin Waterson
A thesis presented for the Degree of Doctor of Philosophy of the University of London and the Diploma of Membership of Imperial College.
Department of Physics Blackett Laboratory Imperial College London SW7 2BZ.
August 1987 -2-
To my Parents ABSTRACT
Kac-Moody and Virasoro algebras provide the mathematical tools for understanding the structure of two dimensional con formal field theory. These algebras are intimately related in that it is possible to construct Virasoro generators from Kac- Moody ones by means of the Sugawara construction. Both alge bras also possess various supersymmetric extensions. It is possible to associate vertex operators with a variety of integral lattices. For example, it is known that Kac-Moody generators can be realised from vertex operators corresponding to lattices of points of length squared 2, while fermionic fields can be realised from lattices of points of squared length 1. Here we find another type of vertex operator associated with a lattice of points with square length 3, which provides a realisation of the supercharges for an N=2 super-Virasoro algebra. The second part of this thesis is concerned with the realisation of Kac-Moody and Virasoro algebras in terms of symplectic bosons rather than the well known fermionic con structions. Again there exists a Sugawara construction for the Virasoro generators and the conditions for this to equal a Virasoro algebra constructed from free symplectic bosons are provided by a ’superalgebra theorem’. Both fermionic and bosonic constructions can be combined to provide realisations of certain Kac-Moody superalgebras and the corresponding ’ super-Sugawara construction’ can be performed. The condition for this to equal the sum of free fermion and^ free symplectic boson Virasoro generators is provided by a ’supersymmetric space theorem’. -4- These results are then applied to the specific case of the ghost system of fields arising in covariant string theory, It is found that extra derivatives of u(l) Kac-Moody generat ors have to be added to the various Sugawara constructions in order to reproduce the string theory results. PREFACE
The work presented in this thesis was carried out in the Department of Physics, Imperial College, London between Oct ober 1984 and July 1987, under the supervision of Professor D.I. Olive. Unless otherwise stated, the work is original and has not been submitted before for a degree of this or any other University.
Chapters 1 to 3 are mainly introductory while Chapter 4 is based on a paper appearing in Phys. Lett. B171 (1986) 77. Chapters 5 and 6 are the result of work done in collaboration with P. Goddard and D. Olive (Imperial preprint TP/86-87/15 to appear in Communications in Mathematical Physics). In Chapters 7 and 8, our formalism is applied to the BRST string ghosts though no new results are obtained. I would like to thank David Olive for providing ever helpful advice and guidance throughout. I am grateful to Peter Goddard for discussions and the Science and Engineering Research Council for financial support. Finally, I would like to thank all my friends and colleagues in the Theory Group at Imperial College. -6- CONTENTS
ABSTRACT...... 3 PREFACE...... 5 CONTENTS...... 6 CHAPTER 1 - INTRODUCTION 1.1 Motivation...... 10 1.2 Classical Conformal Symmetry...... 13 1.3 Radial Ordering and Quantum Conformal Symmetry..16 1.4 Conformal Fields...... 18 1.5 The Bosonic String...... 20 1.6 Layout of Thesis...... 24
CHAPTER 2 - KAC-MOODY AND VIRASORO ALGEBRAS 2.1 Virasoro Algebras...... 27 2.2 Kac-Moody Algebras...... 34 2.3 The Sugawara Construction...... 37 2.4 The Quark Model and Quantum Equivalences...... 41 2.5 The N=1 Superconformal Algebra...... 46 2.6 Virasoro Algebras with 'u(l) current' terms ....49
CHAPTER 3 - VERTEX OPERATORS 3.1 Basic Construction...... 52 3.2 Vertex Operators and Lattices...... 55 3.3 Lattices with points of square length 1...... 59 3.4 Lattices with points of square length 2...... 61 3.5 Vertex Operators for Virasoro algebras with
1 'u(l) current' terms 65 -7- CHAPTER 4 - VERTEX OPERATORS AND SUPERCONFORMAL GENERATORS 4.1 The N=2 Superconf ormal Algebra...... 69 4.2 Vertex Operators for points of square length 3...71 4.3 Further Realisations of N=2 Superconformal Algebras...... 77 4.4 Vertex Operators for points of square length 4?..84
CHAPTER 5 - SYMPLECTIC BOSON CONSTRUCTIONS 5.1 Realisations of Kac-Moody Algebras...... 88 5.2 Two Virasoro Algebras and a Superalgebra Theorem...... 92 5.3 Examples of the Superalgebra Theorem...... 99 5.4 Critical Representations...... 102 5.5 Symplectic Fermions...... 104
CHAPTER 6 - SUPERAFFINE ALGEBRAS AND SUPERSYMMETRIC SPACES 6.1 Affine Superalgebras...... 107 6.2 Realisations of Affine Superalgebras...... 109 6.3 The Super-Sugawara Construction...... 114 6.4 A Super-Symmetric Space Theorem...... 119 CD LO • Examples of the Super-Symmetric Space Theorem...124
CHAPTER 7 - BRST METHODS 7.1 Motivation and Applications in String Theory.... 128 7.2 BRST Approach to Quantisation...... 130 7.3 Application to Virasoro Algebras...... 132 7.4 Superconformal Ghosts and Fermionic Strings... 134 7.5 Applications to Affine Algebras...... 137 - 8 - CHAPTER 8 - GHOST FIELDS IN COVARIANT STRING THEORY 8.1 Conformal Ghost Field Constructions...... 141 8.2 Superconformal Ghost Field Constructions...... 145 8.3 Algebraic Structure of Combined Ghost System.... 148
CHAPTER 9 - CONCLUSIONS AND OUTLOOK...... 153
REFERENCES 156 -SI-
CHAPTER 1
INTRODUCTION -10- 1.1 Motivation
Two dimensional conformal field theory has recently attracted much attention due mainly to the resurgence of int erest in string theories, which at present are the most prom ising candidates for providing a unified theory of all fund amental forces, including gravity (see Scherk,1975 and Schwarz 1982 for reviews). The basic object occurring in these theories is a one dimensional string, which sweeps out a two dimensional worldsheet as it propagates through a higher dimensional spacetime. Conformal properties are guaranteed, as will be seen in the next section, by the existence of a symmetric, traceless energy-momentum tensor, 0a^, (for string theory Qa^ = 0 on shell). The conformal group in more than two dimensions is finite dimensional but, as we shall see, in two dimensions it is infinite dimensional and corresponds to analytic (and anti- analytic) transformations of the complex plane. Thus the powerful and elegant mathematical techniques of complex anal ysis become applicable, enabling a very rich mathematical structure to be developed for such two dimensional conformal theories. This leads to a better understanding of string theories as well as simplifying some of the calculations. Since string theory contains quantum gravity, the strings should, in principle, determine the geometry of the background spacetime through which they propagate. This unsolved problem will not be addressed here as we shall assume a Minkowski 1 background, though the techniques we shall be using could be useful in the more general case. The ultimate goal is to -11- elucidate whatever fundamental symmetry underlies string theory. Of course, the formalism we shall be using can be applied to any conformally invariant two dimensional theory and there are several other physically interesting applicat ions as well as strings. One notable example is in the case of various two dimensional statistical models, which exhibit conformal invariance at second order phase transitions (see Cardy, 1985 for a review). Conformal invariance determines the various correlation functions of these models more or less uniquely and provides an explanation for the observed critical exponents. Other applications include current algebras and non-linear sigma models with Wess-Zumino terms. The formalism is also of interest to pure mathematicians as, for example, it provides connections between modular forms and the Fischer-Griess Monster Group (see e.g. Frenkel, Lepow- sky and Meurman, 1984 and references therein). It seems remarkable that such apparently disparate areas of mathematics and physics should, in some sense, be related. One common feature of all these applications is the widespread use of vertex operators, which were originally introduced as emission vertices in string theory. It was later realised that these operators are intimately related to the theory of lattices, providing further mathematical structure and insight. Having motivated a study of the algebraic properties of two dimensional conformal theories, we shall discuss their fundamental properties and give an account of the two dimen sional conformal algebra in the rest of this chapter. In section (1.5) a brief account of the bosonic string is given, 12-
partly as an example of a two dimensional conformal model and partly to establish notation. The layout of the rest of the material is given in section (1.6). -13-
1.2 Classical Conformal Symmetry
Conformal field theories possess energy-momentum tensors 0^v , which can be taken to be traceless (0*1 = 0) as well as symmetric (0^v= 0V^). These properties enable us to construct conserved currents of the form
= e ^ a . ( 1 . 2 . 1 ) provided 5a + 5a - —2 g,5a^=0, p ( 1 . 2 . 2 ) |1 v V |i \iv p
where g v is the metric in d dimensions. The transformation corresponding to this Noether current is
-► x^ + a^(x)s = x^ + 6 a . (1.2.3)
In two dimensions, using Euclidean space with coordin- ates (x 1 ,x 2 ), equations (1.2.2) have solution
— (a- + ia9) = a+(rei0) = a+(z) , /2 1 2
—1 (a1 , - iaQ) x = a (re -i0x ) = a ~~\(z) , (1.2.4) /2 1 2
where we have transformed to polar coordinates, (r,9), by
x = r cose ,
x = r sinG , (1.2.5) -14- i 0 and written the complex variable z = re together with its _ _ -| Q complex conjugate z = re . Performing two consecutive transformations of the kind (1.2.3) leads to
t6a ’ 6b^ x<* = 6cx0C ’ a =1’2 , ( 1. 2 . 6 ) where + 1 + + + + c = — (c., ± ic0) = a 5, b - b 5, a (1.2.7) /2 1 ^ ± ±
We see the conformal algebra splits into the + and - compo nents separately, which, for the most part, can be treated independently. Here we concentrate on the + component, where the fields depend only on the complex variable z i.e. they are analytic functions and can be expanded in a Laurent series.
Consider now a term in the Laurent series of a(z)
n+1 _ a n = e -n z , n e , ( 1 . 2 . 8 ) where e_n is a small parameter and the corresponding generator of (1.2.3), denoted by L , has the form n
T n+l * L n = - z 5 z (1.2.9)
Equations (1.2.6), (1.2.7) then read
[L , L ] = (n-m) L 1 2 10 L n m J n+m * ( . . ) which is the classical form of the two dimensional conformal algebra. Since the indices n,m in (1.2.10) take all integer -15- values, this is an infinite dimensional algebra as anticipated in section (1.1). Note that the generators L and L . gener- o — 1 ate dilations and translations respectively, and the conformal transformations (1.2.3) correspond to analytic transformations of the complex plane. The - component of the algebra we are disregarding corresponds similarly to anti-analytic transform ations of the complex plane and gives rise to generators Ln , also obeying (1.2.10) but commuting with the L^. -16-
1.3 Radial O r d e r i n g and Quantum Conformal Symmetry
In conformal field theory it is more natural to quantise on surfaces of the form
X X constant , (1.3.1) rather than on surfaces of constant time as in conventional field theory. The main advantage of this is that (1.3.1) is a covariant expression whereas picking out a time coordinate is not. The details of this quantisation scheme were developed by Fubini, Hanson and Jackiw (1973), who noted that time ordering of quantum operators should be replaced by a radial ordering
R ♦(»)*(?) = 9(x2-y2) + e(y2-x2) *(y)*(x). (1.3.2)
The Noether charges in this quantisation scheme should be radially invariant and, in terms of the complex variable z of the previous section, they can be written
Q = -i- i dz e++(z) a+(z) , (1.3.3) a 2ui C where the contour c encircles the origin once in a positive sense. (The fact that 0++(z) may be expressed in terms of z follows from d 0^v = 0.) That (1.3.3) is radially invariant follows from Cauchy’s theorem assuming no poles of the inte- \ grand for 0 < |z| < ®. Classically the charges Q would obey a + the conformal algebra (1.2.10) for a suitably chosen a (z), -17- though quantum mechanically there is an anomaly term ( a
Schwinger term) arising from the radial ordering required for two energy-momentum tensors. As shown by Fubini, Hanson and + Jackiw (1973), with a (z) expanded into a Laurent series as before, the analogous result to (1.2.10) for the conformal algebra is
tLn- hJ = (n-m> Ln+m + (n3_n> 6 n+m, o (1.3.4) where c is a central charge. The algebra (1.3.4) is known as the Virasoro algebra (Virasoro, 1970), which was first en countered as the gauge algebra of the bosonic string. -18-
1.4 Conformal Fields
As we saw in section (1.2), the conformal transformat ions of interest correspond to analytic transformations. Con sider such a transformation z -*> £(z) and a field,
where h is known as the conformal weight of 4>(z). Writing one such infinitesimal transformation as ^£ = z + e -n z = (v 1 — £ -n L)z,n' * (1.4.2) where s__n is a small parameter, gives [Ln , 4> (z) ] = 6n4)(z) = zn(zdz + (n+l)h) This may be written in the alternative form, [Ln , zh<|>(z)] = zn(zdz + nh) (zh or, by expanding z^4>(z) as z (Kz) = l [L fn(h-l) m) . (1.4.6) L n , <}> m ]J = v — 3 <|> n+m -19- We shall mainly be interested in generators with the hermitian property (Ln )+ = L_n - (1-4.7) 1 and with fields defined on the unit circle |z|=l, so z = z From (1.4.4) we see that (z 4>(z)) is then also hermitian. We shall take eq. (1.4.4) to be our definition of a conformal h. field, (z In this section we review the properties of the bosonic string which will be relevant for our purposes. For further details see e.g. the review article by Scherk (1975). The action for the bosonic string can be taken to be S = - — S i do J dt /=g"ga|3 aax|1 , (1.5.1) 2tz where (a,x) parametrise the two dimensional worldsheet swept out by the string, x*1, l Minkowski space, though in principle the spacetime metric should be determined by the dynamics. The equation of motion for g p from (1.5.1) is d d Qx - — g QgYS b x^1 doX = 0 , (1.5.2) a p \i 2 aP Ybn ' x 7 and substituting this back into eq. (1.5.1) yields the Nambu (1970) action, S = - — J^da Jdx / -det(5ax^ d^x^) • (1.5.3) 71 The integrand of (1.5.3) is just the area of the world- sheet and so gives a geometrical interpretation for the action. 1 The action (1.5.1) is invariant under reparametrisations of the worldsheet, a = a(a,t ), t = x(a,x), as well as under -21- rescalings of the metric g ocp -> A(a,T) . . g apr . (1.5.4) These invariances allow the gauge choice ga^ = t]0^ i.e. g 11 = 1» g 0(5 = -i. g lO = 0 . (1.5.5) Substituting this gauge choice into eqs.(1.5.1), (1.5.2) lin earises the action and yields the constraints 5 x 5 x = 0 , a i {i * dx^dxa a \i +5x^dx i i \i =0, 9 (1.5.6) together with the equations of motion ( ”—? ~ ~— ) x^(a,x) - 0 . (1.5.7) b i 5a For an open string, the solution to (1.5.7) can be written as H ji n a n\i -m x x (a,x) = q + p x + i J — e cos na , (1.5.8) n^o n where q*1, p^, a^ are classical variables. The constraint n equations (1.5.6) can be more compactly written as fd x^ ± 5 x*M^= 0, which together with the expansion (1.5.8) ^ T O' ^ ' yields -in(x+a) l ~0 a„ n-m m . am m 2e = 0 , (1.5.9) n ,m 2 - 2 2 - u n where a£ is identified with p . This produces the infinite set of constraints L = 0 , ( . . ) n 1 5 10 where ^n ^ ^ an-m* m (1.5.11) 2 m When the system is quantised by imposing the canonical commutation relations = n t/ V5 (1.5.12) [an ’ n+m, o pv IT} (1.5.13) the constraints (1.5.11) have to be written in terms of a normal ordered product of oscillators in order to make sense quantum mechanically. As will be seen in section (2.3), the resultant constraints, Ln , obey the Virasoro algebra (1.3.4) with c=d. It will be shown in section (2.1) that it is no longer possible to impose = 0 for all n if the central charge of the Virasoro algebra does not vanish. The best that can be achieved is to impose <4>2|Ln- = 0 (1.5.14) for any two physical states |4^> and |<}>2>» where h is a poss ible c-number resulting from the normal ordering prescription. i ' 4 - Since we have L = L , we shall satisfy (1.5.14) by demanding that -23- fv L n - h6 n , o 1; 1'1|>> = 0 , n> 0 , (1.5.15) for all physical states J4^>. The mass spectrum of the open string can be obtained from (1.5.15) with n=0 giving masses M obeying M2 = 2 (v T“ a^ -n a np. - h) ' (1.5.16) n>o The spectrum of states potentially contains many negative norm states, or ghosts, due to the Minkowski metric and for a physically viable theory these must be removed. It was shown by Brower (1972) and Goddard and Thorn (1972) that ghost decoupling requires d=26 and h=l. (In fact the result could be extended to d<26 for h 1.6 Layout of Thesis In this thesis we shall be concerned with the various realisations of some of the operators and fields arising in two dimensional conformal field theory in terms of bosonic and fermionic fields. Various equivalences between apparently different constructions will also be explored. Chapter 2 is introductory and expands on the material of this chapter. Virasoro algebras are discussed in more detail and Kac-Moody algebras are introduced. The Sugawara construc tion, whereby Virasoro generators can be written in terms of Kac-Moody generators is also described. The concept of quan tum equivalence arises in the context of Virasoro generators constructed via the Sugawara method or out of free fermion fields. Further details of these and other related topics appear in the review article by Goddard and Olive (1986). Superconformal algebras, the supersymmetric extensions of the Virasoro algebra, are introduced in section (2.5). In chapter 3 the concept of vertex operators is intro duced and these are related to various integral lattices. A review of the use of vertex operators in constructing level 1 Kac-Moody algebras or fermionic fields is given. A further application of vertex operators is given in chapter 4 where they are used to realise supercharges for a certain N=2 super- conformal algebra. Chapter 5 is devoted to a construction of Kac-Moody and Virasoro generators out of symplectic bosons rather than the fermionic constructions reviewed in chapter 2. Many of the results of chapter 2 have direct analogues, including the Sug- -25- awara construction. The conditions for this to be quantum equivalent to a Virasoro generator constructed from the free fields are given by a 'superalgebra theorem’, analogous to the 'symmetric space theorem’ of chapter 2. The fermionic and bosonic constructions of chapters 2 and 5 are combined in chapter 6 to yield representations of Kac-Moody superalgebras. There is an associated super-Suga- wara construction for a Virasoro algebra and the conditions for this to equal the sum of the bosonic and fermionic free field Virasoro algebras are provided by a 'supersymmetric space theorem'. Chapter 7 contains a review of the BRST approach to quantisation and its applications to string theory. The Vir asoro generators corresponding to the various ghost fields are given and nilpotency conditions for BRST charges are dis cussed. The ghost fields are reinterpreted in chapter 8 as pseudo-orthogonal fermions and symplectic bosons, so the results of chapters 2,5 and 6 can be applied. It is found that extra terms have to be added to the Sugawara or super- Sugawara constructions in order to identify them with the various ghost Virasoro generators. Conclusions are given in chapter 9 and possible further developments are discussed. -26- CHAPTER 2 KAC-MOODY AND VIRASORO ALGEBRAS -27- 2.1 Virasoro Algebras As we saw in chapter 1, the Virasoro algebra can be written as fL , L ] = (n-m) L + — (n^-n) 6 L n ’ m J v ' n+m 12 n+m, 0 ( 2. 1. 1) where c is known as the central charge or anomaly and, for our purposes, it is a real number. The indices m,n in (2.1.1) take integer values. It is the anomaly term that is respons ible for interesting quantum mechanical effects and it is absent classically as we saw in chapter 1. The anomaly term arises due to the requirement of normal ordering quantum operators and its form is uniquely determined (up to redefin itions of L ) by the Jacobi identities. Two commuting copies of the Virasoro algebra generate the conformal group in two dimensions which consists of (anti) analytic transformations of the complex plane. Thus the Vir asoro algebra is a fundamental part of any two dimensional conformal field theory and provides the mathematical frame work for our discussion. The operator L is of special 0 importance as it is the generator of dilations and, using radial ordering as in section (1.3), this leads to the ident ification of the Hamiltonian with L since dilations are the o analogues of time translations in the usual approach. This identification means that, for physical models, states should be eigenstates of Lq and Lq should have a spectrum which is bounded from below. For a physical state |\>, we find from (2.1.1) that -28- Lo(Ln|\>) = (\-n) (LnIX>) , (2.1.2) where Lq |\> = X |X> . (2.1.3) Eq.(2.1.2) says that L^|\> has a lower L eigenvalue than |X.> for n>0 but since L is bounded below, there must be a lowest o eigenstate |h> obeying L |h> = 0 , n>0 , n L |h> = h !h> . (2.1.4) o States satisfying (2.1.4) are, for perverse historical reas ons, known as highest weight states and give rise to highest weight representations of the Virasoro algebra, which are generated by the action of L_q (n>0) on |h>. We shall mostly be interested in unitary representa tions for which t Ln L-n 9 (2.1.5) or, in terms of the field L(z) defined on the unit circle, L(z) = L(z) where L(z) = l L z n , |Z| =1 . (2.1.6) ne» For such unitary highest weight representations, the effect of the anomaly term in (2.1.1) becomes clear. Using (2.1.4) we -29- have, for n>0, || L_n | h> I]5" = = 2nh + — (n3-n) , (2.1.7) 12 assuming we have normalised the states by Highest weight representations are characterised by their c and h values, which are restricted if we wish to consider the physically relevant unitary representations, which correspond to there being no negative norm states in the spectrum nk nl . ..(L_k ) --- (L_x) |h> ♦ (2.1.8) By examining the matrix of inner products of states (2.1.8) for each value of the level £ kn, , Friedan, Qiu and Shenker k * (1984) found the allowed spectrum of (c,h) values. In partic ular, unitarity requires c>0, h>0. Consider a conformal field (z*$(z)) of weight h and assume that 1 ... L Lim f n z+o„ „ Ln Lim (<}>(z)l|0> = h Lim ( Thus we can write a highest weight state |h> as lh> = Lim (4>(z)) |0> . (2.1.11) z-*o An example of this is provided by a realisation of the Vir- asoro algebra in terms of free fermions, Ha(z), where Ha (z) = J b“ z“r , l Ha (z) can be periodic or antiperiodic in z i.e. Ha(ze2711) = ± Ha(z) (2.1.13) and this corresponds to r z £ (Ramond case) or r e 2T+1/2 (Neveu-Schwarz case) respectively. Hermiticity of Ha (z) implies a br (2.1.14) and we impose the canonical anticommutation relations {b“ . b^} = 6a|36r+s, o (2.1.15) These fields are normal ordered according to , a . p r <0 , b„r bl.: s = b“r ** s , _ 1 wP [ br > bg] ’ r=0 > .0 .a - b^s b^ r , r>0 , (2.1.16) -31- which leads to the operator product expansion Ha(z)HP(C) = :Ha(z)HP(5): + 6af3A(z,e) , |z| > |5 | , (2.1.17) where (zH)/2 A(z,C) Ramond (R) case, z-^ /z? Neveu-Schwarz (NS) case. (2.1.18) z-C It is then possible to construct Virasoro generators from these free fermions by t H, x v -n L (z) = l Ln z = - :z4§'dz Ha (z): + — (2.1.19) n 16 where e=0 or 1 depending on whether the fermions are of NS or R form respectively. The proof that (2.1.19) does indeed generate a Virasoro algebra follows from considering the H H operator product expansion L (z)L (£ ) , |z| > |£| , using Wick’s theorem and (2.1.17) to write this in terms of fermion fields. One may worry that it is not legitimate to take |z|> |£| since the fields are defined on the unit circle. However, analytic continuation provides the justification for this kind of calc ulation. It is found that the operator product expansion H H L (£)L (z), |£| > |z| yields the same expression. Since l JJ11 = U dz z11"1 LH( z ) , (2.1.20) o where henceforth the integration symbol incorporates the 2ni -32 factor, and c is a closed contour encircling the origin, we o have tH tH / , / n-1 „m-l t H/ w Ln Lm = ^c dz ?c z ^ L (Z)L ^ » lz< > ISI , o o Lm L n = d5 K dz 5m_1 zn_1 L”(5 ) L H (z ) , I5l>lz|. ( 2 . 1 . 2 1 ) o o The integrands in these two expressions are the same and denoting them by f(z,£) we have [l " l “] = ( j dz f a t - f d5 f dz ) f(z,5) • (2.1.22) lz| > 151 |5| > |z| Fig.l shows the contour integrals in the z-plane. Using Cauchy's theorem and the fact that f(z,£) is non-singular away from z=£, we can deform the contours as in Fig.2 to obtain [l “ I*] = i d5 $ d z f(z,5) , (2.1.23) O t where c is a closed contour encircling z=£. The inner inte- gral can now be performed by residues. This is the basic method of calculating commutation relations -33- and in the present context establishes that the L obey the Virasoro algebra with a central charge c=d/2. Each free fermion contributes 1/2 to the overall anomaly. We also find [l", h“(z )] = zn(zSz + n/2) Ha(z) , (2.1.24) which says that Ha (z) is a conformal field of weight 1/2. Thus we should have a highest weight state 11/2> = Lim (z 1/2Ha (z)) |0> . (2.1.25) z->o In the NS sector this implies 11 /2> 0> as we have b® |0> = 0 , r>0 (2.1.26) which is consistent with the normal ordering definition in (2.1.16). In the Ramond sector, (2.1.25) is ill-defined and we have only the Ramond vacuum 10>D , which is 2 ^ ^ ^ - f o l d it degenerate as the Ramond zero modes generate a Clifford algebra. These states have h values of d/16 from eq.(2.1.19). -34- 2.2 Kac-Moody Algebras Given a compact finite dimensional Lie algebra, g, with generators, ta , obeying, r , a , b-. .„ab ,c [t , t ] = if c t , (2.2.1) we can construct an untwisted Kac-Moody algebra, g, given by [Ta , Tb] = ifab TC + nk6 , 6ab , 2 2 2 L n m J c n+m n+m,o ( . . ) which is an infinite dimensional Lie algebra. Here the ind ices m,n z H and k is a central charge. Just as in the case of the Virasoro algebra, it is the non-vanishing of the central charge that leads to interesting quantum effects and prevents the imposition of constraints such as Ta |(P> = 0 , Vn , (2.2.3) on physical states | Many of the features of compact finite dimensional Lie algebras can be generalised to these affine algebras. Con cepts such as root systems, Weyl reflections, simple roots and character formulae are still valid leading to a classification of such algebras (see Kac, 1985 and Goddard and Olive, 1986 for a review). Just as for the Virasoro algebra, we can write the T as Laurent coefficients of an operator T (z) n where \ Ta(z) = l Ta z~n , |z| = 1 , (2.2.4) -35- cL and requiring Tc (z) to be hermitian gives (T„)f= . (2.2.5) The Kac-Moody algebra is important because, as will be seen in the next section, it is possible to construct Virasoro generators out of Kac-Moody ones by the Sugawara construction (Sugawara, 1968). This leads to a semi-direct product struc ture of the two algebras given by the commutation relations (2.1.1), (2.2.2) and [LL n* , T*] m J = “m t m+n , • (2.2.6) v 7 Applying (2.2.6) with n=0 to a highest weight state, I h>, of the Virasoro algebra gives L0(Tm |h>) = i.e. Tffi |h> is an eigenstate of Lq with eigenvalue (h-m), which is less than h for m>0. This can only be consistent with the definition of highest weight states if we have Ta |h> = 0 , ra>0. (2.2.8) States obeying (2.2.8) are known as highest weight states of the Kac-Moody algebra and a representation theory has been developed for such unitary representations (see Goddard and Olive, 1986 and references therein). One important result from this representation theory is that the central charge, k, -36- in (2.2.2) has to be quantised according to 2k x non-negative integer, (2.2.9) where x is known as the level of the representation and ct is the highest root of g . Tq generate the subalgebra g of g and the highest weight states correspond to a representation, \i, of g. In fact, the highest weight representations generated from the highest weight states by the action of T_n (n>0) are uniquely determined by the level x and the representation \i. -37- 2.3 The Sugawara Construction As mentioned in the previous section, the Sugawara con struction (Sugawara, 1968) relates the Virasoro and Kac-Moody algebras. Given a Kac-Moody algebra, g\ based on a compact, finite dimensional Lie algebra, g, define a l T (2.3.1) n 2p m n+m where n,m e 1 , p is a normalisation constant to be determined and the normal ordering defined by crosses is given by ^ TaTa ^ = TaTa n<0 , x n m x n m = TaTa , n> 0 . m n (2.3.2) A somewhat tedious calculation involving splitting up the sums in (2.3.1) (see e.g. Gomes, 1987) shows that satisfy the Virasoro algebra with 2k dim g c = ----- — 2 (2.3.3) 2k + Q 4> a^a where Q, is the quadratic Casimir operator, t t , of g eval- 4> uated in the adjoint representation, i.e. „ cab _ ^acd-bcd V ~ f f (2.3.4) The coefficient p is also determined to be P “ Qc}; / 2 + k . (2.3.5) -38- Given the form of { in (2.3.1) and the desired commutation n cL relation (2.2.6) with T^, p can be determined by applying Ji ^ to a highest weight state |(p> to give \<\>> = - Ta 1 > ta , (2.3.6) P where Ta |(i> = | Acting on (2.3.6) with and commuting it through until it annihilates we find l which reproduces (2.3.5). This argument, due to Knizhnik and Zamolodchikov (1984), can be extended to confirm that c has the value given by (2.3.3) if we act on \<\,> with [£ , £ ] Ci Ci assuming (2.1.1) holds giving L 2 L _ 2 (<('> = 4 £ o |(|>> + c/2 | . (2.3.9) Expanding i2, l. 2 in terms of the Ta and commuting operators which annihilate \<\>> through to the right reproduces c = k._dimg . . 2k. dimg , x_dlgg _ (2.3.10) 8 2k + Q x + h ~ , where h is the dual Coxeter number, 2 , which is an int eger. Since x is also an integer this shows that c is ration al. We see from (2.3.10) that c is bounded above by dim g and further analysis reveals that it is bounded below by rank g -39- (see Goddard and Olive, 1986). The simplest example of this construction corresponds to taking g = [u(l)]b , where the Kac- /V SL Moody generators for g, Tn (l [Ta , Tb] = n 6ab6 L n mJ n+m,o (2.3.11) where for convenience we have set k=l in (2.2.2) since this is arbitrary in the u(l) case. Equations (2.3.11) are just the commutation relations appearing in section (1.5) for the bosonic string oscillators in d dimensions (though with a Euclidean metric here). The Sugawara construction yields a Virasoro algebra with c = dim [u(l)] = d. We see that a single free bosonic field contributes c=l to the anomaly just as in section (2.1) it was found that free fermions contribut ed c=l/2. If g has a subalgebra h, then it is also possible to /N construct Sugawara Virasoro generators corresponding to h, where the sum over a in (2.3.1) is restricted to those gener- /s ators lying in h. We have H B t II a I T 1 mJ m+n U n - Ta]mJ = ”m T m+n , , (2.3.12) where Ta is a generator of h and , «Cn denote the Sugawara m constructions corresponding to g and h respectively. Since £ cl is constructed from T , this gives m LcLnfis - £ <*-n» h <^mJ f h l = V o 3 (2.3.13) -40- so we can form another set of generators K = / n n ^n ^n (2.3.14) which obey the Virasoro algebra with a c value = c 'K (2.3.15) As we are assuming we are dealing with unitary highest weight representations of i s and j}1, the same must be true for K n representations and hence c >0. K \ 41- 2.4 The Quark Model and Quantum Equivalences An important realisation of the Kac-Moody algebras is provided by the 'quark model' , whereby the Kac-Moody generat ors are written as bilinear products of fermionic fields of the kind introduced in section (2.1). Consider a represen tation of a compact finite-dimensional Lie algebra, g, as in & Si Si (2.2.1), given by real antisymmetric matrices, (t -> iM ). Then define Ta (z) = l Tn z n = 1 MaS h“(z)hP(z) . (2.4.1) ne 1 2 H where 1 < a , (3 < d. Note that there is no need to explicitly normal order the expression on the right as M c l is an anti symmetric matrix. From (2.4.1) it is possible to deduce (by a b considering the operator product expansion T (z)T (£) and using the contour integral methods employed in section (2.1)) that the T^ obey the Kac-Moody algebra (2.2.2) with k = -K6ab = Tr(MaMb) . (2.4.3) ct a As H (z) are hermitian, T (z) are also hermitian and hence k appearing in (2.4.2) should be quantised as in (2.2.9). This 2 is indeed the case because the Dynkin index, < / cp , of real representations is always an integer. \ With the Kac-Moody generators defined as in (2.4.1), there are two kinds of normal ordering defined, one with respect to the Kac-Moody generators themselves and one with -42- respect to the constituent fermion fields. Comparing operator product expansions written with both kinds of normal ordering leads to powerful and surprising identities known as quantum equivalences (Goddard and Olive, 1985). For example, with respect to the Kac-Moody normal ordering (2.3.2), we have k dimg z£ Ta(z)Taa ) = * Ta(z)Ta(£) l + , |z| > Id . (2.4.4) However, normal ordering with respect to the fermion fields in (2.4.1) using (2.1.17) leads to Ta(z)Ta U ) = :Ta(z)Ta (5): - Ma pMafi:Ha(z)H6(5): A(z ,?)6Py (2 .4 .5 ) where :: denotes fermionic normal ordering. Equating (2.4.4) and (2.4.5) in the limit leads to *Ta(z)Ta(z)* = :Ta(z)Ta (z): + 2QmLH(z ) , (2.4.6) where -Qm 1 = MaMa (2.4.7) SO Tr(MaMa) = - Qm dimM = -k dimg , (2.4.8) and we have used e A2( z ,£ ) (2.4.9) 4 -43- for e=0 or 1 according to whether Ha (z) are NS or R fermions respectively. L (z) is given by (2.1.19). Expanding out the first term on the right of (2.4.6) as :Ta(z)Ta(z): = M ^ M a6 : h “ ( z ) HP ( z ) Hy (z )H6 ( z ) : , (2.4.10) and using the cyclic symmetry of a,p,y,6 under the normal ordering, we find a necessary and sufficient condition for this term to vanish is a a a a M Q M ^ + M -Mq Ma Ma = 0 (2.4.11) ap yo a6 (3y ay 6(3 If (2.4.11) holds, (2.4.6) can be rewritten as L (z) — * Ta (z)Ta(z) * , (2.4.12) 2Qm and the right hand side looks like a Sugawara construction with normalisation factor (3=Q^. For the correct Sugawara form this should equal the expression in (2.3.5), i.e. QM + k (2.4.13) The c number of the Sugawara construction is then given by 2k dimg Q.,dimM dimM M c = (2.4.14) 2k + Q, 2k + Q, which agrees with that for L (z) as it must. Conversely, if 1 d* (2.4.13) holds, then - --- :T (z)T (z): is equal to the diff- 2Qm erence of two Virasoro algebras with the same c-value. If we -44- denote this difference by _ LH - / Kn n ’ (2.4.15) [ K , K ] = we find L n * m J [£„’ 4 J + [ Ln ’ - [l£tL n m J " [inL n , L„] mJ • (2.4.16) However, [Ln -■ JL Tal = 0 , (2.4.17) so ^n* An] 0 , (2.4.18) and (2.4 .16) becomes [K , + [l h, l h] = (n-m)K , KnJ " - tin- 1 n* mJ v 7 n+m (2.4.19) can be shown (Gomes, 1986) that for K Now it n acting on a positive space, the only allowed representation of the algebra is the trivial one i.e. to all intents and purposes we can take K(z)=0. Then we have the quantum equivalence LH( z ) = £(z) , (2.4.20) and the conditions (2.4.11) and (2.4.13) are equivalent cond itions. At first sight (2.4.20) seems rather bizarre because L (z) involves expressions bilinear in fermion fields while iX z) involves quantities containing four fermions. However, this equivalence must be interpreted in the context^of unitary highest weight representations and is a purely quantum effect. -45- We will encounter many other examples of these quantum equi valences in later chapters whenever composite operators enable different kinds of normal ordering to be defined. In the present case, the representations for which conditions (2.4.11) hold have been enumerated by Goddard, Nahm and Olive (1985). If G is the Lie group whose Lie algebra, g, gives rise to a Sugawara construction based on the affine algebra g, constructed from fermions in a real representation, M, of g, then the quantum equivalence conditions are met if there exists a symmetric space G ’/G, where the tangent space gener ators transform like the fermions. In other words, we can add ct a generators t to the generators t of (2.2.1) obeying [ta , tb ] = if a bc t c (2.4.21) r a bn ab c The Jacobi identities reproduce [M , M J = f M as well as v the condition (2.4.11) for the quantum equivalence to occur. -46- 2.5 The N=1 Superconformal Algebra Both Virasoro and Kac-Moody algebras have possible supersymmetric extensions, where extra fermionic generators are introduced into the algebra. In the case of the Virasoro algebra, the simplest such superalgebra can be written as (2.5.1) [L , G ] = (n/2-r) G (2.5.2) l n ’ rJ v ' n+r {Gr , Gs} = 2Lr+s + 2. (r2-l/4)6 r+s,o (2.5.3) 3 The indices n,m e # but, being fermionic generators, the supercharges, Gr , come in two forms depending on whether r e 1 or #+1/2. The former case corresponds to the Ramond form of the algebra (Ramond, 1971) and the latter to the Neveu-Schwarz algebra (Neveu and Schwarz, 1971). The Jacobi identities determine the form of the anomaly term in (2.5.3) relative to that in (2.5.1). These structures arose originally in the fermionic string model of the early 1970’s, where fermionic fields were defined on the string in addition to the bosonic oscillators. This corresponds to the realisation (in Euclid ean space) L(z) = - l Ta (z)Ta (z) (2.5.4) 2 A G(z) = Ta(z)Ha (z) , (2.5.5) -47- where the Ta (z) generate a [u(l)]^ Kac-Moody algebra as in (2.3.11) and L (z) is the free fermion Virasoro generator (2.1.19) for d fermions. These L(z), G(z) have Laurent coefficients which satisfy the superconformal algebra with c=3d/2. Note that equation (2.5.2) says that G(z) is a conformal field of weight 3/2 and hence physical states will also be highest weight states, |h>, obeying Gr lh> = 0 , r>0 . (2.5.6) The highest weight representations are again characterised by their (c,h) values and the spectrum of unitary highest weight states has been investigated by Friedan, Qiu and Shenker (1985), who found allowed values of c > 3/2 and h > 0 , or c = — ( 1 - — --- ) , m=2,3.... , 2 m(m+2) h=h = r (m±g)PTm dim g c (2.5.8) 2 -48- (This is also the c value for a free fermion construction and we have the quantum equivalence (2.4.20) arising as the condition (2.4.11) is nothing but the structure constant Jacobi identity.) The G(z) generator can then be written (Goddard and Olive, 1985), G(z) = l G_ z"r = — -- fab° Ha(z)Hb(z)HC(z) , (2.5.9) r /T8Q", where Q is the quadratic Casimir evaluated in the adjoint abc representation, f are the structure constants of g and G(z) a has the same periodicity properties (NS or R) as the H (z) fields. This realisation of a superalgebra in terms of only fermionic fields is a surprising feature of these infinite dimensional superalgebras, which avoid the necessity for the number of bosonic and fermionic degrees of freedom to match. We shall see in the next chapter that in two dimensions there exists, in certain cases, a fermion-boson equivalence relation so which fields are taken as fundamental is often a matter of taste. In chapter 4 we shall also find realisations of the supercharges, G(z), in terms of purely bosonic fields and we shall encounter a larger superalgebra with more supercharges. -49- 2.6 Virasoro Algebras with 'u(l) current* terms Given a serai-direct product structure of a Virasoro algebra, of central charge c, with a u(l) Kac-Moody algebra specified by (2.1.1) and -mT [Ln m+n (2 .6 .1 ) [T T ] = n6 ( . . ) n m J n+m, o 2 6 2 it is sometimes convenient to construct another Virasoro alge- Q bra, l£, given by l nJ! = L n + inpT r n + r/p2/2 6„ n , o , (2.6.3) v ' which has central charge c^ = c + 12p2 (2.6.4) and has the hermitian property (Ln)+ = L^n . (2.6.5) provided LQ^ = L_n , TQ^ = T_n and (3 is real. These types of Virasoro algebra have been used in the context of the two dimensional statistical models by Dotsenko and Fateev (1984) and will be seen to have a role to play in the ghost system of fields present in covariant string theory to be discussed in chapter 8. Note, however, that with respect to L^, T(z) is no - 5 0 - longer a conformal field as we have defined them. A similar construction can be made for the superconform- al algebra of the previous section if, as well as the u(l) generator, T , we also have an hermitian fermion field, H(z), where T ] = -mT , [L , H ] = (-n/2-r)H , , L[L n . m J n+m l n’ rJ v ' ' n+r T ] = -nH , , (G , H } = T . , [Gr. nJ n+r 1 r sJ r+s H ] = 0 , t V r j t V V = 5r+ s, o • (2-6*6) Then define = L + inpT + n n v n P2/2 Sn,o ' = Gr ♦ 2ierHr , (2.6.7) which obey (Gr)+ = G-r • (2.6.8) and satisfy the superconformal algebra (2.5.1-3) with cP = c + 12(32 . (2.6.9) The system of equations (2.6.6) is, for example, realised by (2.5.4) and (2.5.5) for a single fermion and a single boson giving c=3/2. -51- CHAPTER 3 VERTEX OPERATORS -5 2 - 3.1 Basic Construction Vertex operators originally made their appearance as emission vertices in string theory (Fubini, Gordon and Veneziano, 1969). Define the Fubini-Veneziano (1970) field, Q1(z), by i a_ Q1(z) = q1 - ip1lnz - ij — - zn , l where an ’ a^ n6 5n+m,o ’ (3.1.2) p \ qJ] = -16ij . (3.1.3) (These are the Euclidean version of the commutation relations (1.5.12) and (1.5.13).) The vertex operator is defined by V a (z) = za /2 0 eia-Q(z) 0 K J o o 0 r 1 n 1 -n 2 / o / “ C£ • ct „ Z • ■ »%**"/ "" CC • SL I a 2 £ -n t* n>o n la.q a.p n>o n n z e e z e , (3.1.4) where the normal ordering defined by 0 0 has been explicitly • o o unravelled. The a* are assumed to have the hermitian proper ties (an)+ = a-n ’ (3.1.5) and p1 , are assumed hermitian. Using (3.1.2), (3.1.3) together with the useful result -53- A B B A [A, B] e e = e e eL J (3.1.6) which holds if [A, B] is a c-number, it can be shown that Va (z) has the hermitian property (Va(z))+ = V_“ (z) . (3.1.7) From (3.1.1), define the momentum field PX(z) by P1(z) = iz— Q1(z) = l z ~ D , (3.1.8) dz n where p i has been relabelled aQ i . Note that the generators aR i generate a [u(l)]°* Kac-Moody algebra as in (2.3.11) so we can perform the Sugawara construction of section (2.3) to obtain Virasoro generators L (z) = l Lnz"n = i * Pi(z)Pi(z) l , (3.1.9) n 2 which obey the Virasoro algebra with c=d. With respect to this algebra, the vertex operators transform as [Ln , v“(z)] = zn(zaz + n cc2/2) v “(z ) , (3.1.10) which says that Va(z) transforms as a conformal field of 2 weight a /2. From the point of view of string theory, the vertex operators of interest are the ones of conformal weight 1. This is because the L are gauge conditions so the n emission vertices are required to have nice commutation relat ions with them. (Indeed, these requirements were the original - 5 4 - motivation for introducing vertex operators.) Thus, for example, the tachyon emission vertices correspond to setting 2 a = 3L i n eq . ( 3 . 1 . 4 ) . In the following sections we shall see that choosing the a1 to correspond to various vectors on a lattice leads to interesting algebraic structures. i -55- 3.2 Vertex Operators and Lattices We shall be using vertex operators to construct various operators with which we want to construct various highest weight representations. Thus it is natural to expand the vertex operators, Va(z), in terms of a Laurent series v“ (z) = l V “ z"“ , (3.2.1) n with the index n running over 2? or ff+1/2 depending on the type of operator we wish to construct. The inverse of (3.2.1) is „ct e j n-1 TTa, N V = ♦ dz z V (z) n 7 c v y o m a .a z n - 1 /2 -m i a .q a .p m>o m = f d z z e e z J c 0 -m a .a z - 1 1 m m>o m x e ( 3 . 2 . 2 ) In order for (3.2.2) to be well defined for the contour encir cling the origin, we require o n-l+a /2+a .p e 'S , ( 3 . 2 . 3 ) and so we have two cases, (i) neZ (R) o 2 /2 + a .p z > 2 (ii) neff+1/2 (NS) , a / 2 + a . p € Z + l / 2 (3.2.4) - 5 6 - Moreover, acting on a highest weight state, |p >, which satis- o t i e s i a n> 0 , n 1 a o V ( 3 . 2 . 5 ) we find the condition for this to be also a highest weight state of V®, i.e. n>0 , n IP0 > ° 0 ’ ( 3 . 2 . 6 ) t o be n-l+a /2+a.pQ > 0 , n>0 . ( 3 . 2 . 7 ) Actually, since (Va(z))+ = V a(z), it is better to demand that |Pq > is also a highest weight state of v”a(z) and evaluating the condition (3.2.7) together with the analogous condition f o r a + - a leads to the results, for cases (i) and (ii) above, 2 2 (i) |a.pQ| (ii) |a.p | <(a2/2-l/2) , a2/2+a.pQ e ZT+1/2 (NS) . (3.2.8) If the vertex operators are being used to construct some closed algebraic structure, they must have well defined (anti) commutation relations with each other. Consider the operator product expansion of Va(z)V^(£), which can be evaluated using the definition of normal ordering given by (3.1.4) and the commutation relations (3.1.2), (3.1.3). The result is - 5 7 - 2 2 Y® / 2 ) (£) = za ® gioc«Q(z) o o gip»Q(5) o s o oo o = za2//2£p2/2 0 eia -Q(z)+iP-Q(^) 0 (z-£)a*P (3.2.9) 0 0 for |z|> |£| . Evaluating VP(^)V8 cc (z) leads to the same express ion multiplied by (-l)a ’^ and valid for|zJ<|£| . Thus, by the contour integral methods of section (2.1), we obtain VaVP -(-l)a *P VPVa = f dl l ^ 1 f dz z11-1 za /2 n m m n Jc ;c_ ^ o l x 0oeia*Q(z)+ip,Q(5)o o (z-5)a,p, (3.2.10) where c is a closed contour encircling z=£ and c is a cont- t, o our encircling the origin. In order that (3.2.10) has a pole structure that makes sense, we must have a.p e . (3.2.11) Regarding a and 0 as vectors of a lattice, eq.(3.2.11) shows that the lattices of interest are integral ones (Goddard and 2 Olive, 1984). Setting p=a in (3.2.11) shows a z'S and, in this 2 chapter and the next, we shall be considering a = 1, 2 or 3. 2 For these values of a , eq.(3.2.8) determines the allowed values of la.p I to be ' o' a2=l , |a.po| = 1/2 (R) , 0 (NS) , (3.2.12) -58- o a =2 , ja.p | = 1 or 0 (R) , = 1/2 (NS) , (3.2.13) a2=3 , |a.p |= 1/2 or 3/2 (R) , = 0 or 1 (NS) . (3.2.14) To fully establish the connection with lattices, note that the states built up from Ip > by the action of the vertex o operators, have p eigenvalues of the form p +A , where A is o eia#q|P0> = lPQ+ ct> . (3.2.15) -59- 3.3 Lattices with points of square length 1 In section (3.1) it was seen that the vertex operator Va(z) corresponds to a conformal field of weight a /2. It was also noted in section (2.1) that when Virasoro generators were constructed out of free fermion fields as in eq.(2.1.19), then the fermion fields were also conformal fields of weight 1/2. This suggests that if we consider lattices with points of squared length unity, it should be possible to construct fermionic quantities from vertex operators. Considering two e f 2 2 such v e r t e x operators, V (z) and V (£), where e = f = 1, we find from eq.(3.2.10) that if e.f > 0 , V®r V* s -(-l)® ' 7 'f v* s V® r = 0 * = 6 ^ if e.f=-l . (3.3.1) r+s,o v 7 Here we have used the fact that (in Euclidean space) e.f = -1 only if e=-f. These are almost the desired fermion anticom- mutation relations apart from the troublesome (-1) * factors. As shown by Goddard and Olive (1984), these signs can be 0 corrected for by multiplying Vr by a 2-cocycle c q and defining e H (3.3.2) r The c can be constructed for the unit vectors defining a e cubic lattice so as to retain the hermitian properties of the vertex operators, i.e. (H®)+ = Hi® (3.3.3) -60- and also produce the desired anticommutation relations e (3.3.4) (H r ’ — 6 r+s,o 6 e,-f o To obtain hermitian fermions of the kind encountered in chapter 2, define the linear combinations 1 1 e —e H_r = ^— (H„ r + H_ r ) , = — (H® - H~®) , (3.3.5) which obey (H^, H^} = 6lJ6r+S;0 , l and have the hermitian property (»J:)+ = H i r . (3.3.7) Note that we have r,s e 7 (R) or 7+1/2 (NS) and the corresponding spectrum of momentum values is given by (3.2.12) and (3.2.15). In the Neveu-Schwarz case this means that p lies on the cubic lattice while in the Ramond case it is dis placed from the lattice by a vector of the form (±— ,±— ...±— ). 2 2 2 In d dimensions the cubic lattice generated by the basis vect ors e., l 3.4 Lattices with points of square length 2 For the case of a =2, the corresponding vertex operator, Va (z), has conformal weight 1, which is the same as that of a Kac-Moody generator as shown by eq.(2.2.6). When constructing the Sugawara form of the Virasoro algebra in section (2.3), it was noted that the resulting central charge, c, was bounded below by rank g. As shown by Goddard and Olive (1986), this lower bound is attainable in the case where the Lie algebra g is simply laced (all the roots are of the same length) and the level of the corresponding Kac-Moody algebra, g, is unity. However, the Cartan subalgebbra, h, of g gives rise to a [ u ( 1) ] B Kac-Moody algebra and the resulting Sugawara construction also has c=rank g. We then have the possibility of constructing a third Virasoro algebra which is the differ ence of the above two and has vanishing central charge by eq.(2.3.15) and hence is trivial. Thus we have the quantum S\ equivalence of a Sugawara Virasoro algebra constructed from h and a Sugawara Virasoro algebra constructed from g. This suggests that it must be possible to write all the generators A A of g in terms of the h generators and this is achieved precisely by the vertex operator construction. The vertex operator construction for simply laced, level 1 Kac-Moody algebras was first discovered by Frenkel and Kac (1980) and Segal (1981), though the notation used here follows that of Goddard and Olive (1984). For two vectors a1 and p1 of square length 2 in Euclid ean space, we have ^ |oc.p| < 2 , (3.4.1) -62- so equation (3.2.10) reads, if a.p > 0 , VavPn m v ' mvpv“ n = 0 if a.p = -1 , = vn+m “ + P = a.a + n6 if a.8 = -2 , (3.4.2) n+m n+m,o using the fact that a.p=-2 only if p=-a. If a and p are roots of g, then a+p is also a root for a.p=-l as the Weyl reflect ion of p in a gives a+p. Thus we again have to correct for the (-1) *p signs and this is achieved in a similar way to the previous section, defining cocycles c and a a c (3.4.3) En Vn a Again the cocycles can be constructed to maintain the hermit- ian properties (En)+ = E-n > (3.4.4) and the resulting commutation relations are = 0 i f a .p > 0 , _ a + 8 = e (a ,p ) E , H i f a .p = - 1 , n+m = lx .a , + n6 , if a.p = -2 , n+m n+m, o where s(a,p) is another cocycle taking values ±1. Together -63- with the remaining commutators, [a1 , a"5] = n L n * mJ 6 1 ^6 n+m,o ’ r i „a i i t^oc laL n . E m l J = a E n+m , (3.4.6) o this defines a level 1 Kac-Moody algebra (4, =2 so the level is x=k=l), written in a Cartan-Weyl basis. The analysis leading to (3.2.13) is a little more complicated here because, as well as imposing the constraints En iJV = 0 - n>0 > (3.4.7) we also want to impose e“ |p > = 0 , a>0 . (3.4.8) 0 0 (a>0 means a is a positive root.) Equation (3.4.8) is equiv alent to the condition a.pQ > 0 , a>0 . (3.4.9) Equation (3.2.13) then shows (since we only consider the Ram- ond case), that p^ must be zero or a minimal weight of the Lie algebra g (a minimal weight is a fundamental weight obeying eq. (3.4.9) and also 4>.po=l). These are in fact all the level 1 representations of g (see e.g. Goddard and Olive, 1986). ^ In this section we have noted the quantum equivalence of a Sugawara construction based on the Kac-Moody algebra and -64- the Sugawara construction based on the affinised Cartan sub- /\ algebra h in cases where g has level 1 and g is simply laced. We can regard h, defined in eq.(3.4.6), as being constructed from rank g free bosons, a1(z) . In section (2.4) we noted the quantum equivalence of a Sugawara construction and a free fermion Virasoro algebra and gave the conditions under which this equivalence obtained. These two results overlap only in A the case of level 1 representations of so(2r), where we have a fermion-boson equivalence since the representations can be realised in terms of r free bosons (vertex operator construct ion) or 2r real fermions (quark model construction). Such equivalences have been known for a long time (Skyrme, 1961) though have only been understood more recently in terms of Kac-Moody algebras (Witten, 1984, Goddard and Olive, 1986). In fact, the equivalence is rather more subtle than one might initially imagine on account of so(2r)A possessing 4 inequiv alent level 1 representations. It is possible to express the 2r fermions in terms of the r bosons by means of the vertex operator construction of the previous section. By considering operator products of the fermion fields so written, the boson ic fields can be recreated. Thus whether one writes a fermion field per se or a vertex operator is often a matter of conven ience. For example, the calculations of Green and Schwarz (1981) to establish spacetime supersymmetry of the superstring using the fermion emission vertex can be much simplified by writing the fermionic part of the vertex as a vertex operator itself. We shall use similar bosonisation procedures in chap ter 8 when discussing the ghost fields ^of covariant string theory. - 6 5 - 3.5 Vertex Operators for Virasoro algebras with 'u(l) current1 terms The vertex operators of section (3.1) were constructed so that they transformed as conformal fields with respect to the corresponding Virasoro algebra (3.1.9). We saw in section (2.6) that it is possible to to add extra derivatives of u(l) currents to the Virasoro generators to obtain another Virasoro algebra. However, these additional terms spoil the conformal properties of the vertex operators and the definition (3.1.4) has to be modified if we wish to retain vertex operators as conformal fields. To see how this works, consider the u(l) Kac-Moody algebra a 1 = n6 , (3.5.1) [a n m J n+m,o and the c=l Virasoro algebra x x a a . (3.5.2) L n = ~ l x - m n+m x 2 m The vertex operators are given by eq.(3.1.4), where v -n . dQ a(z) = ) a z = i z —— (3.5.3) n dz Now, by eq.(2.6.3), the modified Virasoro generators are given by l |! = L + in(5a . + p2/2 6„ , (3.5.4) n n r nl r ' n,o v 8 2 and we recall that cp = l + 12p . We find we can then write -66- down a conformal field 2 , v a /2+ipa o iaQ(z) o „V ct (z) = z 7 K e ' (3.5.5) v y o o 2 which has conformal weight a /2+ipa with respect to the Virasoro algebra (3.5.4). As noted in section (2.6), for unitary representations a and p are required to be real so this form of the vertex operators seems to have little relevance. However, for statistical models, the unitarity condition can be relaxed, p can take imaginary values and vertex operators of the form (3.5.5) do play a role (Dotsenko and Fateev, 1984). Another way of avoiding these difficulties is to use u(l) Kac-Moody generators of level -1 (though this also requires a relaxation of our previous unitarity condit ions) . We define (3.5.6) [p*, q *] = i (3.5.7) from which we can construct the c=l Virasoro generators -m’ an+m 1 x (3.5.8) We can then form the vertex operator (3.5.9) which is hermitian, i.e -67- (VX (z))+ = VX (z) , (3.5.10) 2 and defines a conformal field of weight X /2 with respect to (3.5.8), where dQ *(z) a ’ (z) 1Z (3.5.11) dz The analogue of eq.(3.5.4) is ,P -= L ’ + in p a * - p /2 6n (3.5.12) n n K n r ' n , o 2 which defines a Virasoro algebra with e=l-12p . With respect to this Virasoro algebra, define the vertex operator, z \2/2-|3\ o e\Q*(z) o (3.5.13) VX(z) 0 o which has conformal weight X /2-pX. The V^(z) are hermitian for p and X real and the conformal weight is also real. How ever, for a well defined vertex operator, p'=a^ must have imaginary eigenvalues despite being an hermitian operator. These results will be required for our discussion of the vertex operator realisations of the ghost fields occurring in string theory. -68- CHAPTER 4 VERTEX OPERATORS AND SUPERCONFORMAL GENERATORS -69- 4.1 The N=2 Superconformal Algebra A supersymmetric extension of the Virasoro algebra was given in section (2.5), where extra fermionic generators, G , were added to the Virasoro generators, L^. The algebra occur red in two forms depending on whether vz’S (Ramond) or re2?+l/2 (Neveu-Schwarz). This algebra has an N=1 supersymmetry as there is a single supercharge in each (R or NS) sector. How ever, it is also possible to construct superconformal algebras with a larger number of supersymmetries (Ademollo et al. ,1976 and Ramond and Schwarz, 1976). The N=2 superconformal algebra was originally introduced as a gauge algebra of a string theory, but the critical dimension of such a theory turned out to be d=2 and hence of limited interest. However, these models could be of relevance when considering compactification of superstrings to four dimensions (Candelas, Horowitz, Strom- inger and Witten, 1985). They also occur in the covariant quantisation of superstrings (Friedan, Martinec and Shenker, 1985) and are also relevant for certain statistical models. The defining relations for these N=2 superalgebras are n+m,o * < 0 = {°r- = 0 > (G+ , G-} = 2L + (r-s) + (r2-l/4) 6 - r+s,o * r s r+s 3 + [Ln , Gy] = (n/2-r) Gn+r * [Ln > Gr] = (n/2_r) Gn+r ’ - 7 0 - = -m T , = n6 , , Tml n+m tT n ’ Tm] n+m,o f Tn ’ - / T C r • tT n- G r] . (4.1.1) <1 J c -J f C v where r,s e 7 (Ramond) or r,s e 7+1/2 (Neveu-Schwarz) and n,m e 7. The usual hermiticity conditions assumed are L ^=L , n - n ’ T "*"= T and f G“ ) "*"= G+ . We see that the L , T generators n -n ^ r ' -r n m form a subalgebra which is the semi-direct product of a Vir- asoro algebra and a u(l) Kac-Moody algebra. Unitary highest weight representations of (4.1.1) are characterised by c, h (the L q eigenvalue) and t (the T q eigenvalue). These eigen values have a spectrum which has been determined by Di Vecchia, Petersen and Zheng (1985), Yu and Zheng (1987) and Boucher, Friedan and Kent (1986) analogous to the results of Frie'dan, Qiu and Shenker (1985) for the N=1 case. Their results for the N=2 superalgebra are more complicated than for the N=0,1 results in that above the threshold of discrete values at c=3, it is not true that all representations with c>3, h>0 are unitary. However, we shall be interested in c<3, for which the unitary representations correspond to c = 3(1 - -) , m=3,4. . . , (4.1.2) m h = — [p2-l - (s-r)2] + , (4.1.3) 4m 8 t = P 1 [ — + - ] . (4.1.4) J m-2 m 2 ■ where p = l,2...(m-l), |S| < p-1, (p-s) is odd, r = ±1 (R) or 0 (NS). -71- 4.2 Vertex Operators for points of square length 3 In the previous chapter, we saw that vertex operators could be associated with points of squared length 1 or 2 giving rise to closed algebraic structures. It is natural to ask whether any further interesting structures arise from considering lattices with points of square length 3,4.. etc. In general, such constructions will not lead to closed alge bras due to the higher order poles appearing in equations like (3.2.10). One exception occurs for the one dimensional latt ice of points of square length 3 (Waterson, 1986). In this case we have two vertex operators, which we shall denote suggestively by x+ z 3/2 o ei/"3Q(z) o G + (z) O 0 ^"z3/2 o e”i/3Q(z) o G “ (z) = l G~z (4.2.1) r where Q(z) is the 1-dimensional Fubini-Veneziano field and X + and X are normalisation coefficients to be determined. If G+ (z), G (z) are hermitian conjugates of each other as we have assumed, then (X J = X . The field Q(z) can be used to construct a c=l Sugawara Virasoro algebra by the construction given in eqs. (3.1.8) and (3.1.9), namely *. ( z ) = i t y D = - ^ > t (z ^ . (4 -2 -2 > n 2 where -n dQ T(z) iz— (4.2.3) = I v n dz -72- Using the results of section (3.1), we see that G+(z) both have conformal weight 3/2 with respect to the Virasoro algebra, which is the same weight as the supercharges of the superconformal algebras given in sections (2.5) and (4.1). This suggests that we may be able to construct a superconform al algebra out of the G (z), £(z) and T(z). Actually, because we have two supercharges and the u(l) current generator, T(z), we may expect to obtain a realisation of the N=2 superconform al algebra given in the previous section. Since the super charges can be integrally or half integrally moded, we should be prepared to let the index r in eq.(4.2.1) run over H (R) or ff+1/2 (NS). Most of the commutation relations given by eq.(4.1.1) are straightforward to check. For example, since we know G± (z) are conformal fields of weight 3/2, we have (n/2-r) G*+r , (4.2.4) and since T(z) is a conformal field of weight 1 we have (4.2.5) n+m The defining commutation relations for T , namely (4.2.6) enable us to compute [T , G*] = +/3 G* (4.2.7) L n’ rJ n+r -73- + + It remains to check the {G- , G"} anticommutators and the usual 1 r s ‘ arguments involving operator product expansions and contour deformations lead to + s+1 {G \ V f d£ f dz zr+1 0 1/5(Q(z )-Q(6))o (z_5)-3 r ’ J c o (/3(r-s)Tr+s + 6i.p+s + 0-2-l/4)6r+S;0) . (4.2.8) Comparing this expression with the corresponding one from eq. (4.1.1), we find we must choose + — . X \ = 1 (4.2.9) 2 3 and then we have the N=2 superconformal algebra with c=l. (The {G , G } =0 and {G , G } =0 relations follow immediately as in the relevant operator product expansions there are no poles at z=£ so the contour integrals vanish.) This construction is somewhat simpler than the construction of fermions or Kac- Moody generators discussed in the previous chapter in that the 2-cocycles used there to ensure correct (anti-)commutation relations are not needed here. It seems remarkable that a model containing a single bosonic field defined on a circle can possess such a rich structure as an N=2 superconf ormal symmetry. This bosonic construction of a superalgebra should be contrasted with the purely fermionic construction given in section (2.5). - \L The value c=l for the N=2 superconformal algebra corres ponds to taking m=3 in eq.(4.1.2) and we should check that the spectrum of h and t values for the unitary highest weight rep reservations agrees with (4.1.3) and (4.1.4). The highest weight states, |h,t>, are characterised as before by Tn lh,t> = 0 , n>0 , T |h,t> = t |h,t> , (4.2.10) o L |h,t> = 0 , n>0 , n L |h,t> = h |h,t> . (4.2.11) o 1 2 Since L q = — TQ + £ T_nTn » the highest weight states have 2 n>o h _ 1 + 2 h — t • (4.2.12) 2 Highest weight states also obey G* Ih,t> = 0 , r>0 , (4.2.13) so we can apply the results of section (3.2 ) to find the allowed values of t and hence h. Using eq.(3.S 1.14) with a=/T and p identified with t, we find o /3t = ±1/2, ±3/2 (Ramond) , /3t =0, ±1 (Neveu-Schwarz) . (4.2.14) The corresponding h values are then given by - 7 5 - h = 1/24, 1/24, 3/8, 3/8 (Raraond) , h = 0, 1/6, 1/6 (Neveu-Schwarz) . (4.2.15) The values of t and h given by (4.2.14) and (4.2.15) agree with the results (4.1.3), (4.1.4) for m=3. + In the Ramond sector, the generators G q intertwine the vacuum states | 3/8, ±/cT/2> and, from (4.2.1), we find G " | 3,- £3" > = 0 . (4.2.16) 8 2 The other Ramond vacua, |l/24, ±l/2/lT>, are G* eigenstates with zero eigenvalue. It should be noted that the N=2 algebra contains the N=1 algebra of section (2.5) as a subalgebra. The value c=l is the only c value common to the discrete spectra of allowed c values for unitary highest weight representations of the N=2 and N=1 algebras given by eqs.(4.1.2) and (2.5.7). This N=1 algebra can be realised, for example, by the generators £ and G ’, where r (4.2.17) which obey eqs. (2.5.1-3) with c=l. The allowed spectrum of h values for the N=1 algebra, given by (2.5.7), is -76- h = 1/24, 1/16, 3/8, 9/16 (Ramond) , h = 0, 1/6, 1, 1/16 (Neveu-Schwarz) , (4.2.18) and we see all the h values appearing in (4.2.15) also appear in (4.2.18) as indeed they must. - 7 7 - 4.3 Further Realisations of N=2 Superconformal Algebras A more general realisation of the N=2 superconforraal algebra has been given by Di Vecchia, Petersen, Yu and Zheng (1986) using just fermionic fields to construct the generat ors. In this section we shall show that their construction for the c=l case is the same as the bosonic case of the previous section. This provides an example of the use of vertex operators and the power of quantum equivalences. Other realisations will also be mentioned. The fermionic construction involves an su(2) level 1 Kac-Moody algebra constructed by the quark model of section (2.4). However, in section (2.4), the representation matrices McL were taken to be real whereas for su(n) we can have complex representations. Following Goddard and Olive (1986), we can always restrict attention to real representations by doubling the number of fermions. In the present case the level 1 rep resentation corresponds to using 4 real fermions and defining J X(z) = l A " n = — (H1(z )H4 (2)-H2 (z )H3 (z )} n ii 2n J2 (z) = l A n = - i (HX(z )H3 (z )+H2 (z )H4(z )) n 2 J3(z) = l J3z"n = (H1(z)H2 (z)-H3(z)H4(z)) (4.3.1) n 2 which can be shown to obey c n+m 2 -7 8 - using the operator product expansion (2.1.17). Here the structure constants fal3c take the values „123 231 312 f = f = f (4.3.3) a b c with all other f vanishing. Since the su(2) subalgebra generated by has roots of unit length, the level of the representation as given by eq.(2.2.9) is 1 as claimed. It is also possible to construct a u(l) Kac-Moody algebra commuting with the su(2) one by defining I(z) = — (H1(z )H3(z )-H2(z )H4(z )) , (4.3.4) which obeys [ I , I ] = n6 , , (4.3.5) L n* m J n+m,o v ' and [In > J„] = 0 • (4.3.6) Constructing a Sugawara Virasoro algebra for this u(l) Kac-Moody algebra and adding it to the su(2) Sugawara Virasoro algebra, we have the required conditions for a quantum equiv alence to the free fermion Virasoro algebra L(z) = i :z— H1 (z) : + - . (4.3.7) 2 dz 4 This corresponds to a symmetric space su(3)/su(2)xu(1) as discussed in section (2.4)^ and the Virasoro algebra L(z) (or 3L x ' ) has c = 2. Since we are dealing with a level 1 A representation of the simply laced algebra su(2), the vertex -79- operator construction implies that the su(2) Sugawara con struction is quantum equivalent to a Sugawara construction 3 using just the Cartan subalgebra generators, J (z) , so L(z) can be rewritten as L(z) = 1 ? (z) + £ :(z) , (4.3.8) where 3 l J (z) = *J3(z)J3(z)* , (4.3.9) and (4.3.10) £ J(Z) = -2 x*I(z)I(z)* x . To be more explicit, the vertex operator construction cor responds to taking H1(z) —i z-'z i/2foBiei*Q(z)0' J(°e e c^c- + 0; e ier Q(z) o 1 o 0°c - 1i) J /2 0 H2(z ) . zl/2foeiel*Q(Z)o,- o / ie1#Q(z)o (v o O l 0 oc-l) . n 1 l/2f o ie ^ o -ie2‘Q(z)o , H3(z ) 2 *Q (Z)0 ^ z U e oC2+ oe o°-2) ' , 1 /9 ^ ie9.Q(z) -ie .Q(z) H4(z ) — z1/2(°e 2 o °c9- 2 o°e 2 o-2;, °c 9) , v (4.3.11) /2 ' where the cocycles c+ ^, c+g are required as in section (3.3) to ensure the fermion fields anticommute correctly. Here and are two vectors obeying i , j = l , 2 , (4.3.12) and the cocycles can be chosen consistently so we have the -8 0 - /N usual vertex operator construction for the su(2) algebra, namely +i(e +e ).Q(z) J-(z) z °e o o o ’ t 3 - J (z) - (e1+eo).P(z) . (4.3.13) 2 1 Z Similarly, we can add an extra component to the Fubini-Venez- iano field to write the u(l) current, I(z), as I(z) = e3 .P(z) , (4.3.14) where e^ is a unit vector orthogonal to e^ and e ^ . It proves 3 advantageous to define linear combinations of the J (z) and I(z) generators by T(z) = l T z"n = i- (l(z) + 2J3(z )) , n / 3 R(z) = I Rnz n = (I(z) - J3(z)) , (4.3.15) n 3 which obey [ T , T 1 = n6 , L n ’ m J n+m,o [RL n ,’ R mJ] = n6 n+m,o , , [TL n ,* R mJ] = 0 . (4.3.16) These u(l) Kac-Moody generators provide two more c=l Sugawara Virasoro algebras defined in an analogous way to (4.3.10) and denoted by X and £ • From eq.(4.3.8) we find the alternative -8 1 - R T expression for L(z) in terms of £ and £ to be L(z) = £ R ( z ) + tT(z) • (4.3.17) It turns out that £ (z) is the Virasoro generator for which the N=2 construction is possible and from (4.3.13), (4.3.14) we have the expression for T(z) , T (z) = — (e1+e9+eJ.Q(z) (4.3.18) / 3 1 * 3 + The relevant G (z) given by Di Vecchia et al. becomes, in this notation, ± i (e + z3/2 o l+e2+e3)-Q(z)o G±(z) X" e (4.3.19) 0 0 where X* are the same normalisation coefficients as appeared in section (4.2). Thus we see that the fermionic construction reduces to the bosonic construction of the previous section provided we identify /*5" Q(z) there with (e +e +e ).Q(z) here. JL Ci O The general construction for the discrete unitary highest weight representations of the N=2 superconformal algebra corresponds to taking n copies of the su(2) level 1 algebra /V A together with the u(l) algebra generated by I(z). A u(l) subalgebra (corresponding to Rn above) is subtracted from the resulting Sugawara construction giving a Virasoro algebra with an overall c given by 3n 2 c + 1 1 = 3( 1 - ) , n = l ,2 . . . (4 .3 .2 0 ) n+2 n+2 -8 2 - which is the same as eq.(4.1.2). Di Vecchia et al. show that the corresponding spectrum of h and t values is that given by eqs.(4.1.3) and (4.1.4). Although this is an explicit con struction for a general c value on the list (4.1.2), the con struction does not necessarily make the most economical use of the fields as shown by the example of the previous section. Another interesting construction of these algebras has been given by Schwimmer and Seiberg (1987), who show how to construct the c=3/2 representation (corresponding to m=4 in eq.(4.1.2)) using a single boson and a single free fermion. Actually, their construction has a larger, N=3, algebra for which the N=2 superalgebra is just a subalgebra. Other relat ed constructions have been discussed by Schoutens (1987) and Aratyn and Damgaard (1986). The original discussion of the N=2 superconformal alge bra (Ademollo et al. , 1976) was in terms of a construction involving 2 fermions and 2 bosons. Denoting the bosons by P1(z) and the fermions by Ha(z), where (4.3.21) and defining canonical commutation relations r+s,o * (4.3.22) we write the N=2 superconformal generators as x 1 dHa a £ L(z) = I xp1(z)P1(z) + — : z— H (z): + - (4.3.23) 2 X 2 dz T(z) = -iH1(z)H2(z) , (4.3.24) - 8 3 - G+(z) = — fp1(z)H1(z)+iP1(z)H2(z)+P2(z)H2(z)-iP2(z)H1(z)) , n G~(z) = — (P1(z)H1(z)-iP1(z)H2(z)+P2(z)H2(z)+lP2(z)H1(z)) . /2 (4.3.25) As usual, e=0 or 1 depending on whether the H (z) are Neveu- Schwarz or Ramond fermions. It can be shown that the above fields generate the N=2 superconformal algebra with c-3. In chapter 7, where the BRST approach to quantisation is considered, it will be shown that for the various conformal algebras critical values of c arise for which it is possible to cancel the central charges of the algebra by adding extra ghost fields. This is necessary for a consistent quantisation scheme and leads to the well known critical dimensions for the various string theories. For example, the bosonic string has a critical c value of 26 and, since each bosonic field con tributes c=l, this leads to a critical dimension of d=26. In the present case of N=2 superalgebras, the critical value of c is c=6 and the original realisation of this algebra given above has c=3 leading to a critical dimension of d=2. This is a widely quoted result but is rather misleading as we have seen there are alternative realisations of the algebra in terms of different numbers of fields giving different c values and hence different critical dimensions since c ..=6 still. crit Thus, for example, the c=l realisation of section (4.2) will have a critical dimension of d=6, and the Schwimmer-Seiberg construction will have critical dimension d=4. -8 4 - 4.4 Vertex Operators for points of squared length 4 ? As was seen in section (4.2), it is possible to realise a closed algebraic structure using vertex operators corres ponding to points of square length 3 (for a 1-dimensional lattice) . It is natural to enquire whether we can extend the construction to vertex operators for points of square length 4 or more. We will find no new structures emerge, at least for the commutator or anticommutator brackets we have been using, This is because high order poles occur in the operator product expansion of two such vertex operators giving rise to terms x 3 x like xP (z)x , which cannot form a closed algebra because they x 4 x commute to terms in P (z) etc. Of course, one can define a x ' ' x more complicated bracket for the vertex operators in order to cancel the unwanted higher poles. The most notable example of this occurs in the context of the Monster group and the associated (but non-associative!) Griess algebra (see Frenkel, Lepowsky and Meurman, 1984 for a review). This is related to an assignment of vertex operators to points on the Leech lattice (a 24 dimensional lattice with points of minimum square length 4). However, here we shall restrict ourselves to conventional (anti-)commutation relations and see whether we can construct anything interesting from the vertex operat ors + + -n 2 o i2Q(z)o V (z) = 7 V z = vz e v ' L n r o n 2 o^-i2Q(z)o v (Z) = l V nz n = y z “ °e (4 .4 . 1 ) o ’ n where y is a normalisation constant -8 5 - We have restricted attention to a 1 dimensional lattice as we already found that that had to be the case for points of square length 3 in order to get a closed algebraic structure. Since the vertex operators (4.4.1) have conformal weight 2, they should behave somewhat akin to Virasoro generators. Forming the hermitian linear combination of V± (z) by L(z) = l L nz " n = V + (z) + V"(z) , n L(z) = l Lnz_n = i(V+(z) - V (z)) , (4.4.2) n and restricting attention to L(z) for simplicity, we compute + — (n3-n)6 (4.4.3) [Ln. = (n-m) £ n+m 12 n+m,o ’ where (4.4.4) with P(z) = iz-^— Q(z) (4.4.5) dz Another Virasoro algebra can be defined by (4.4.6) which has c=l/2 provided the coefficients a,b are chosen to be a = ±1/2 , b = 1/2 (4.4.7) However, the algebra is nothing new; it is just the single free fermion Virasoro algebra of section (2.1) rewritten in -86- terms of bosonised fermions. To see this define HX(z) = i. zl/2(oeiQ(z)o 0 -iQ(z)0) V V 0 0 O n J * H2(z) = L. zl/2,o iQ(z)o _ o -iQ(z)0) (4.4.8) v ^ o o o J We then have the quantum equivalence relations 2 dH , x £ _ x^2 , Nx , z 1 : z— ^H .,1 (z): 1 p (z) + _ A A a 2 dz 16 4 4 1 dH2„2/ N e — 1 Xt.2o , nx z 2 ,oe2iQ(z)o + oe-2iQ(z)o^ : z— H (z): P (z) - — ^ A rv n ' ’ 2 dz 16 4 x x 4 0 0 0 (4.4.9) which shows that Kq is just a free fermion Virasoro algebra as stated. Thus we see that the vertex operators defined in (4.4.1) do not lead to any new algebraic structure. \ -8 7 - CHAPTER 5 SYMPLECTIC BOSON CONSTRUCTIONS -88- 5.1 Realisations of Kac-Moody Algebras In section (2.4) we discussed the quark model construct ion of a Kac-Moody algebra, g, starting from a real orthogonal representation of the corresponding Lie algebra g. The rep resentation matrices M preserve the symmetric form 61J , l 1 M it = -M1 , (5.1.1) and the construction involves dim M real fermion fields. Here we shall show that there is a similar realisation of certain Kac-Moody algebras starting from a real symplectic represent ation of a Lie algebra g (Goddard, Olive and Waterson, 1987). SL The representation matrices N obey [Na, Nb] = fabcNC . (5.1.2) jj g where f are the structure constants of g and the N pres- V ct 8 erve the symplectic form J , l J N a J = - ( N a )t . (5.1.3) Ja^ is a real antisymmetric matrix and we shall choose a basis so that it has the canonical form of along the diagonal and zeros elsewhere. Instead of fermion fields, the construction makes use of symplectic bosons, £a(z), defined by S a (z) l (Ramond) or s e #+1/2 (Neveu-Schwarz) respectively. Here 2n S i is the dimension of the representation given by N , which, oi course, must be even for a symplectic representation. The fields £a (z) are taken to be hermitian on the unit circle so U g ) + = lts • (5.1.5) We also impose the canonical commutation relations a iJa p 6 (5.1.6) [5r ’ r + s , o and take the fields to act in a Fock space with vacuum |0> obeying 5“ |0> = 0 , r>0 . (5.1.7) The commutation relations (5.1.6) then give rise to the oper ator product expansion 5a(z)5p(5) = :5“(z)CP(5): + ij“PA(z,5) , lz| > |?| , (5.1.8) where A(z,£) is defined as in (2.1.18) and the normal ordering is defined by r<0 , ^rs s ^rs s - U “ . t£} . r=0 , 2 r S = F P F a (5 .1 .9 ) q,s^r ’ r>0 , -90- analogous to (2.1.16) in the fermionic case. cl 3. Equation (5.1.3) shows that R =JN is a symmetric mat rix. We can use this to construct a realisation of a Kac- Moody algebra, g, by defining Ta (z) = I Tn z_n = " - Ra„g5“ (z)5P(z) . (5.1.10) n 2 ap This should be compared with the corresponding fermionic field construction given by (2.4.1). Note again there is no need cl for explicit normal ordering in (5.1.10) as R is a symmetric matrix. The usual methods of operator product expansions and £1 contour deformations show that the T q obey the Kac-Moody alge bra a i a b c [T if ab t c + nkr) 6 (5.1.11) n' c n+m n+m, o with central charge k = k /2 , (5.1.12) where Tr(NaNb) = Kt)ab • (5.1.13) ab ab The metric ri is less explicit than the 6 used in the ferm ionic case. We shall be interested in metrics of indefinite signature and (possibly) non-compact algebras g. Thus the normalisation implied by (5.1.13) is not very explicit, but we ab ab can take t) to be 6 on the maximal compact subgroup of g if g is simple. Note the change of sign between eqs. (2.4.3) and (5.1.13); this is in order to preserve the relationship between k and < given by (5.1.12) and it will appear more -9 1 - natural in the next chapter when we combine fermionic and bosonic constructions in a supersymmetric framework. We pres erve the definition of the level, x, of the representation, i ,e. x = 2k/ 2 cito where cj, is a long root of g calculated using the metric ti ab Thus x is independent of the normalisation used for tj but it is no longer restricted to positive integer values as was the case for the unitary highest weight representations. Unitar- ity is lost because the symplectic bosons do not act in a positive space, as evidenced by the fact that iJa^ has eigen- ab values ±1 and r\ possibly has an indefinite signature. From a physical viewpoint this may seem a major drawback, though, as will be seen, such systems have connections with the BRST ghost fields arising in the covariant quantisation of string t h e o r i e s . \i -9 2 - 5.2 Two Virasoro Algebras and a Superalgebra Theorem From the Kac-Moody algebra (5.1.11), we can construct a Virasoro algebra by the Sugawara method just as in section (2.3) though now using the metric rather than 6 Define £ 5 (z ) = l n l a b (5.2.1) n 2k+Q 3. where T (z) are the generators of the Kac-Moody algebra g given by (5.1.11) and is the value of the quadratic Casimir of g evaluated in the adjoint representation. The quadratic Casimir is given in general by (5.2.2) - 11 ab” 3^*3 - 1 for an irreducible representation defined by the symplectic matrices N3,. Thus, for the adjoint representation (assuming g to be simple for the moment) ~ac _bd _ ab (5.2.3) f df c = “ V ’ el b where f are the structure constants of g. The normal c ordering given by the crosses in eq.(5.2.1) is defined, by analogy with (2.3.2), by x TaTb x Ta Tb n<0 , ^ab x n m x 'ab n m Tb T a n>0 . (5.2.4) 'ab m n -9 3 - Similar arguments to those used in section (2.3) show that (5.2.1) defines a Virasoro algebra with central charge 2k dimg _ x dimg C “ “■ (5.2.5) 2k+Q x + h where h=Q^/(J> is the dual Coxeter number of g. We can also construct a free field Virasoro algebra analogous to the fermionic construction given by eq.(2.1.19). This takes the form L5(z ) = l |» z n = i J :Z^ V ( z ) : n l where again e=0 or 1 depending on whether £a (z) are Neveu- Schwarz or Ramond fields respectively and normal ordering is defined as in eq.(5.1.9). It can be shown that (5.2.6) defines a Virasoro algebra with c=-n. Since there are 2n symplectic bosons used in the construction, we can say that each contributes -1/2 to the overall c number and also con tributes -1/16 to the Ramond L q eigenvalue. These values are just the negatives of the corresponding results for the ferm ionic free field construction. We now wish to establish the criteria for the free field construction (5.2.6) to be quantum equivalent to the Sugawara construction (5.2.1). This will provide the analogue of the symmetric space theorem discussed in section (2.4), which gave the conditions under which the free fermionic construction equals the corresponding Sugawara construction. Since we can normal order either with respect to the symplectic bosons -94- themselves or with respect to the Kac-Moody generators con structed from them, there are two ways of writing the operator ct b product expansion ti (z)T (£)> namely „abTa(z)Tb(5) = X bTa(z)Tb (5)^ + ; dimg - ^ - 5 , (5.2.7) 2 (z-£) or TlabTa(z)Tb(C) = :nabTa (z)Tb(C): + :?a(z)5P(?): A(z,5) 2 ---- dimN A (z,£) , (5.2.8) 2 w i t h k given by (5.1.13) and by (5.2.2). We now take the limit l+z using the relations (2.4.9) and k dimg = -Qn dimN , (5.2.9) which follows from the trace of (5.2.2), to obtain - Hab *Ta (z)Tb(z)* = - ilab:Ta (z)Tb(z): + QNL5(z) . (5.2.10) Dividing by we see that the term on the left could be identified with fj* (z) if 2QN = K + % * (5.2.11) This condition also ensures that £,^(z) and L^(z) generate Virasoro algebras with the same c value, namely -9 5 k dimg QNdimN dimN c = ------= ------= ------, (5.2.12) < +Q(t) 2Q n 2 making use of (5.2.9). Thus (5.2.11) is a necessary condition for the quantum equivalence of L^(z) and i5(z), though a sufficient condition is that _1 ^ a b :Ta(z)Tb (z) 0 . (5.2.13) 2Q N Expanding this out in terms of the £a(z) fields and using the symmetry properties of the fields under the normal ordering operation, we find (5.2.13) is equivalent to + Ra Rb + Ra ,Rb ) = 0 . (5.2.14) W B\ p R T6 ay op ao (3y' In the fermionic construction of chapter 2, the analogous conditions to (5.2.11) and (5.2.14), namely (2.4.13) and (2.4.11), were in fact equivalent conditions. In the present case with condition (5.2.11), it is still possible to con struct a third set of Virasoro generators, K (5.2.15) n which generate a Virasoro algebra with c=0. However, the argument of section (2.4), whereby only the trivial represent ation of K could occur, breaks down here because the operat ic 1 ors do not act on a positive space. Thus eq.(5.2.14) is a stronger condition and it is equivalent to the statement that there exists a Lie superalgebra with fermionic generators -96- transforming under the representation of g given by the matrices Na . In other words, it is possible to extend the Lie algebra g, generated by t , to a superal gebr a, a, by the addition of fermionic generators sa obeying [ta , tb] = ifabctc , [ta , s ] = iNaP s , L a J a p II +b (s » (5.2.16) 1 a a p z 11 ab ’ where we have written R a = JNa as before. This algebra has the important property that it possesses a quadratic Casimir operator given by ii j (5.2.17) SLD a p The Jacobi identities for (5.2.16) reproduce eqs.(5.1.2) and (5.2.14), whilst the requirement that Q commutes with all the Si Si Si generators confirms that R = JN . For cases where this quantum equivalence holds, the metric ri . has necessarily to be of indefinite form. To see ab this define Xa = Ra . x V (5.2.18) a{3 and multiply (5.2.14) by xax^x^x 6 to obtain 3^abXaXb= 0. If n K were positive (or negative) definite, this would imply a d X a = 0 for all xa ,x^ and hence R a a= 0, contrary to assumption. -9 7 - This means that g has to be non-compact or has to be non simple with the metric taking different signs for the differ ent factors, which could all be compact. We will see in the next section that the algebras we are interested in are non-simple in general so we should allow for this minor complication. Suppose g is the direct product of simple algebras and u(l) factors, written as g = ® gA . (5.2.19) A A Equations (5.1.13) and (5.2.2) now apply for each factor g so we write Tr(NaNb) = KAr)ab , (5.2.20) nabNaNb = - QA 1 . (5.2.21) Similarly, eq.(5.2.3) becomes _ac „bd _A ab (5.2.22) f d f c - - % 1 ’ where the indices a,b,c,d are those indices corresponding to A the factor g of g. The metric T)ak now has one free normal- isation parameter for each factor g . Taking the trace of (5.2.21) yields k A dimgA = - QA dimN . (5.2.23) Equation (5.2.10) again holds in this more general setting, where now -9 8 - Qjq I Qjg • (5.2.24) A Thus a necessary condition for iS (z) = L^(z) is that 2Qn » k A + , (5.2.25) in order for the left hand side of eq.(5.2.10) to be express ible as a sum of Sugawara constructions, one for each factor A . g of g, i.e. = l -PT-K x ^labTa(Z)Tb(z) l , (5.2.26) A k +Q. where the summed indices run over values corresponding to the A relevant factor g . Again condition (5.2.25) is sufficient to show the c values for £^(z) and L^(z) are both -dimN/2, though it is not sufficient to prove quantum equivalence. The cases where this equivalence holds are in 1-1 correspondence with the superalgebras (5.2.16) possessing quadratic Casimir oper ators (5.2.17). -99- 5.3 Examples of the Superalgebra Theorem The simple superalgebras possessing quadratic Casimir operators introduced in the previous section have in fact been classified by mathematicians (Kac, 1977) and are denoted su(p |q), osp(p|q), g(3), f(4) and d(2|l;a) (see e.g. Freund, 1986 for details of these algebras). By the results of the previous section, each case on the list corresponds to a quantum equivalence of {J* (z) and L^(z). Unfortunately, writ ing an arbitrary superalgebra in terms of simple superalgebras is more complicated than the corresponding result for ordinary Lie algebras. Hence it is not possible to conclude that all r r the cases where j^(z)=L^(z) correspond to direct sums of the simple superalgebras listed above. Nevertheless, these simple superalgebras provide examples to illustrate the results of section (5.2). We shall choose to normalise the generators of g such A that appearing in (5.2.22) can be written as A where h ^ is the usual dual Coxeter number for the factor g g A 2 of g and X plays the role of c}> used to normalise the gener ators in chapter 2. However, because of the indefinite A m e t r i c , X can be either positive or negative, but the relat- ive normalisations of each g factor are (almost) fixed. The level of the representation of is given, as before, by ^ A A, A x K /X J (5.3.2) 100- and so we can rewrite eqs.(5.2.23-25) in the form A A A A X x dimg = - dimN , (5.3.3) 2«N = 2 = ^A(xA + h A) . (5.3.4) A g A These provide a homogeneous set of equations for the X show ing that in general only one of them can be set arbitrarily. A With a convenient choice of normalisation for the X 's (namely setting X^= 1 below), we find the following values for the parameters occurring in eq.(5.3.4): a) su(p|q), p^q, g=su(p)@su(q)@u(1), dim N = 2pq, QN=(p-q)/2, hsu(p)= p ’ xl= " “ ,2q ’ xl= 1 ’ QN = qCP2-1) / ^ » hsu(q)= q » x2= “ ~-2p ’ x2= _1 ’ Q n = “P ( q 2"l)/2pq , hu(l) = ° * x3= (p_q)/^ * x3= » Qn = > b) su(pip) , g=su(p)@su(p), dim N = 2p , QN= 0 , hsu(p)= P ’ xl= _ i,2p ’ ^1= 1 * 4 =' (P^O/Sp , hsu(p)= p ’ x2= - -*2p • _1 ’ < 4 ='-(P2-D/2p ; 2 -101- c) osp(plq) , g=so(p)©sp(q), dim N=2pq, QN= (p-2q-2)/2, = (p-l)/2 , hso(p)= p_2 ’ xl= _2q ’ xl= 1 ’ 2= " ~ , X2= -2 , Q2 = -q-1/2 ; hsp(q)= ^ ' x d) g(3) , g=g2©sp(l), dim N=14, QN=1, h 2 = 4 , x 1= l.(-2) , \ 1= 1 , QN - 2 , hsP(i)= 2 ’ *2= - - x2= -4/3 - «n = - 1 ; e) f(4) , g=so(7)©sp(1), dim N=16, QN=3/2, hso(7)= 5 * xl= > x*= 1 - Q„ = 21/8 , = 2 , x 2= - - . 8 , \ 2= -3/2 , Q 2 = -9/8 ; sp(l) 2 f) d(2 |1 ;g) , g=su(2)©su(2)0sp(1), dimN=8, QN=0, hsu(2)= 2 ' xl= 1>(-2) ’ Xl= 1 ’ Qn = 3/4 , hsu(2)= 2 ’ x2= 1'<-2> ' x2= I* ’ % = 3p/4 , hSp(i)= 2 • x3= - - - 4 • x3= ~(1+n) » Qn = -3(l+n)/4 • 2 In cases a) and f) \i is an arbitrary constant showing that there is still some choice in the normalisation of the differ ent factors -102- 5.4 Critical Representations r In establishing the quantum equivalence of L ( z ) and i/ (z) in section (5.2), we divided eq.(5.2.10) through by with the implicit assumption that was non-zero. However, looking through the list of examples given in the previous section, we find that for the cases su(p|p), osp(2q+2|q) and d(2 |l;a), QN does in fact vanish. Thus for these cases eq.(5.2.10) reads - nab *Ta(z)Tb (z)* = 0 . (5.4.1) 2 We call the corresponding representations of the Kac-Moody /\ algebra, g, critical in these cases (Kac, 1987). Define the *unrenormalised' Virasoro generators by t V ) *Ta(z)Tb(z)* , (5.4.2) 2 ”* b A where the sum over a,b corresponds just to the factor g of g. By (5.4.1) we then have £ / = 0. This must be interpreted A in the context of highest weight representations since the are quantum operators and hence on a highest weight state, | l I n = 0 > Vn - (5.4.3) where )(}>> obeys T* |c|i> = 0 , n>0 . (5.4.4) -1 0 3 - Since [IAn- < 1 = Qj)/2 Tn+m = 0 - (5-4.5) we have [ £ A ] = 0 . (5.4.6) and this confirms that it is consistent to impose the con straints (5.4.3) for negative as well as positive values of n. This seems a rather strange result on the face of it, but it must be remembered that we have an explicit construction for cL the Tr generators in terms of symplectic bosons as given by eq.(5.1.10) and the state J<|>> is also a highest weight state for the symplectic bosons, i.e. c“ 1<1>> = o , r>0 . (5.4.7) By acting on a few states constructed from the vacuum by the operators , r>0, one can convince oneself of the validity of eq . ( 5.4. 3) . -1 0 4 - 5.5 Symplectic Fermions In section (5.2) we saw that free symplectic bosons contributed -1/2 to the Virasoro c number whereas the free fermions of section (2.1) contributed +1/2. Similarly, it was noted in section (2.3) that free bosons contributed c=l so one may naturally wonder if there is a symplectic fermion con struction where the fermions each contribute -1 to the c value. This is indeed the case as will be demonstrated in this section for completeness. The symplectic fermions, - -raP c i J rn6 (5.5.1) n+m, o where (z) = J ^ z"n , l and we assume the usual hermiticity properties U a (z))+ = With normal ordering defined analogously to eq.(2.1.16), we have the operator product expansion 4>a(zHPU) = :(j,a(zHPa): + iJap — . (5.5.4) ( z - O The free field Virasoro algebra is then defined by L (z) - - (z) (z) , (5.5.5) -105- which can be shown to obey the Virasoro algebra with c = -2d. Since we have 2d of these symplectic fermions, this shows that each such fermion does indeed contribute -1 to the overall c value as anticipated. The symplectic fermions are conformal fields of weight 1 with respect to L^(z). Like the symplectic bosons, the symplectic fermions would appear to have few physical applications, though they do arise when writing the superconformal ghosts of string theory in terms of vertex operators. By analogy with the bosonic vertex operator construction of chapter 3, it should be possible to construct ’vertex oper ators’ from these symplectic fermions. This appears to be the case, with a.Q(z) in the exponent of eq.(3.1.4) being replaced by iJ ^ O Qp(z), where Qp(z) now has an expansion in terms of symplectic fermion modes. Thus 0a must be grassmannian co ordinates. In order to have a well defined algebraic struc ture for these vertex operators, the 0a must be quantised and one would have to deal with 'grassmannian lattices’. -1 0 6 - CHAPTER 6 SUPERAFFINE ALGEBRAS AND SUPERSYMMETRIC SPACES -1 0 7 - 6.1 Affine Superalgebras In this chapter we shall combine the results of the F previous chapter concerning the quantum equivalence of Lr(z) a n d (z) with the analogous fermionic results of section (2.4) . We shall find both sorts of quantum equivalence are particular cases of a more general result. The framework for our discussion will involve an affinisation of the superalge bra given by eq.(5.2.16). In section (6.2) an explicit real isation of these affine superalgebras is given in terms of fermions and symplectic bosons. The generalisation of the Sugawara construction to a super-Sugawara construction is made in section (6.3). The circumstances under which the super- Sugawara construction is quantum equivalent to the correspond ing free field Virasoro algebra constructed from the same fermions and symplectic bosons are investigated in section (6.4) . This results in a 1 supersymmetric space theorem’, various examples of which are given in section (6.5). We begin by associating an affine superalgebra, a, with the superalgebra a given in (5.2.16). This affine superalge bra is defined by .-ab mC , . ab„ [T a , Tb ] = if „ T , ^ + nkri , (6.1.1) L n* m J c n+m 1 n+m,o [T*. S“ ] = iN%a s£+r , (6.1.2) (6.1.3) {s“ , sl) = Ra“P Tr+s + ikrjaP6r+s,o * where k is a central charge, n,m e 2T and r,s e % (Ramond) or -1 0 8 - r,s e 2T+1/2 (Neveu-Schwarz) . The bosonic part of the algebra given by eq.(6.1.1) is just an affine algebra, and the indices a 8 a,b,c and a,p are raised using r\ and J p respectively. Thus cc 8 b 8 R p is related to N p as in the previous chapter by a y D a (3 _ Tay ,, b (5 (6.1.4) Ra = J I’a b N Y The Jacobi identities determine the form of central term in (6.1.3) relative to that of (6.1.1) as well as requiring that the matrices N form a real representation of the Lie algebra ab given by the structure constants f . The condition (5.2.14) c is also reproduced from the Jacobi identity involving just the . Note that in the Ramond sector the superalgebra (5.2.16) is a subalgebra of the affine superalgebra (6.1.1-3) obtained by taking n=m=r=s=0. -1 0 9 - 6.2 Realisations of Affine Superalgebras We would like to be able to construct realisations of the affine superalgebras of section (6.1) in terms of free fermions and symplectic bosons. Because the bosonic part of the superalgebra need not be composed of compact algebras, we shall generalise the definition of free fermions given in section (2.1). Define HX(Z) = I bj: z -r , (6.2.1) r where K' bs> = ^r+s.o ’ <6-2‘2> and (»r)+= b-r > (6.2.3) where r,s e ZT (Ramond) or r,s e 5?+l/2 (Neveu-Schwarz). The m e t r i c t)1 "* replaces the used in chapter 2. Symplectic bosons, £a (z), are defined as in section (5.1) with Ja^ being replaced by Jap for notational convenience. We shall assume we have 2n symplectic bosons and m=m^+m2 fermions, where has m^ positive eigenvalues and negative eigenvalues. Since the fermions and symplectic bosons all have con formal weight 1/2, it is natural to try to construct the Ta(z) and Sa (z) generators bilinearly in these fields. Indeed, we <3l already have expressions for T (z) in terms of such quantit ies, namely eqs.(2.4.1) and (5.1.10). Sa(z) is a fermionic quantity and hence must contain one H 1 (z) and one £a(z). Thus we are led to the following ansatz, -110- Ta (z) = V Taz"n = - Ma . . H1(z) iP (z) ?p (z)5°(z) * n " 2 1J S“ (z) = I S“z_r = Xai(j H1(z)5 °(z) . (6.2.4) Normal ordering is not required in these expressions as we ~ a a assume M . . and K are respectively real antisymmetric and 1 j p a symmetric matrices. Here the index a runs over dim g values and a over dim N values, where N 21 is the representation of the bosonic part, g, of the superalgebra a, under which the fermionic generators transform. Note that the Ta generators are always integrally moded whilst the S® generators are integrally moded if the bosons and fermions are of the same (NS or R) kind and half integrally moded otherwise. We shall write K = JL as before, where J is a real symplectic form and 2L ^ ^ also define M = M n , which is again a real matrix. In order for T S L (z) to obey the affine algebra ^g, we require M q to provide a real representation of g and L £L to provide a real symplectic representation of g. We find that the ansatz (6.2.4) provides a realisation of the affine superalgebra a, given by eqs.(6.1.1-3) provided Xa. XP. ^ + Xa. XP . tJiJ = - R ap Ka lcr jp ip jo 1 a ap (6.2.5) Xa. Xp . Jap - Xa . Xp. Jap - Ma .. R “P , ( 6. 2 . 6) io jp jo ip ij a fta..X® - Ka X®. = Na “ XP. (6.2.7) ij ka ap ±i p la These equations are not very transparent as they stand but we note that a representation, r, of a is provided by -111- Ma ta -> i 0 'l 0 Laj a 0 x“ s ■> ( 6. 2. 8) Ya o where pa (6.2.9) (Note there is no significance to the position of indices in the ~ space.) The conditions for (6.2.8) to satisfy the superalgebra a are just the above conditions (6.2.5-7). Thus we are guaranteed an satisfying these conditions with the assumed form of M ct and L £1 . The central charge of the resulting superalgebra, a, is then given by k = tc/2 , (6.2.10) where - Ti^ m V 3) + Tr(LaLb ) = Kt)ab , (6.2.11) and -2i Tr(XayP) = . (6.2.12) These expressions can be more succinctly written by defining the supertrace in the usual way, namely Str (^g) = Tr(A) - Tr(B) , (6.2.13) \t from which we see we can write (6.2.11), (6.2.12) as Str(t t J = jct) , Str(sasP) = iKJ“P (6.2.14) -112- where t 3 * , s CC denote their matrix representation (6.2.8). We can evaluate the quadratic Casimir (which we are assuming to be unique) given by eq.(5.2.17) for the represen tation r of a with the result - + iJ apX“Yp “ «r* 1 ' " ' z b L* hb + iJapY“xP = Qr 12n * (6.2.15) Taking the trace of these relations and subtracting using eqs.(6.2.11) , (6.2.12) then gives k sdima = Q^, sdimr , (6.2.16) where sdima = dimg - dimN and sdimr = m-2n. An important example of this construction is provided by the adjoint representation of a, which corresponds to taking t 7 L . = N »a\ Y a - ~ 1 J r (6.2.17) X\x X icT 71 Rj and identifying r\ , ti and J,J. If these values are inserted into eqs.(6.2.15) we find Qr= Qa= + *A = 2Qn , (6.2.18) where KAT)a^= Tr(NaN^) for the component g^ of g and the other quantities are similarly defined as in chapter 5. Equation (6.2.18) reproduced eq.(5.2.25), a necessary condition for L^(z) = *5(z) and this is consistent with the 1-1 correspond- -113- ence with superalgebras established in chapter 5. The level of the adjoint representation of a is obtained from k = k /2 = Qa/2 = Qn . ( 6 . . 19) -114- 6.3 The Super-Sugawara Construction The generators Tn3* and SrOL of section (6.1) can be used to construct a supersymmetric version of the Sugawara Virasoro algebra. To establish this we first need to define a normal ordering for the generators analogous to that given for the T^'s in (2.3.2). This is given by x sasp x r<0 , x r s x Sar SPs = - spsas r r> 0 (6.3.1) (One may worry about the somewhat ad hoc definition of xSaS^xX O OK but this term will give unambiguous contributions in the application below since it is multiplied by the antisymmetric symplectic form, J _). The form of the quadratic Casimir, (5.2.17), for the superalgebra a suggests the form of a possible generalisation of the Sugawara construction (2.3.1). Define £(z) = — ix7>abTa(z)'I'b(z:)x + xiJaBS“(z^sP(z)x} ’ (6-3-2) where y is a renormalisation constant to be determined. As usual, £(z) is expanded into its Laurent series by l(z) = I l nz_n - (6.3.3) n and the algebra of the *s can be computed. To this end it is easiest to compute the commutation relations of £ with Ta -115- and Sar first. We find a (k+ k A/2 + QA/2) tin- TJ = Tm+n ’ (6.3.4) Y where the S i occurring in (6.3.4) are those corresponding to A the factor g of g. Since we know from chapter 5 that the superalgebras correspond to a quantum equivalence L(z) = £(z), for which a necessary condition was 2 Qn = we see that if we choose Y = k+QN , (6.3.7) then TffiS i and Sr (X will have the required property that they transform as conformal fields of weight 1 with respect to £ . It can then be shown that, for of Ramond form, the £ obey the Virasoro algebra with 2k sdima c = (dimg-dimN) = (6.3.8) k+Q 2k + Q N where the superdimension, sdima, is defined in the usual way and is the value of the quadratic Casimir (5.2.17) eval uated in the adjoint representation. As was shown in the previous section, this takes the value 2Q^ as assumed. We see eq.(6.3.8) is the natural generalisation of the central charge -116- for the Sugawara construction given by eq.(2.3.10). A minor complication arises when the are of Neveu- Schwarz form i.e. the index r takes half integer values. To see the nature of these difficulties, consider the usual Sugawara construction for which the calculation of the central term results in the expression (see e.g. Gomes, 1987) k dimg ?/ 2k dimg 1 ^n+m,o -----& I (nr-r ) 6 2k+Q r=l 2k+Q^ 12 n+m, o (6.3.9) However, if the Kac-Moody generators were half integrally moded (corresponding to a twisted affine algebra), the sum in (6.3.9) would be replaced by k dimg 2 n 2k dimg 1 6 ------i (nr-r ) (n +n/2)6 n+m, o 2k+Q^ r=l/2 2k+Q 12 n+m, o 4> (6.3.10) which does not have the canonical form for a central charge of a Virasoro algebra. However, if we redefine i 0 by t . t + b, for some constant b, we find a further contribution to the anomaly of -2nb6 . . Thus b has to take the value n+m, o A k dimg (6.3.11) 8 2k+Q, in order to reproduce the anomaly in the form (6.3.9). Adapting these results to the present case where a' are assumed to be integrally moded but can be half integrally moded, we see that the definition of £(z) given by (6.3.2) needs to be modified to -117- #t , x j , x (1-c) k dimN £.'(Z) = i(z) - ---- ^ ------, (6.3.12) 16 k+QN where e=0 or 1 according to whether S® is of Neveu-Schwarz or Ramond type respectively. The form of the factor y and the value of the central charge can be checked by arguments analogous to those used in section (2.3) given the form of the Virasoro generators in (6.3.2). Considering the case of the adjoint representation of a given at the end of the previous section, for which k=Q /2, we a find the super-Sugawara construction has sdim a c = ------(m-2n)/2 , (6.3.13) 2 in the notation of section (6.2). This is the same c value as that of L^(z) + L (z) for the corresponding free field con structions. In section (6.5) we will see that there is indeed a quantum equivalence of the free field and super-Sugawara Virasoro algebras in this case and in the next section we will investigate the general circumstances under which these sorts of equivalences obtain. We conclude this section with the observation that for the adjoint representation of a“ it is also possible to construct super-Virasoro generators of the kind encountered in section (2.5) out of the same boson and fermion fields used to construct a\ We define 1 -118- G(z) = l Grz — ( 1 fab° H (z)H.(z)H (z) r /Q^ 6 a + i (jNa)apHa(z)5a(z)5P(z) } . (6.3.14) 2 H F J Together with L (z)+L (z) (or jl(z) as we shall see this is an equivalent operator in section (6.5)), the Gr generators obey the super-Virasoro algebra given by eqs.(2.5.1-3) with c = — sdim a. G(z) has the same Ramond or Neveu-Schwarz 2 properties as its constituent fermion fields, independent ol the periodicity properties of the symplectic bosons. -119- 6.4 A Super-Symmetric Space Theorem We saw in the last section that when we constructed the super-Sugawara Virasoro algebra from fermions and bosons in the adjoint representation of a superalgebra, a, the central charge was equal to that of the free field construction. We shall show that the two Virasoro constructions are actually quantum equivalent (the equality of central charges being a necessary but not sufficient condition for this equivalence). In this section we establish general criteria for these sorts of equivalences to occur. The results will subsume the ’symmetric space theorem’ of section (2.4) and the ’super- algebra theorem’ of section (5.2) as special cases. Proceeding in the usual manner, we note that the expres sion 1 tiabTa (z)Tb(C) + i JapSa(z)SP(5) , (6.4.1) 2 2 can be normal ordered either with respect to the generators themselves as in section (6.3) or with respect to the con- stituent fermion and boson fields since we assume T cl (z), S C£ (z) are constructed as in section (6.2). Equating the two result ing expressions and taking the limit z->£ in the usual way leads to Y l(z) = i nab:Ta(z)Tb(z): + i JafJ :Sa(z)SP(z): + QrL(z) , 2 2 (6.4.2) where y is given by eq.(6.3.7) and L(z) is given by L ( z ) = LH( z ) + L ^ (z ) , (6.4.3) -120- f or T H, v 1 ~ dH1.. i , . , me ^ ,. L (z) = - ri • . : z— HJ (z): + — , Ki,j L^(z) = - J : z— £p (z): - — , l The c value for L(z) is (m-2n)/2, whereas that for ^(z) is given by eq.(6.3.8), where now k= 2Q^ = k +Qa = 2k+2Q^ . (6.4.6) From eq.(6.3.7) we see that this is equivalent to y=Qr so the normalisation of the Virasoro generators occurring in (6.4.2) works out correctly if we have this quantum equivalence. A sufficient condition for L(z) = £(z) is provided by 1 tlab:Ta(z)Tb (z): + i Jap:Sa(z)SP(z): = 0 . (6.4.7) 2 2 Using the explicit realisation of a given by (6.2.4), we find eq.(6.4.7) is equivalent to the conditions />~a ~a ~b . ~a ~b % _ 11 ab ^ ijM kl + M ikM lj + M ilM jk) = 0 (6.4.8) T> K (Ka.. Kb + Ka Kb + Ka Kb 1 = 0 , (6.4.9) 'ab^ vp \p pv \v pp; * v ' J Xa . XP . + J Xa . XP . - t] ,Ma . .Kb = 0 . ap la jp ap ip ja 'ab lj pa (6.4.10) -121- Just as for the fermionic construction an expression like (6.4.8) enabled us to associate the quantum equivalence state ment with a symmetric space and for the symplectic boson con struction an expression like (6.4.9) associated a superalgebra with the quantum equivalence, so here also we have an algebraic construction associated with the equality of L(z) and i(z). In fact, the equations (6.4.8-10) are precisely the conditions which allow us to associate a supersymmetric space a f/a with the superalgebra a. To see what is meant by a supersymmetric space, first consider embedding a in a larger superalgebra, a', and split each of these into their even (bosonic) and odd (fermionic) generators denoted by g,g* and s,s’ respectively, i.e. a = g+s , a f = g'+s’ . (6.4.11) Assuming the quadratic Casimir operator, Q , can be extended a to one Q , for a*, we can write a ’ g ’ = g+'c , s ’ = s+a , (6.4.12) where t and a are respectively orthogonal to g and s with respect to the quadratic form on a* corresponding to Q a Thus we have [g, t] CZ t , [g, a] d a , [s, t ]CZ In other words, i and a provide representations of g. The -122- special feature of supersymmetric spaces is that the generat ors a and x close on g and s, i.e. [t , t ] Cl g , [x, a ] d s , (a, a} CZ g . (6.4.14) This means the algebra is invariant under the automorphism g->g, s-*s, x->—x, a->-a, and a is the subalgebra of a ’ which is even with respect to this involution. In the present context, if we add generators x^, l X i . T a Li [ta , xJ] = iMakJxk , [ta , ° ] = lL nX° * [xJ , s“] = i(^Xaj)JxaX , {s“ , 0X} = X“ .. , (6.4.15) J A- corresponding to (6.4.13) and [t1, tJ] = l(flMa)j l ta , {oX. o'1} = Ka X(jta . [xj , oX] = ij (nXP)Jxs“ , (6.4.16) corresponding to (6.4.14). The various Jacobi identities satisfied by eqs.(6.4.15) and (6.4.16) reproduce equations (6.4.8-10) and (6.2.5-7) as well as requiring Ma and La to provide real and real symplectic representations of g respect ively. Thus we have the result that the cases where L(z)=£(z) correspond to the existence of supersymmetric spaces with tangent space generators transforming under the same rep -123- resen tation of the even part of the superalgebra as the ferm ions and symplectic bosons used in the construction. Note that eq.(6.4.8) is the symmetric space condition of section (2.4) and eq.(6.4.9) is the superalgebra condition of section (5.2). The present result thus contains these previous results as special cases. To understand this better, note that both superalgebras and symmetric spaces have 57^ gradings corresponding to involutions. In the one case the involution follows from reversing the sign of the fermionic generators and in the other by reversing the sign of the tangent space generators. Thus the supersymmetric spaces we have been discussing have a richer 2^ structure for which there are three ZTg subgroups. Correspondingly, we have three ^ 2 graded structures, namely the symmetric space g’/g> the superalgebra g+cr and the superalgebra a=g+s. The first of these corresponds to the quantum equivalence L**(z) = z) discussed in section (2.4), where the fermions transform under the representation Ma of g. The second two structures cor respond to the bosonic quantum equivalences L^(z) - i5(z) of section (5.2), where the bosons transform under the represent- ations of g given by L cl or N cl respectively. The involutions of Lie superalgebras we have been dis cussing have been classified in the mathematics literature (Serganova, 1983 and Leites, 1984) for the case of the simple algebras (those possessing no non-trivial ideals). The resulting supersymmetric spaces are the analogues of the type I symmetric spaces, but it is not clear how a general super- 1 symmetric space is related to these. Consequently in the next section we limit ourselves to discussing a few examples of the supersymmetric space theorem. 6.5 Examples of the Super-Symmetric Space Theorem As was shown in section (6.3), the super-Sugawara con struction of the Virasoro algebra gives a c value equal to the free field value of sdima/2 when the fields are in the adjoint representation of a. Also the conditions (6.4.8-10) are sat isfied by the quantities given in eq.(6.2.17) thus establish ing the quantum equivalence of the two Virasoro constructions. This is related to a supersymmetric space axa/a analogous to the type II symmetric spaces discussed by Goddard, Nahm and Olive (1985). The analogues of type I symmetric spaces prov ide further examples of the supersymmetric space theorem. As a first example, consider the superalgebra osp(m|n), for which the corresponding affine superalgebra can be con structed from m(m-l)/2 + n(2n+l) fermions and 2mn symplectic bosons corresponding to the adjoint representation and giving rise to a super-Virasoro algebra with c = sdim osp(m |n)/2. Another possibility is to take m fermions and 2n symplectic bosons and construct the first m(m-l)/2 T El (z) generators out of the fermions in the defining representation of so(m). The a remaining n(2n+l) T (z)'s are constructed from the symplectic bosons in a representation corresponding to the construction of the level 1 representation of osp(l|n). The 2nm S GC(z) gen erators are constructed from all possible bilinear products of the m fermions and 2n bosons. Explicitly, we have Sa (z) = S(la)(z) = — H1(z)5a(z) , (6.5.1) / 2 , using the normalisation of the generators implied by section -125- (5.3). We find the Ta (z), Sa (z) generators form an osp(m|n) representation with k=l/2 in (6.1.1-3). Combining this with the value Q =2Q = (m-2-2n), we find the super-Sugawara con- a I’l struction has a central charge given by (6.3.8) of k sdim osp(m|n) c •i (m-2 n) , (6.5.2) k + Q 2 N which is the same as the free field value. This suggests we may have the quantum equivalence of L(z) and f.(z) and this is indeed the case as we can easily verify that eqs.(6.4.8-10) hold. The supersymmetric space osp(m+l|n)/osp(m|n) is the one associated with this equivalence and we can check that the numbers of tangent space ‘generators (i.e. the dimensions of the representations provided by a and t in the previous sect ion) match the number of fermions and symplectic bosons used in the construction. Note that k+Q^= 0 when m=2n+l giving rise to critical representations analogous to those discussed in section (5.4) and in this case it is not possible to form the ’renormalised’ super-Sugawara generators. The particular case of the above construction with m=l was previously given by Feingold and Frenkel (1985), though not in the context of supersymmetric spaces. The case osp(l,lJl) with the fermions transforming under the defining representation of so(l,l), rather than so(m) above, will be seen in chapter 8 to play a role in the ghost system of string theory. Another slightly more complicated example involves the supersymmetric space su(m12n)/osp(m |n), which provides us with (m-1)(m+2)/2+(2n+l)(n-1)+l fermions and 2mn symplectic bosons with which to construct an osp(m |n) representation. (We -126- assume m^2n, since m=2n corresponds to a critical represent- a ation). The T (z) generators are now a sum of two pieces, one constructed from the bosons as in the adjoint representation case and the other constructed from the fermions. The first S i m(m-l)/2 T (z)’s involve a quark model construction with fermions in a (m-l)(m+2)/2 dimensional representation of so(m) S i while the remaining n(2n+l) T (z)’s involve a construction with fermions in a (2n+l)(n-l) dimensional representation of sp(n). With the normalisation of section (5.3), we find the resulting affine superalgebra has k=(m+2-2n)/2 so the corres ponding super-Sugawara construction has k sdim osp(m |n) — (2n-m-2)(2-m+l) , (6.5.3) k + Qn 4 which is the same as the value obtained by a free field con struction as it must be. Other examples can be read off from the tables given by Serganova (1983), though the above examples embody the general features. In view of the fact that we do not know how to write a general supersymmetric space in terms of the simple ones, we cannot in any case give a comprehensive list of cases where the supersymmetric space theorem holds. 127- CHAPTER 7 BRST METHODS 7.1 Motivation and Applications in String Theory An elegant and powerful method of quantising gauge theories has been developed by Becchi, Rouet and Stora (1975) and Tyutin (1975) based on the Dirac theory of constrained Hamiltonians and Faddeev-Popov ghosts. Classically, a natural object arising in this formalism is a nilpotent BRST charge, Q, but when applied to the first quantised string theory, there is an anomaly term in {Q, Q} , bilinear in the ghost fields which only vanishes if the string is taken in its critical dimension. The BRST operator is hermitian so its nilpotency may appear strange at first sight. However, be cause of the ghost fields that have been introduced, one is no longer working in a positive space. The ghost fields are of crucial importance in covariant string theory and are required, for example, to complete the covariant fermion emission vertex (Friedan, Martinec and Shenker, 1985). Of course, it is possible to work in a light cone gauge where ghost fields are not needed, though then manifest Lorentz invariance is lost and this presumably ob scures whatever symmetry principle underlies string theory, The second quantised covariant actions for string theory make use of the BRST charge, Q, and the free field action can be written as (1986) has exploited the analogy between Q and the different ial operator d to reformulate string field theory in terms of a non-commutative differential geometry. -129- As well as these string field theory applications, the BRST methods can be used to simplify certain calculations such as the proof of the ’no-ghost theorem’ originally given by Brower (1972) and Goddard and Thorn (1972). It can be shown (see Spiegelglas, 1987) that any physical state |X>, for which Q|\>=0, can be written as |X> = |p> + Q | v> , (7.1.1) where | \i> is a physical state with positive norm. (There is also the technical requirement that for physical states, one of the two-fold degenerate ghost vacuum states has to be decoupled. This is discussed further in section 7.3.) Thus physical states with positive norm correspond to the cohomol ogy classes of Q. Projection operators onto the physical subspace can also be defined using BRST methods (Freeman and Olive, 1986). Having motivated the study of the BRST approach to string theory, we shall develop the relevant formalism in this chapter, paying particular attention to the BRST charges and the ghost field structures. In the next chapter we will see that the ghost fields arising in string theory can be inter preted as examples of the pseudo-orthogonal fermions and sym- plectic bosons occurring in chapter 6. -1 3 0 - 7.2 BRST Approach to Quantisation Consider a classical theory with constraints L n= 0, ne^, obeying the Poisson bracket algebra [L , L ] = f P L , (7.2.1) L n ' mJ nm p * v J where f p are structure constants. (We shall use the same notation for Poisson and quantum brackets.) The BRST method associates a pair of ghost, antighost fields, denoted cn , cr respectively, with each L^. (In the context of string theory the term ’ghost* is rather overworked since the ghosts dis cussed here have nothing to do with the negative-norm state ghosts of section (1.5).) It is then possible to construct a BRST charge Q = \ L n c -n - —0 l f nm -n -m p ' (7.2.2) n el 2 n,m,pL ^ and, imposing the canonical anticommutation relations, {c1 n ,’ c m }J = 0 ’, 1{c n 7, c mJ} = 0 , {c , c } = 6 , , (7.2.3) 1 n* mJ n+m,o v J we find the nilpotency condition {Q, Q} = 0. Modified con straints are defined by 1 L'n = {Q, c nJ l = L n - l „ f nm pc -mmc„ p 7, (7.2.4)v J m,p v which obey the same constraint algebra (7.2.1) and also have -131- the property that [ Q, L;] = 0 , (7.2.5) which follows from the definition of L' and {Q, Q} =0. Turning now to the quantum theory, we have Poisson brackets replaced by quantum brackets and the constraint alge bra (7.2.1) can pick up an anomaly term through normal order ing prescriptions as in the case of Kac-Moody or Virasoro algebras. The ghost fields have also to be normal ordered and we shall define this by : c n c m c n c m n<0 , n=0 , = - c m c n , n>0 . (7.2.6) The BRST charge is now written as Q^ = YL L n c -n — — _ YL f nm P :c -n c-m cp : + ac o * , (7.2.7) n 2 n ,m ,p ^ where a is a constant required to ensure Q is nilpotent (see e.g. Schwarz, 1986). If we have the hermiticity properties c -n * (7.2.8) then Q is also hermitian. Again modified constraints, L/, obeying the same algebra as the L n , but without a central extension, can be defined by the analogue of (7.2.4). -132- 7.3 Application to Virasoro Algebras Letting fnmP = (n-m)6n+m p in section (7.2) reproduces the Virasoro algebra, which acquires a central extension in the quantum version. The BRST charge is given by eq.(7.2.7), which is nilpotent if the Virasoro algebra has central charge given by c = c . = 2 6 and a=-l in (7.2.7). This reproduces the well known critical dimension of the bosonic string, as we saw in section (1.5) that the Virasoro algebra was the rele vant gauge algebra. Physical states )(j>> are defined by Q | <|>> = 0 , (7.3.1) which is equivalent to the infinite set of conditions (1.5.15) with h=l. Here we have assumed that c | which is motivated by the normal ordering prescription of (7.2.6). The zero modes, c q and c q , require more careful treatment since they give rise to two degenerate vacua, only one of which is retained in the truncation to physical states. One way of achieving this is to impose co |4>>=0, though this requires an extra factor of c q in the definition of the scalar product of states. An alternative approach (Freeman and Olive 1986) is to define linear combinations of the two vacua and retain only the combination with positive norm. The modified constraints, , given by (7.2.4) can be written as -133- L'n = L n + L n ’ , (7.3.3) where L^ = l (n-m):c__c - 6 (7.3.4) m n+m -m' n,o 0 The Ln obey the Virasoro algebra with c=-26 so 1/ also obeys a Virasoro algebra but with c=0. This vanishing of the anomaly term for the modified constraints is a general feature and here follows from [L l£,n c mJ l = (n-m)v 7 c n+m , , v(7.3.5) 7 and CLA' Lml = CLi» IQ- = {Q. [K> 5mN = (n-m) {Q, on+m} = (n-m) L^+ffl . (7.3.6) We also have cml = -(2n+m) cn+m > (7.3.7) and can define ghost and antighost fields on the unit circle by c(z) = l cnz”n » °(z) = l ^nz"n • (7.3.8) ne% ne? Equations (7.3.5), (7.3.7) then say that c(z) and c(z) are conformal fields of weight 2 and -1 respectively. These weights differ from the canonical weight of 1/2 for a free fermion field and the origin of this discrepancy will be seen in the next chapter 7.4 Superconformal Ghosts and Fermionic Strings The fermionic string gauge constraints satisfy the Neveu-Schwarz-Ramond algebra given by eqs.(2.5.1-3). The ghost fields associated with the fermionic generators Gr will need to be bosonic in nature to maintain Q as a fermionic operator. These ghost and antighost fields are conventionally denoted e(z), e(z) respectively (though we include an extra factor of i in e(z)), where e(z) = l erz r , e(z) = l erz r , (7.4.1) r r and the fields have the same NS or R nature as the G(z) gener ators. We have the canonical commutation relations [er> es] = 0 , [er, ej = 0 , [ V 5sl - i 6 r+s,o ’ <7-4-2^ and the hermiticity properties (er)+ = e-r > (®r)+ = ®-r ‘ (7.4.3) In the next chapter we shall use the fact that these fields are just examples of the symplectic bosons introduced in chapter 5. Indeed, this observation was the original motivat ion for studying such symplectic boson fields. For the moment however, we shall retain the conventional notation and write the BRST charge for the superconformal algebra as -135- Q = J L c T (n-m):c c c , : + ac + Y G e ^ nL n -n 2n n ,m L -n "m n+m o rL r -i y e e c ^ - i J (n/2-r) :c e e . (7.4.4) u — r* — c: r* 4- o u ' • ' —q _ j » n + T ' ' r , s ■r -s r+s n , r A straightforward but lengthy calculation shows that Q is nilpotent provided the central charge of the superconforma1 algebra is c=15 and a takes the values -1/2 and -5/8 in the Neveu-Schwarz and Ramond sectors respectively. For the superstring, we have one free boson and one free fermion for each spacetime dimension, giving a contribution to the central charge of 3/2 and hence the critical dimension is d=10. The modified constraints are given by (7.4.5) K = = Ln + LS + Ln ’ ^ = “i[Q. 5r] - Gr + G°'e , (7.4.6) where L® = -i I (n/2-r) :e_rSn+r: + (l/2-£/8)6Q , (7.4.7) r * and Gp’6 = -i I (n/2-r) 5n+pc - 2 \ c_n+ren . (7.4.8) n n As usual, e=0 or 1 for the Neveu-Schwarz or Ramond algebras respectively. The 1/ , obey the superconformal algebra with Cg C 6 c=0 so Lq + Lr and G * also obey this algebra but with c=-15. We also have the commutation relations L[Le n* , e m ]J = -(3n/2+m) v ' ' e n+m , , [^n* eml ( n/2 m) en+m > (7.4.9) -136- which show that e(z), e(z) are conformal fields of weight -1/2 and 3/2 respectively. The results of this section can clearly be generalised to superconformal algebras with more supersymmetries just by adding extra fermionic or bosonic ghosts corresponding to each generator of the algebra. However, for the N=4 superconformal algebra considered by Ademollo et al. (1976), the nilpotency of Q requires the central charge to take the value c=-12 so we cannot construct unitary highest weight representations of the Virasoro algebra for any positive dimension. For the N=3 superconformal algebra discussed by Ramond and Schwarz (1976), it is not possible to construct a nilpotent BRST charge. The only other possibility for the algebras allowed by Ramond and Schwarz is the one for which the gauge algebra is the N=2 superconformal algebra given in section (4.1). It turns out that this can have a nilpotent BRST charge provided the central charge takes the value c=6. This particular string model was discussed in section (4.3). It should be noted, however, that by relaxing some of the conditions imposed by Ramond and Schwarz it is possible to construct other superalgebras. For example, using fermions H SL in the adjoint representation of a Lie algebra g, we have a seen how to construct corresponding Kac-Moody generators, T , by the quark model construction and Virasoro generators, Ln , by the Sugawara construction. Super-Virasoro generators, , are constructed as in eq.(2.5.9) and the Ha ’s, Ta,s, L ’s and G ’s form a closed superalgebra. Although it is not possible to construct a nilpotent BRST charge in this case, it is clear that there are other interesting superalgebras than the ones considered by Ramond and Schwarz. -137- 7.5 Applications to Affine Algebras In section (7.3) we saw that it was possible to con struct a nilpotent BRST charge for the Virasoro algebra provided c=26. We have also seen that the Sugawara construct ion of the Virasoro algebra involves a semi-direct product of Kac-Moody and Virasoro algebras. An interesting question is whether it is possible to construct nilpotent BRST charges for either the Kac-Moody algebra or for the semi-direct product algebra (Hlousek and Yamagishi, 1986). We consider the Kac- Moody algebra [T*L n7 Tml mJ = c n+m + 1 n+m,o > (7.5.1)v 7 as in section (5.1). The corresponding BRST charge is written (Kawai, 1986) -c ^ ^ Tnca,-n _ f c ^ ‘ca,-ncb,-m^n+m**c. (7.5.2) n~ ’ 2 n ,m * ’ * where a bi = 0r\ , | f c a , c —b) = 0 _ , c n , c m tJ ’ 1 a’ mJ ’ a -bi abc c , (7.5.3) n ’ c mJ 1 ^ ^n+m,o * ab and the metric ti is used to raise indices. We then find (Q, Q} = (k+Q^) l n„abc!nc£ , (7.5.4) 1 where Q, is given by eq.(5.2.3). For the unitary highest -138- weight representations considered in section (2.2), both k and Q, are necessarily positive so it is not possible to construct a nilpotent BRST charge in these cases. In the case of the symplectic bosons of chapter 5, k could take negative values and it should be possible to construct nilpotent BRST charges, though the physical significance of this is not clear. It should be noted that the nilpotency condition, k+Q^=0 , is a different condition from the critical representation condit ion, 2k+Q^=0, of section (5.4). This difference becomes apparent when the semi-direct product of a Kac-Moody and Virasoro algebra is considered with the Virasoro generators being the sum of a Sugawara construction and an extra piece which commutes with the Kac-Moody generators and has c=c o . The nilpotency conditions are (Hlousek and Yamagishi, 1986) k+Q^ = 0 , (7.5.5) c + 2k dlm? = 26 + 2dimg . (7.5.6) 2k+% Using (7.5.5), eq. (7.5.6) can be rewritten as c o = 26 , (7.5.7) y so we see that the central charge corresponding to the Suga wara piece of the Virasoro generator cancels out if we use the nilpotency condition of the Kac-Moody algebra. A similar mechanism applies to the affine superalgef^ras and the super- Sugawara construction of chapter 6. The nilpotency condition on the BRST charge for the affine superalgebra (6.1.1-3) is -139- k + 2Q_. = 0 . (7.5.8) N For the semi-direct product with a Virasoro algebra formed from the sum of the super-Sugawara construction and a piece which commutes with the affine superalgebra generators and has c=c o , there is an extra condition for nilpotency, namely 2k sdima c + = 26 + 2sdima (7.5.9) 0 2k+2Q N Inserting (7.5.8) into (7.5.9) reproduces the condition c o =26. -1 4 0 - CHAPTER 8 GHOST FIELDS IN COVARIANT STRING THEORY -1 4 1 - 8.1 Conformal Ghost Field Constructions To conform to the notation of chapter 2, we shall define linear combinations of the c(z), c(z) ghost fields defined in eq. (7.3.8) by H1(z) = ncSl H^z n = — (c(z)+c(z)) , H2(Z) = TIC'Sl H2z"n = ^— (c(z)-S(z)) . (8.1.1) From eq.(7.2.3) we have the anticommutation relations i (8.1.2) {Hn * 6n+m, o where Tii^= diag(l,-l). Thus the Hi(z) are just the usual pseudo-orthogonal fermions. These fields can be used to con struct an so(l,l) algebra by the quark model construction if we define T(z) = l T z_n = i M •H1(z)HJ(z) (8.1.3) n 2 1J where M = M (8.1.4) ij -i a • In the usual way we find [T -n6 (8.1.5) n ’ n+m,o ’ and the T(z) are hermitian and proportional to the convention al ghost current. The Sugawara construction arising from this -142- so(l,l) Kac-Moody algebra is given by 1 H(Z) = - - x T(Z)T(Z) X ■ (8.1.6) 2 which generates a Virasoro algebra with c=l. By the usual method of normal ordering T(z)T(£) with respect to the or the , it can be shown that (z) is quantum equivalent to the free field Virasoro generator given by LH(z ) = — t)j j :z—~ HJ(z): + i . (8.1.7) 2 J dz 8 In section (2.4) we saw that such a quantum equivalence corresponded to a symmetric space. This remains true here also for the non-compact so(l,l) as we see by computing H*] [ v mJ = -iH2n+m ,* [V Hm]mJ = -iH^n+m . (8.1.8) Thus , denoting T by t3 0 , the corresponding symmetric space is 1 2 . given by t2 and the tangent space generators t , t obeying CO • i ii H* [t3, «-* . 1 [t3, *2] = “IT CO •H -p II 1 I * 1. *2] • (8.1.9) 2 3 This defines an so(2,l) algebra for which t , t are non compact generators and is a compact generator. Hence we -143- have the symmetric space so(2,l)/so(1,1), which guarantees TH, n fH, . L (z) = i (z). The pseudo-orthogonal fermions can also be realised in terms of vertex operators of the kind given by eq.(3.5.9) (see e.g. Friedan, Martinec and Shenker, 1985). Taking X=±l in eq.(3.5.9), we have 1/2 (oQ ?(z)o o - Q r(z)o H X (z) V O /2 z = I _ 1/2 f ° Q'(z)o + o -Q'(z)o H2( ) o J) (8 . 1 . 10) n Using the vertex operator expansion z 1/2 o Q'(z)o .1/2 oe± Q ’(5)o = zl/2 1/2 o Q'(z)±Q>(5)o, -.±1 A A ^ A /A = Ao oA ' ^ / > IZI > ISI , (8.1.11) we can verify that the Laurent coefficients of H1(z) satisfy the correct anticommutation relations, (8.1.2). Since the H^z) are Ramond fields, the vacuum is 2 fold degenerate and, using results analogous to eq.(3.2.8), we find the Fubini-Veneziano field, Q ’(z), has p ’ eigenvalues ±i/2. It may seem strange for an hermitian operator to have imaginary eigenvalues but this is just a reflection of the fact that we have a non-positive space. The vacuum states j±i/2> are intertwined by the according to 1 i 1 i. 1 i 1 i 1- > = — H = — 0 1- -^ > > 0 1 - - > !- 2 /2 2 2 ✓2 2 2 i v 1 i 2 i -1 i H = — H = — 0 I" > 1- - > , 0 1- - > 1- (8 . 1. 12) 2 /2 2 2 n 2 -144- The so(l,l) generator T(z) can be identified with the Fubini- Veneziano momentum field, P'(z)=izd z Q'(z), since Q'(z)o o -Q'(z)o T(z) = iH1(z)H2(z) Lim — z ^ V /2 (°e z->£ 2 c> Q '(£)o , o -Q*(£)o (-'e ^ o = izd Q f(z) (8.1.13) z v 7 where we have used (8.1.11). L (z) and £ (z) both have c=l and H (z) are conformal fields of weight 1/2 with respect to these Virasoro algebras. However, we saw in section (7.3) that the ghost Virasoro algebra had c=-26 and the ghost, antighost fields had conform al weights -1 and 2 respectively. To see the source of these discrepancies, we rewrite Lc (z) of section (7.3) in terms of the H^(z) fields. We find LC(z) = LH(z ) + - i za„T(z) - - . (8.1.14) 2 8 This has the structure of the Virasoro + u(l) current derivat ive term discussed in sections (2.6) and (3.5). Equation (8.1.14) reproduces (3.5.12) with p=-3/2 so the central charge of L (z) is correctly given by l-12(-3/2) =-26. The vertex operators given by (3.5.13) with \=±1 have conformal weights c i l/2±3/2 with respect to L (z) as required. Note that H (z) are no ^longer conformal fields with respect to L (z). Because T T T T L (z)= £ (z), the right hand side of (8.1.14) can be written entirely in terms of T(z), which would imply a vertex operator construction had we not already found one. -145- 8.2 Superconformal Ghost Field Constructions As we mentioned in section (7.4), the superconformal ghosts e(z) and e(z) are examples of the symplectic bosons of chapter 5. To conform to the notation of chapter 5, we shall 1 2 henceforth denote e(z), e(z) by £ (z), £ (z) respectively. These fields have the expansions C1(z) = I , 52 (z ) = l ij2 z ~ r , ( 8 . 2 . 1 ) r r and from (7.4.2) we have iJ°P6 A U“, tji - r+s,o a , (3=1,2 , (8.2.2) where J=Ja^= ) • Ca(*) are hermitian fields and we can form the currents a -n Ta (z) = l T*z - Raap:§a(z)cp(z): (8.2.3) n as in section (5.1), with the matrices R chosen to be 1 1_ /-I 0 R R' 1/0 1 (8.2.4) ✓3 ' 0 - 1 ✓5 V1 ° Defining matrices Na by Na= -JRa , we find the Na obey the commutation relations „ab Mc f N 9 c (8.2.5) where f 1 2 3 = - / 2 " , f312 = /? , f23x = - / z . ( 8 . 2 . 6 ) -146- tL Thus the N define a real symplectic representation of a non- a compact sp(l) algebra. The T (z) obey the Kac-Moody algebra (5.1.11) with k = 1/2, p = diag(1, -1, 1) . (8.2.7) A The level of the sp(l) representation given by (5.1.14) is 2 -1/2 since with the choice of metric made in (8.2.7), c}> = -2. As in section (5.2), we can construct a Sugawara Vir- asoro algebra from the Kac-Moody generators yielding L l (z) = - - riab *Ta(z)Tb(z)* , (8.2 .8) 3 which obeys the Virasoro algebra with c = -1 from eq.(5.2.5). This is the same c value as for the free field construction (5.2.6) for a single pair of symplectic bosons. Indeed the two Virasoro constructions are quantum equivalent as can be verified directly or by appealing to the superalgebra theorem of section (5.2). In this case the relevant superalgebra turns out to be osp(l|l). Just as for the conformal ghosts, it is possible to write the superconformal ghosts in terms of vertex operators (Friedan, Martinec and Shenker, 1985). In our notation we have ,.1, x -1/2 o Q(z)o , 1, N l (Z) = Z ' Qe v '0 <|> (z) , ?2 (z) = z 1/2 °e_Q(Z)° 4.2 (z ) , (8.2.9) where Q(z) is the usual Fubini-Veneziano field. Since we have taken the symplectic fermions to be of Ramond form, the vertex . -1/2 o ± Q ( z ) o operators z e must be chosen to have the same ^ o o periodicity properties as the symplectic bosons. From the results of section (7.4), we deduce that the Virasoro algebra given by the superconformal ghost fields, e ? L (z) , has c=ll whereas L (z) has c=-l. Just as for the con formal ghosts, the difference of these two Virasoro algebras is a u(l) current derivative term and we find Le(z) = L^(z) - /2 izbzT3(z) + I . (8.2.10) 2 This is of the form (2.6.3) with p=l giving correctly the central charge for Le (z) of -1+12=11. Note that we could obtain a Virasoro algebra with c=ll by replacing /2T ( z ) by B Ta (z) for 8 pa = 2, but the form (8.2.10) is picked out by the property that it commutes with the ghost number operator, 3 T . The conformal weights of the ghost fields are modified o P 6 from 1/2 with respect to L (z) to 1/2+1 with respect to L (z). -148- 8.3 Algebraic Structure of Combined Ghost System In the previous two sections we have seen how the con formal and superconformal ghosts, written as fermions and symplectic bosons, could be used to construct Kac-Moody alge bras and their associated Sugawara Virasoro generators. These Sugawara generators were found to be quantum equivalent to the respective free field constructions corresponding to examples of the symmetric space theorem of section (2.4) and the super algebra theorem of section (5.2). This suggests that if we consider the combined system of conformal and superconformal ghosts, we may be able to realise an affine superalgebra as in chapter 6, together with a super-Sugawara construction which provides another example of the supersymmetric space theorem. a 1 With this in mind, define currents T (z), l a 4 section, generating the non-compact sp(l), and T (z) is ident ified with T(z)//2", where T(z) is the so(l,l) current occur ring in section (8.1). We then form the fermionic generators, Sa (z), 1< a < 4, out of all possible bilinear products of boson and fermion fields with the notation SX(z) = — H1(z)51(z) , S2(z) = -i- H1(z )?2(z ) , S 5 / 2 S3(z) = i- H2(z )C2 (z ) , S4 (z) = H2(z )51 (z ) , (8.3.1) /2 where the Hx (z) are defined as in section (8.1) and the £a(z) as in section (8.2). a c l Computing the algebra of the T (z) and S (z), we find we -149- obtain the affine superalgebra given by eqs.(6.1.1-3) with k = 1/2 , nafc> = diag(1,-1,1,-1) , (8-3.2) and the only non-vanishing structure constants are those given by eq.(8.2.6). These are the structure constants for sp(l) Q so(ljl), which is the bosonic part of osp(l,l|l). The mat rices N occurring in (6.1.1-3) are here given by a 0 1 0 0 0 -1 0 0 1 _ 1 0 0 0 1_ 1 0 0 0 N N< 1 0 0 0 1 0 0 0 1 > /2 /2 ,0 0 1 0 . 0 0 -1 0 0 0 0 0 0 0 1 1_ 0 -1 0 1_ 0 0 1 0 N 0 N. 3 0 0 -1 0 0 1 0 0 > /2 /2 0 0 0 1 1 0 0 0 (8.3.3) and the symplectic form, is given by /0 1 0 0 0 0 0 J 0 0 0 1 (8.3.4) VO 0 -1 0 We can explicitly check that (8.3.5) and so we have realised an osp(l,l|l) superalgebra using the conformal and superconformal ghost fields. Performing the super-Sugawara construction of section (6.3), it is possible -150- to explicitly check that this construction equals the sum of H £ L (z) given by eq.(8.1.7) and lr (z) given by (5.2.6) with n=l. By the results of section (6.4), it must be possible to associate a supersymmetric space with this equivalence. It turns out that the relevant space is osp(2,111)/osp(1,11 1). The super-Sugawara Virasoro algebra has c=0 so in order to construct the Virasoro algebra with c=-15 required for the superstring ghost algebra it is necessary to add derivatives 3 4 of the currents T (z) and T (z) as before. We find l c = - 15 = £(z) _ izazR(z) _ § ^ ( 8 .3 . 6 ) 8 where £(z) is the super-Sugawara Virasoro generator and R(z) = l R nz~n = ” ^475 (3/272 T 4 (z) - /2* T 3 (z ) ) . (8.3.7) Since [R , R 1 = -n6 , R 1 = -mR ± , (8.3.8) L n ’ m J n+ra,o ’ ’-^n* m J n+m * v ' we again have the construction of (3.5.12) with p=/5/4 giving a central charge of 0-12(5/4)=-15 as required. The relevance of these superalgebra constructions to string theory is not clear and the requirement of adding extra u(l) current derivative terms to the (super-)Sugawara Virasoro generators in order to reproduce the string theory results does not appear natural. This problem does not disappear when we consider gjhost systems for superconf ormal algebras with more supersymmetries. For example, the N=2 superconformal ghost system allows us to realise an o1sp(2,2|2) superalgebra -151- but the corresponding super-Sugawara construction has c=0 rather than c=-6 so again we would need to add the correction t e r m s . Another puzzling feature about these ghost systems was noted by Friedan, Martinec and Shenker (1985), who found that the ghost system for the N=1 superconformal algebra actually possessed an N=2 supersymmetry rather than the N=1 one which was automatically present. Unfortunately, our formalism in terms of affine superalgebras does not appear to shed any light on this mysterious result. It would be interesting to see if the N=2 superconf ormal algebra similarly has a ghost system with an enhanced amount of supersymmetry. -152- CHAPTER 9 CONCLUSIONS AND OUTLOOK -153- Conclusions and Outlook Throughout the preceding chapters we have seen how the basic techniques of two dimensional conformal field theory can be applied to a variety of algebraic structures. The powerful method of obtaining quantum equivalences by normal ordering composite operators in different ways was a recurring feature as was the use of vertex operators. Vertex operators have been used to construct a variety of conformal fields and are related to integral lattices. Here we have restricted ourselves to Euclidean lattices and seen that for lattices of points of squared length 1,2 and 3, the corresponding vertex operators realise fermions, Kac- Moody generators and Virasoro supercharges respectively. The construction of the N=2 superalgebra in terms of a single bosonic field discussed in chapter 4 is a surprising result since canonical constructions of these sorts of algebras match the numbers of boson and fermion fields. However, N=1 super algebras have also been constructed in terms of just fermion fields (Goddard and Olive, 1985). These non-canonical con structions show that the well known critical dimensions of string theory are not in themselves as fundamental as the c values for the corresponding Virasoro algebras. Vertex operators associated with points of square length 4 can be used to realise the non-associative Griess algebra, which is connected with the Leech lattice and the Monster group. Further developments in this area seem likely. In chapter 5, many of the well known fermionic con structions of Virasoro and Kac-Moody algebras were seen to -154- have analogies in symplectic boson constructions. These give rise to non-unitary representations though may still be relevant for systems such as the ghost fields of string theory. The Sugawara and free field forms of the Virasoro algebra constructed from symplectic bosons were shown to be quantum equivalent if there existed a certain superalgebra, which was the analogue of the symmetric space condition for the fermionic constructions. The symplectic boson and fermionic constructions were combined in chapter 6 and these fields could be used to realise certain affine superalgebras. A super-Sugawara construction was found and the condition fox this to equal the free field Virasoro algebra was given by a ’supersymmetric space theorem’, incorporating the ’symmetric space theorem’ and ’superalgebra theorem’ as special cases. These results were applied to the specific case of the ghost string fields in chapter 8. It was found that extra ’u(l) current derivatives’ had to be added to the Sugawara or super- Sugawara constructions in order to reproduce the string theory results. The physical meaning of these extra terms remains mysterious. One topic which we have not mentioned, but which seems likely to produce many interesting new results is connected with Lorentzian lattices and their corresponding vertex operators. The lattices of most interest are the self dual, even ones which only exist in 2 mod 8 dimensions. 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