Free fermions and WZW theories on nonsimple groups

Z. Phys. C Particles and Fields 35, 89-95 (1987) Zeitschdft P a r t k ~ for Physik C

and Fields �9 Springer-Verlag 1987

Free Fermions and WZW Theories on Nonsimple Groups

J. Fuchs t Joseph Henry Laboratories, Princeton, NJ 08544, USA

Received 16 January 1987

Abstract. We consider conformally and Kac-Moody invariant theories based on the groups G=G(N)x G(N) where G(N) is any of the classical groups. For the values k =/V, ~= N of the Kac-Moody central charges, the monodromy problem involved in the computation of the four point function for primary fields in the defining representation of G possesses two distinct solutions. As a consequence, the WZW theory on G (with an additional U(1) factor if G(N)=SU(N)) cannot be equivalent to a theory of free fermions.

1. Introduction

It is well known that in two space-time dimensions various field theories which look rather different at the classical level are actually quantum equivalent. Such an equivalence is certainly interesting from a purely theoretical point of view, but there are also various applications, e.g. in the theory of the quan- tized Hall effect El, 2], based on the fact that some quantity of interest which is difficult to compute in one theory may be more easily obtained in another, equivalent one.

An example of this phenomenon is the equivalence of the critically coupled Wess-Zumino-Witten (WZW) theory [-3] on the group manifolds O(N) and U(N) "-~SU(N) • U(1) with unit Kac-Moody central charge k to the free field theories of N massless Majorana and Dirac fermions, respectively. The proof of this equivalence [3-6] is based on three observations: First, the bosonic and fermionic theories possess the same current (affine Kac-Moody) algebra; second, they possess energy-momentum tensors of the Suga- wara [7] form, i.e. traceless and bilinear in conserved currents (and hence with only two independent com-

1 Supported by Deutsche Forschungsgemeinschaft; address after February 1, 1987: Institut f/Jr Theoretische Physik, D-6900 Heidel- berg, FRG

ponents T=T(z) and T=T(z-)), and therefore the same conformal (Virasoro) algebra. Finally, for k = 1 the bosonic and fermionic fields possess identical correlation functions [4].

In the O(N) or SU(N) case with k arbitrary, the Sugawara like energy-momentum tensor is given in terms of the O(N) (or SU(N)) currents J"(z) by

T(z) = - ( k + C.dj)- 1: J"(z) J"(z): (1.1)

while in the SU(N) x U(1) case there is an additional term containing the U(1) current J(z),

T(z) = -- (k + Cad j)- 1 : j , (z) J" (z): -- (2 N) 1 : j (z) J (Z)" (1.2)

(here Cadj is the quadratic Casimir in the adjoint representation; our normalizations of C,dj and of k are as in [8].)

It is often assumed that the coincidence of the algebraic structures (i.e. current and conformal algebras, with energy-momentum tensor of the Sugawara form similar to (1.1) or (1.2)) is not only necessary but actually sufficient to ensure the equivalence of two theories. In particular it has been argued ([1], see also [6, 9]), that the theory of free massless Dirac fermions in the defining representation of SU(N) x SU(N) • U(1) is equivalent to the WZW model on SU(N) x SU(Af), with central charges k = N and ~'= N, supplemented by a free boson which provides the U(1) degree of freedom. By the same token the theory of free massless Majorana fermions in the defining representation of O(N) x O(]V) or Sp(N) x Sp(N) would be equivalent to the WZW model on O(N)xO(.N) or Sp(N)xSp(bT), again with k= .~ , k'= N. We remark that, up to some exceptional cases, these are all possibilities where the energy-momentum tensor of free fermions in a representation of a group with more than one simple factor is of the Sugawara form [10] and thus an equivalence to a WZW model is conceivable at all.

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In the present paper we attempt to clarify the relation between the above mentioned fermion and boson theories by solving the Knizhnik-Zamolodchikov equation [-4] for the four point function of primary fields in the defining representation of the relevant group; in particular we discuss the monodromy problem [11, 4] involved in computing this correlator. The paper is organized as follows. Section 3 contains the calculation of the four point function for the conformally and Kac-Moody-invariant theory with energy- momentum tensor

T(z) = - (k - Caaj)- 1 : j , (z) J" (z):

--(]~'~- Cadj) 1: J~(Z) J"(Z) : (1.3)

where J"(z) and J~(z) are G(N) and G(N) currents, respectively, with G(N)=SU(N), O(N) or Sp(N). In the case G(N)=SU(N), we do not include the U(1) degree of freedom in this calculation, since its imple- mentation is trivial; we also find it convenient to first discuss the theory (1.3) with k, k arbitrary and after- wards specialize to the values k = N , g = N we are interested in, in which cases

~ N + N

k ~- Cad j = k-t- (~adj ~--" i N + ~T _ 2

I N + N + 1 [SU(N)

for G(N)=~O(N) . (1.4) / [Sp(N)

Before considering the G(N) x G(N) theory in Sect. 3, we first treat, in Sect. 2, the simpler G(N) theory with energy-momentum tensor (1.1), in order to introduce relevant notations and to simplify the discussion of Sect. 3.

In Sect. 4 we use the results of Sect. 3 to argue that the above mentioned boson and fermion theories are in fact not fully quantum equivalent and to discuss the precise manner in which the equivalence fails.

2. The Four P o i n t Funct ion for G= G(N)

The Sugawara form (1.1) of the energy-momentum tensor gives rise to the existence of a null vector [4] of the combined current and Virasoro algebras. Any correlation function containing this null vector van- ishes. Combination of the equation expressing this fact with the Kac-Moody Ward identities yields matrix differential equations for all correlators of primary fields. In particular, the four point functions which because of the projective Ward identities [12] can be expressed as

~ 1 (Z1, Z1) (~2 (Z2, "72) q~3 (Z3,53) ,~4-(Z4, 54))

= lq ( z , - z , ) - ~ ( ~ , - ~ ) -~" p<q

�9 (r (z, ~) ~2(0, o) ~3(1, 1) ~ 4 ( ~ , o0)) (2.1)

(where the exponents Apq, Apq fulfill ~ Apq=2Aq, p:4=q

Apq=Aq, with A,, Ap the anomalous dimensions p4-q of ~bp), obey two ordinary matrix differential equations in the projective variables z=(z 1-z2) (Z3--Z4)/(Z4--Z1)(Z2--Z3) and Z : (Z1 -- Z2)(Z3 -- 54)/ (~4- ~1)(~2 - ~3), respectively.

In the present paper we are interested in the four point functions for primary fields �9 = ~J in the defining G x G-representation and ~b = qS~ in its conjugate representation; here G=G(N) or G=G(N)x G(N), with G(N) any of the classical groups SU(N), O(N) or Sp(N). More precisely, we will consider

~(Z1 , Z1; "'" ; Z4, Z4)

= (~)(Z1, Zl) ~ ( Z 2 , 5 2 ) (~(Z3, Z3) (D(Z4, 54)) (2.2)

in the SU(N) and Sp(N) cases, and

~(zl , zl ; ---; z4, z4)

= ~ ( Z 1 , -71) ~ (Z2 ,52 ) ~(Z3,7~3) ~(Z4 , 54)) (2.3)

in the O(N) case. Using (2.1) and expanding f# in G x G invariant t e n s o r s IA@I- B we have

f f ( z l , 51; ... ; z4 , i4)

= [-(Z1 -- Z4)(Z1 -- Z4)(Z2 -- Z3)(Z2 -- 53))] - 2A

~AA ( Z, ff.) l A ]- A A,A=I

(2.4)

where n is the number of singlets contained in the tensor product of two defining representations of G and two of their conjugates. The differential equation for the z-dependence of f# then becomes [-4]

- - l (k - t - - Cadj) z ( z - - 1) ~ Z (ffA)i(Z' ~)

= E [-(z- 1)Pa~ + zQa,,] %a(Z, ~), (2.5) B

and a similar equation (with the n x n matrices P, Q replaced by their transpose acting from the right) holds for the i-dependence.

The explicit form of the invariants IA and of the matrices P, Q is as follows. For G=SU(N) we have n=2 ,

_ i3 i2 (2.6) I1=6i~2 6i~ ~, I2-6 i l 614

and

p = 1 { U 2 - 1 1 --1 2 N \ 0 N-l)' Q : ~ ( N 0 N 2 - 1) (2.7)

[4]. For G = O (N) we have n = 3,

11 ~'(~ili2 6i3i4' 12 =(~i l i3 (~i2i4 13 : ( ~ i l i 4 6i2i3 (2.8)

and

1 2 ( i l 1 1 ) ~. 0 0 1 1 ) P = 0 1 - - 1 , Q=~- 1 N - 1

- 1 0 - 1 0 (2.9)

[8]. Finally, for G = Sp (N) again n = 3, with

1 1 - i~ i~ I2=Oil ~ 12 "" (2.10) --(~il (~i4 ' (~i4 ' I3.~_ei~i4 g . . . .

(here the 2N x 2N matrix e is the tensor product of the two-dimensional totally antisymmetric tensor and the N-dimensional unit matrix), and we find

1 ( 2 % + 1 1 - -1) 1 ( ! 0 i ) P = ~ 0 ; 1 , Q = 2 N + l .

\ 0 - i 0 (2.11)

The most general solution of (2.5) and the corresponding equation involving the i-dependence is

fgaa(z , z - ) = ~ a ~ f # A ~ P ) ( z ) f r ( 2 . 1 2 ) p,q= 1

where av~ are arbitrary coefficients and the functions f#~P)(z), p = 1, 2...n, form a system of independent solutions of (2.5). We will need the explicit form of these functions; for SU(N), they are given by [4]

~]X ) (z) = z - p + ~/N (1 -- Zy/N F (a, -- a ; -- fl + 1 ; z)

N~2)(z) = z~/N(1 --Z) l -O+~/N F(a+ 1, -- a+ 1 ; fl+ 1 ; z)

fr (z) = k- 1 z a -a+,/N(1 _ z y m F ( e + 1; - - e + 1;

- / ~ + 2 ; z )

fc~2~)(z) = -Nz~m(1-z) -~+~/UF(a, - a ; fl; z) (2.13)

where

1 N a - N + k ' f l = N + ~ (2.14)

and F(a, fl; 7; z) is the hypergeometric function�9 In the O (N) case, we have [8]

91

f#~P)(z) = z- P(1 --z) 1 -~ Hp(cc, c~-- 1, fl-- 1; 1 --fl--3:r z)

~ ) ( z ) = z I -P(1 - z ) -p H p ( ~ - 1, ~ , /~- 1; 1 - / ~ - 3~; z)

~Pl (z) = - z 1 - ~ ( 1 - 0 1 -P

�9 Hv(e-- 1, ~-- 1, fl-- 1; 1--fl-- 3e; z ) (2.15)

where now

1 N--1 ~ = N + k - - 2 ' f l = N + ~ - - 2 (2.16)

and the functions Hp(e, fl, 7; 8; z), first introduced in the context of degenerate conformal theories [11], possess the integral representations

GO oo

H'(a, fl, 7 ;b;z )= S as ~ d t ( s ty 1 1

�9 ( (s - - I ) ( t - - 1))P ( ( s - - z ) ( t - - z ) ) ~ ( s - - t) ~

oo 1

U2(a'fl']:;b;z)=zl+~+~ f ds I dt(st)~ 1 0

�9 ( (s - - 1 ) ( z t - - 1))P ( (s - - z ) ( t - - 1))r (s - - z t) ~

1 1

H3(0~, fl, 7; 8 ; z ) = z 2 + 2 ~ + 2 7 + 0 I ds ~ d t ( s t ) ~ o o

�9 ( (1 - - z s) (1 - - z t)) t~ ( (1 - - s) (1 - - t ) y (s - - t) ~. ( 2 . 1 7 )

In the Sp(N) case, N > 2, the solutions fC)P) are easily found by comparing (2.11) with (2.9): up to an over-all minus sign for N~f), they are again given by (2.15), with the definition (2.16) of ~, fl replaced by

- 1 2 N + l a = 2 ( N + k + l ) ' f l = 2 ( N + k + l ) " (2.18)

Among the solutions (2.12), the physically acceptable ones are singled out by the requirements of locality and crossing symmetry. In more mathematical terms, the requirement is monodromy invariance at z = ~= 0 and at z = ~= 1 [11]. When continued analyt- ically along a closed contour around z=0 , the functions (2.13) and (2.15) just pick up a phase, i.e. the monodromy matrix at z = 0 is diagonal; an obvious solution of the monodromy problem at z = ~ = 0 is therefore

apq = ap 6vq; (2.19)

moreover the behavior of the solutions near z = 0 also shows that, both for the SU(N) and the O(N) and Sp(N) cases, this solution is unique. Monodromy invariance at z = ~= 1 means [8, 11]

~Aa(Z,Z)= ~ aABf~M~(1--Z,I--z) a~A (2.20) B,B=I

92

with

a = ( ~ 10) forSU(N),

o-= 0 for O(N),

0

a = 1 0

0 0 1

for Sp(N). (2.21)

In all cases under consideration it is possible to ex- press ~qa(P)(z) as linear combinations

~q(e)(z)= ~ bvq aAB~B(q)(1 --Z) (2.22) B,q=l

with known coefficients bpq [11, 8]*. Substituting (2.12), (2.19) and (2.22) into (2.20) yields the requirement

• ap bvq bproCOqr. (2.23) p=l

Using the explicit form of the numbers bpq, it turns out that this overdetermined system of equations possesses a one parameter family of solutions, namely

a2 1 F2(--f l)V(a+fl)F(-a+fl) ~ - - N 2 - l F ~ ( D r ( ~ - f l ) r ( - ~ - D

(2.24)

for SU(N) [4] and

a 2 S( f l ) c ( l ( f l - 00)

a I 2c(a) s ( �89

a3 2s(fl) c(�89 3 ~)) s(�89 + fl)) a t s(~) s(2~)

(2.25)

for O (N) and Sp (N); here s (x) - sin (rex), c(x)=-cosfftx), and ~, fl are given by (2.14), (2.16) and (2.18) for SU(N), O(N) and Sp(N) respectively.

In conclusion, in each of the three cases G=SU(N), O(N) and Sp(N), the requirement of monodromy invariance is sufficient to determine the physical solution for the four point function up to an overall constant. In the following section we will investigate whether this is still true for G = G (N) x G (N).

* Actually, for O(N) and Sp(N) the sign of bpq depends on the index A [8]; we suppress this dependence since these signs drop out of the expressions (2.25) and (3.24) we are interested in

3. The Four Point Function for G = G ( N ) x G(fr

Let us now consider the four point functions (2.2), (2.3) in the case where G = G(N) x G(N), i.e. for theories with energy-momentum tensor (1.3). Analogously to (2.4), we can write

~(zl, zt ; ... ; z4, ~4)

= [ (z~ - z , , ) (~ , - ~ , , ) ( z ~ - z ~ ) ( ~ - e 3 ) ] - ~ ~'

fqAa,B~(Z, 5)IAT, raT ~ (3.1) A,~4,B,[I= 1

where I a are the G(N) invariants (2.6), (2.8) or (2.10) and T A the corresponding G(IV) invariants. Instead of (2.5) we have the differential equation

-�89 z(z (9 -- 1) ~z fqa;t'nB(z' 5)

= ~, [(k + C,dj)- 1 [(z-- 1) Pac + z QAC] 3no C , D - 1

+ (~"4- Cad j) - 1 r( Z __ 1) PBD + Z OBD] (~AC]

�9 fqc;~,o~(z, 5) (3.2)

with P, Q as defined in (2.7), (2.9), or (2.11), and P, (~ defined by the same equations with N replaced by

Clearly, the independent solutions of (3.2) are just products of the independent solutions of (2.5) and of the corresponding equation for G(dV). Therefore the most general solution of (3.2) is

%:~,,~(z, 5)= ~ a,,,,rs p,q,r,s= 1

-~qa (') (z) ~B (q) (z) ~qA ~r) (5) ~n x~) (e) (3.3)

with apqr~ arbitrary and with ~qA(P)(z) given by (2.13) or (2.15) and ~A(P)(z) defined analogously, i.e. with the parameters ~:-a(N,k), f l=fi(N,k) replaced by 8=~(N, ~), f l=f l (~ , k'). As before, the physically acceptable solutions within the solution space (3.3) are determined by requiring monodromy invariance.

The monodromy matrix at z = 0 is again diagonal; hence an obvious solution of the monodromy problem at z = 5 = 0 is

apqrs = apq (~ pr (~ qs" (3.4)

Given this solution and the relations (3.3) and (2.22), monodromy invariance at z = 5 = i amounts to the equation

a p q b p r b p r , ~ q s ~ q s , OC(~rr, ass, (3.5) p,q= 1

(with bpq as in (2.22), and ~'pq defined analogously); the solutions of (3.5) are given uniquely by the one- parameter family

apq = a v ~lq (3.6)

with a v as in (2.24) or (2.25), and @ given by the same equations with ~, fl replaced by 5, ft. As a result, a physically acceptable solution of (3.2) is given by

%a.Bs(z, e) = (#aa (Z, i)" f~B~(Z, e) (3.7)

where fqAa and f#n~ are the physical solutions for the four point functions in the cases G=G(N) and G = G (/V), respectively.

It remains to be seen, however, whether the product form (3.7) of the physical solution for G = G(N) x G(b~) is actually unique (up to overall normalization). The step to be checked is the implementa- tion of monodromy invariance at z = i = 0.

In fact, the solution (3.4) of the monodromy problem at z = 5= 0 is unique unless the monodromy matrix at z = 0 is a multiple of the unit matrix. In con- trast, in the latter case monodromy invariance at z = 5= 0 is automatic. Now from the explicit form of the functions fgA (") it can be seen that for generic values of the Kac-Moody central charges k and ~', there is no choice of the coefficients at, q, ~ in (3.3) which results in a monodromy matrix at z = 0 proportional to unity. However, if we consider the special values k = N , k'= N, then such a choice does exist (and is unique); for these values the parameters a, fl in (2.13) and (2.15) obey c~=~, f i + f l = l , and hence we have for SU(N) x sg( f i r )

lim fg<aP)(z) f~(RP)(z)ocz "+~/m+<~/~) z --* O

=z "+I/N~, me{0,_ 1} (3.8)

while for O(N) x 0(P~) and Sp(N) x Sp(N)

3 ) l im( ~ ppq(ff(P)(z)@(q)(z) OCZ m, me{0, + i, +2} z ~ O \ q = 1

(3.9)

with

p = 0 ,

1

(3.10)

i.e. for these combinations the limit z ~ 0 is, up to integer powers of z, independent of the values of A, B, and p, and hence the monodromy matrix at z = 0 is a multiple of the identity. Thus for k=b~, E = N we have, in addition to (3.4), the solutions

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apqr~ = 4,, 5,q 5,s (3.11)

and

apqrs : t~pr ppq ,Ors (3.12)

for SU(N)• SU(N) and for O(N)• O(b~) and Sp(N) x Sp(Ar), respectively.

The coefficients fi,r are further restricted by requiring monodromy invariance at z = ~ = 1; using (3.11), we have

2

E @rbpq~pq'brs~rs'~(~qq'~ ', (3.13) p,r = 1

while in the case of (3.12) the requirement is

3

Z CtprPpp'Prr'bpqbp'q'brs~r's '~ '' (3.14) p,p',r,r'= 1

In both cases, the (up to normalization) unique solution is

apt = @ dr (3.15)

with dp determined by the requirements

2

~tp bpq ~pr OQt~ qr (3.16) p = l

and

3

Z ap,opqbpr~qs~ , (3.17) p,q= 1

respectively. In conclusion, for k = N , ~ = N the monodromy

problem possesses two and only two solutions, given by (3.7) and by

2

p=l

2 �9 ~, ~q%~q)(2)f#~)(2) for SU(N)xSU(~I) q=l

3 %~,~(z,~)= ~ Gp~y)(z)~4~(z)

p,q= l

3

r s $ = l w

for O(N) • O(/V) and Sp(n) • Sp(N) (3.18)

with @ fulfilling (3.16) and (3.17), respectively. Actually, the solution (3.18) yields simple algebraic

functions. To see this, first consider the case SU(N)

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• SU(N); here we find from (3.16)

42 1 d~= NN" (3.19)

But for a = 8, fl + ~ = 1, we see from (2.13) that

~ 1 1 ) (Z) = - - N - 1 Zf1-1 (1 -- z) # if2 (z) (z),

~]2)(z) = Nz ~- ~(1 --z) ~ if2(') (z) (3.20)

and hence

2

ap~(AP)(Z)~(BP)(z)=O for A + B. (3.21) p = l

For A = B, we can use a formula relating the product of two hypergeometric functions to a generalized hypergeometric series [13] to write

2

z. y' a,, ~,')(z) ~?)(z) p = l

2

= ( l - z ) - ~ apff(f)(z)~(f)(z) p = l

= l-al(Z(1-z)UN~'2 3F2 B,-- /~+1; ~zu--z~)

/] + ~ - z ( 1 f l + l , - f l + 2 , 2 ; ' 4 z ( 1 - z ) ;

(3.22)

moreover, a power series expansion of the square bracket on the right hand side reveals that it is actually equal to unity.

Thus we finally have, up to an overall constant,

ff(z,z) = Z (~Aa,,~(z,e)IAI~Ia?~ A,J4,B,B

_ 1 / N ~ / I I T I 121"2 \//-i 71 1212\ =ez,(,-z)(,-z)l

(3.23)

Of course, given the knowledge that a realization of the Sugawara energy-momentum tensor is provided by free fermions in the defining representation of SU(N) x SU(N)x U(1), the result (3.23) could have been guessed right from the beginning. We decided to present the above analysis to emphasize that it is not difficult to derive (3.23) even without this a priori knowledge. In addition, the derivation proved that, apart from the product solution (3.7), (3.23) is the unique solution of (3.2) fulfilling the requirements of locality and crossing symmetry.

In the O(N)x 0(bT) and Sp(N)x Sp(N) cases, the solution of (3.17) turns out to be

4 2 S(fl) S ( f l - 0 0

4, s (2 ~) [ s ( ~ ) - s(/~)] '

a~ s(~)s(~+~) ax s(2c 0 [_s(c0_s(fl)], (3.24)

We should substitute (3.24) together with (2.15) into (3.18) to find the explicit form of if(z, ~). We will short- cut this rather cumbersome calculation by simply pre- senting a local and crossing symmetric solution of (3.2) which is not of the form (3.7); according to the proof of uniqueness of (3.18), this solution must then coincide with (3.18) when appropriately normalized. Both for O(N)x O(N) and Sp(N)• Sp(bT), the solution is

[IlI1 I2T2 ~\[I1~1 I 2 7 2 - 7 3 ~ ) f f(z 'z)=~-+f~z_z-- I313)[~+f~_ ~-- I3

(3.25)

(it is easily checked that (3.25) indeed satisfies (3.2) if k = N , ~=N).

4. Discussion

Summarizing the results of Sect. 3, we have found that the information contained in the algebraic structure (current and conformal algebra with energy-momentum tensor of the form (1.3)) of the G(N) x G(N) model with central charges k = N and ~'= N is not sufficient to determine uniquely the four point function of the primary fields d,J~r D in the defining representation Yii "~ ' J

of G(N)x G(N). Rather, there are two distinct solutions; for one of them, (3.7), the G(N) and G(N) dependence factorizes, whereas for the other one, (3.23) or (3.25), the G(N) and G(/V) dependences are strongly correlated while the z and s dependence factorizes.

At this point it is appropriate to recall [-12, 4] that the leading singularities of the four point function if(z, ~) at z, ~=0, 1, ~ correspond to the contribu- tions of the composite fields in the operator product expansion of ~b(z, ~) ~b(0, 0) (or ~b(z, ~) (~(0, 0)) etc. to if(z, ~). Since these singularities are different for the two solutions (3.7) and (3.23), (3.25), we conclude that two conformal field theories which yield these two different solutions possess different operator product algebras and therefore cannot be equivalent.

More specifically, the disappearing of certain di- vergences in the solution (3.23), (3.25) as compared to (3.7), means the disappearance of the corresponding composite fields from the operator product expan-

95

sion, i.e. the existence of selection rules. Now in a general conformally and Kac-Moody invariant theory there are three types of possible selection rules, namely purely Virasoro algebra, combined Virasoro and Kac-Moody algebra and purely Kac-Moody algebra ones. The first type occurs only in the degenerate conformal theories with Virasoro central charge smaller than one [12, 14] and is therefore irrelevant in the present context; the second type [-4] is ex- hausted by the differential equation (3.2) which is obeyed by both of our solutions; finally, the third type which relates the allowed composite fields to the value of the Kac-Moody central charge [15] is again irrelevant for the representations and k values consid- ered here. We thus conclude that for the solution (3.23), (3.25) to be realized in a field theory, this theory has to provide additional selection rules different from those encoded in its conformal and current algebra structure.

In fact, it is rather obvious which type of theories should account for the two different four point functions found above. The solution (3.7) is reproduced by the WZW model on G(N)x G(b~) with k = N , E=N, with the field ~b(z, 5) given by the operator product of the two elementary fields g(z, 5) and ~(z, 5) of the G(N) and G(N) theories,

q~3~tz 5) = g~(z, e) ~(z, 5); (4.1) i i x ,

the product structure of (3.7) then simply reflects the fact that the G(N) and G(K/) theories do not interact with each other. Furthermore, the solution (3.25) is simply the four point function of bilinears of free massless Majorana fermions in the defining representation of O(N) x O(~/) or Sp(N) x Sp(N) [16, 17] while (3.23) reduces to the four point function of bilinears of free massless Dirac fermions in the defining representation of U(N) x U(N) ~- SU(N) x SU(N)x U(1) 1-18] after, similar to (1.2), the U(1)

piece is reinstalled in the energy-momentum tensor (1.3) (leading to an additional diagonal term in (3.2) and thus swallowing the square bracket prefactor in (3.23)). The factorization of the z- and 5-dependence in (3.23), (3.25) is then just a consequence of the factorization of the primary field ~b (z, 5) itself,

c/)JJ(z Y.) = ~a(z) tiffS(5) (4.2) i i ~

and the additional selection rule restricting the operator product ~b(z, 5)qS(0, 0) is nothing but the Pauli principle.

We thus finally conclude that the bosonic and fermionic G(N)x G(]V) (x U(1)) theories introduced in Sect. I are distinct quantum theories*. They differ in

* In the particular case Sp(N) x Sp(1) this has also been concluded, without explicit calculation of the four point function, in [17]

that the fermionic theories possess selection rules provided by the Pauli principle which are not shared by the bosonic WZW theories. This has to be con- trasted with the boson-fermion equivalence for SU(N) x U(I) or O(N) with k = 1 also mentioned in the introduction; in these cases the Pauli principle of the fermion theory is exactly copied [15] by the pure current algebra selection rules of the WZW theory.

As a final comment let us recall that there are important applications of the SU(N) x SU(N) x U(1) boson-fermion correspondence [1, 6, 9]. For these applications a full equivalence of the bosonic and fermionic theories is in fact not necessary. What is necessary is that there exists in the bosonic theory some field which is equivalent to the free fermion bilinear ~ . As we have seen, and as was already noticed in [1], the product of the elementary WZW fields g, ~ and a free boson vertex operator does not possess this property, although it transforms in the correct SU (N) x SU (N) x U(1) representation.

Moreover, it is highly unlikely that the ordinary WZW theory contains an additional field which transforms in the same representation as this product. What seems to be needed to obtain a field with the required property is a modification of the WZW theory, e.g. by introducing twisted boundary condi- tions which intertwine the SU(N) and SU(N) dependence.

A c k n o w l e d g e m e n t s . It is a pleasu-r0 to :thank D. Gepner and I. Affleck for helpful conversations and f0r reading the manuscript.

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Free fermions and WZW theories on nonsimple groups