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H conformal field theory from gauged WZW model
Volume 215, number 1
PHYSICS LETTERS B
8 December 1988
G/H CONFORMAL FIELD THEORY FROM GAUGED WZW MODEL K. G A W I ~ D Z K I lnstitut des tlautes Etudes Scientifiques, CNRS, F-91440 Bures sur Yvette, France and A. K U P I A I N E N Research lnstitute jor Theoretical Physics, Helsinki University, SF-O0170 Helsinki 17, Finland Received 22 July 1988
We show that the coset construction for affine algebras ~ = h can be realized by coupling a group G WZW model to a .gauge field taking values in the Lie algebra h. The partition function of the coset models is computed exactly in terms of the branching functions of~ ~ ft. Correlation functions may be expressed in terms of those of the G-valued WZW model and of the He/H-valued one, also exactly soluble. The special cases include unitary, superconformal, parafermionic and other discrete series.
The coset construction [ 1 ] o f representations o f the Virasoro algebra has led to the identification o f several interesting families o f c o n f o r m a l field theories [ 2 - 1 0 ] . F o r a general c o m p a c t Lie group G and a subgroup H, it has been conjectured [ 11 ] that a G / H theory exists with a p a r t i t i o n function expressed in terms o f the branching functions [ 12 ] o f the representations o f the affine algebras ~ and fi and correlations in principle d e t e r m i n e d in terms o f the G and H W Z W m o d e l [ 13,14 ] correlations. In this letter we present a functional-integral a p p r o a c h to a G / H theory formulated as a gauged W Z W model. Previous attempts in this direction [ 15-17 ] were not quite conclusive as witnessed by ref. [ 18 ]. Here we achieve an exact solution o f the group G W Z W model, with subgroup H gauged, by transforming it to the essentially decoupled (non-gauged) G W Z W model, a new W Z W - t y p e H C / H n o n - c o m p a c t sigma model and free (ghost) fields. F o r simplicity, we shall consider G and H simple and simply connected. This can be easily relaxed. F o r the general analysis and m o r e details we refer to ref. [ 19 ]. Let us illustrate our argument on the example o f the G / H partition function on the torus. This will be given by the functional integral Z~;/H (Z) = J exp[ - k S ( g , A ) ] D g D A .
( 1)
Here g is a G - v a l u e d field on the torus T~ with m o d u l a r p a r a m e t e r r. A, the gauge field, is an i m a g i n a r y he-valued one-form where h is the Lie algebra of H. The gauged W Z W action is given by
if Tr
-~
B
[(Am, J)+(J,A°')+(gA'°g-',A°')-(A'°,A°')],
(2)
Tr
where B has T~ as the b o u n d a r y , ~ extends g to B a n d the currents are J = g O g -~, J = g - ~ Og. A m and A°~= 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
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Volume 215, number 1
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8 December 1988
- (A ~°)* are the dz parts of A. The Killing form is normalized so that (qb, q~) = 2 for the long roots of G. The level k is a positive integer. Notice that the coupling to A is not of the fJ~'A~,type discussed in ref. [ 17 ]. We shall perform the A integral in ( 1 ) in the following steps. First, on a dense open set of connections, A may be written [20] as a chiral gauge transform of a fiat connection, i.e.,
A°~=h-l(A°~+O)h,
(3)
with h: T<-.H c, the complexification of H. The flat A's are parametrized by their holonomies 7~, 72 around the two cycles to T~. We may take 7~ = exp (2niqb), 72=exp(2niO) in T, the Caftan subgroup of H, and write A ° ' = ( h . h ) - ' 0 ( h . h ) = h - 1 (A°ul + 0 ) h ,
(4)
with u = e x p [ 2 n i ( O - rq~) ] in T c and h,,=u ~-:)/~-:~ Inserting (4) to (2) and using the Polyakov-Wiegmann formula (S(g) =S(g, O) )
if
S(gt,g2)=S(g~)+S(g2)+ ~n
(g(-Z OgL, (Og2)g~ ~) ,
(5)
we get
S(g, A) =S(h.hgh*h]) -jS(h.hh*h*)
(6)
(it can be shown, that the WZW action is well defined for twisted fields as in ( 6 ), see ref. [ 19 ] ). j is the Dynkin index of the embedding h ~ g and occurs in the last S we use the h normalization of ( , ) . Next, we want to make a change of variables in ( 1 ) to h and O and q~. h~H c may be uniquely decomposed as h = e x p ( ~,~>oVc~e~)e~U-bU'
(7)
with o~ roots of h, e, the step operators ~et, the Cartan subalgebra of h, UeH. Eq. (7) is the analogue of the unitary-upper triangular decomposition of an h e GL (N, C). Given A o~ as in (4), with O, q~ generic, h is fixed (up to an element in the center) if we fix ~0(0) and e.g. phases o f v , ( 0 ) for o~ simple roots. With such gauge fixing, we may perform the change of variable and ( 1 ) becomes Z(;/H (r) = ~ " j d g d ~ d O D h d(~0(0) )F(A) exp[ - k S ( g , A) ] ,
(8)
with r = rank (H),
F(A) = d e t ' (13~D ,) [det(a~, eJ) det(a~, aj) ] -~ ,
(9)
e, and a, being the basis of ker IbA and coker 13A respectively transforming by Adh under (3) and orthonormal for A =A,. Computation of (9) (by chiral anomaly) is standard:
F(A ) =exp [2h"S( h,hh*h *) - 2h~S( h,,h * ) ] F ( A , ) ,
(10)
exp[ - 2 h ' S ( h . h * ) ]F(A.) = z~r (qc~) -2+(l/12)dimH[U(q,U)14 ,
( 11)
with
H(q,u)= II (1-ea)(q,u),
(12)
&>0
where ~ = ( o~, n ) are the affine roots of fi acting on C* × T c as ea (qe z) = q"exp ( - ( o~,Z) ). h" is the dual Coxeter number of H, and p half the sum over its positive roots. Combining now eqs. ( 8 ) - ( 11 ) and (6), we obtain ZG/H('C)='~ r
120
j dqbdOZG('C, u) ~eH(Z, U) exp[-2h~'S(h.h*) lF(A.) ,
(13)
Volume 215, number 1
PHYSICS LETTERS B
8 December 1988
where Zc; is the WZW partition function [GW, FGK]
Zc,(r, u)= j exp[-kS(h.gh*.) ] Dg= (q~/)--(l/24)kdinaG/(k+g")~A IZA(r' U)
(14)
with g" denoting the dual Coxeter number of G and Z. being the characters of the level k integrable representations A of~. With E=jk+ 2h ~', Sen(z, u ) = j Dhexp[FcS(h.hh*h.)] .
(15)
Since only hh* appears in (15), the integral can be restricted to HC/H valued fields and becomes the partition function of an exactly soluble WZW-type model based on the coset space HC/H [ 19 ]. For example, for H = SU (2), we obtain the sigma model with values in the three-dimensional hyperboloid. As realized in ref. [21 ] for the latter case, Sen can be computed essentially by gaussian integrals. The H c invariant measure on HC/H becomes in the variables of (7) d~01~.> odu. dd~, where u. =exp( - (o~, ~0) ) v . .
(16)
The WZW action in (15) may be computed using (5) repeatedly. The result is
S(h,,hh*h*)= -
27~ f d2z(( 0-~,a-~>+2 a>O
I I~)a it. + f a I 2) _ 27cr 2 ( qO, ~iO) ,
with
(17)
I)~ =exp( - (c~, fD+Z) )~z exp( ( a , {0+Z) )
and f . certain polynomials in (derivatives of) u. It turns out, that there is an ordering of positive roots ~1 C~ds.t.f., only depends on u.j with j < i. Thus, since ker I ) . = 0 = c o k e r l ) . except for a discrete set ofq~, O, we may perform the u.-integrals in (15) getting ....
~n(r,u)=(qO) ~/2)<¢'¢'> fD~,~(~(O))exp( -
~ f)IJ ( d e t - * ~>o
D~D.)
-~
.
,
(18)
The determinant ( 18 ) is again standard and we get finally ~fi)fH(Z, U) ='r2"/2(qq) {l/2)~'c-h')+ -{i/24) aimHlH(q ' U)[-2
(19)
We now combine ( 13 ), ( 12 ), (14) and ( 19 ) with
z,,(r, u)= E2 b~(r)Z~(r,u),
(20)
where 2 are level jk weights of fi and b~ (r) are the branching functions [ 12 ] from representation A to 2. Using also the Kac formula [22] for the character Z~.we obtain ZG/H(r'
bl)-~'T~/2(q4)(I/24)tdirnH--kdirnG/(l"+g")]E E bXA'(r)b~2(r)m~.,~.2,
(21)
with ~,~2 w~,w2
× f dcbdO exp{ (w, [2-~ + (jk+h")~ ] - p , Z) + (w2 [2-2 + 0 k + h")~2] - p , 2 ) } , where ~, runs over the co-root lattice of h, w~ over its Weyl group, c(2, ~)= ½1112-+(jk+h'~)~ll 2_ Ilpll 2](Jk + h'3 ~, and 2-=2 +p. Z= 2hi (~9- rqb). The integrals dO, dqb are over t/co-root lattice and Weyl alcove Co of t, 121
Volume 215, number 1
PHYSICS LETTERS B
8 December 1988
respectively. The O integrals w~ = w2, (~ =~2 and 2L = / ~ 2 and the qMntegral combined with the sum over w~ and ~ yields a gaussian integral over the Cartan subalgebra of h. We get rn; ,>.2= c~a,a2 z ~-r/2( q c ] ) -,pll 2/2 (J/" + hv) Finally, since ILPt[2= Zc;/n(r) = with
~2 h z, dim
( 22 )
H, (21 ) and (22) yield
(q(l) -(*/24~C<'/~Y'. Ib~(r)12 A,2
C(~/Hthe
,
(23)
closet construction central charge [ 1 ]
kdimG
CG/H= k+g"
jk dim H jk+h"
(24)
Relation (23) generalizes to arbitrary semisimple G and H with (24) involving the sums over simple factors. U ( 1 ) factors in H are easily accommodated too, for example the level k parafermionic partition functions SU(2) = U ( 1 ) come out asin [23] k
= (qcT)-k/~gk+ 16) Y',
Zst,~2),/u~,~
/=0
2k-- I
Z
(25)
I Cg,(r) I21r/(T) 12 ,
m=0 m = / rood 2
where the C}I~(z) are the so-called functions [24] and q(T) is the Dedekind function. Also the assumption that G is simply connected can be easily relaxed, with the help of the results of ref. [ 25 ], leading to partition functions given by non-diagonal expressions in branching functions. For example the variants of the classical coset construction [26] of the unitary series, S U ( 2 ) × S U ( 2 ) ~ d i a g S U ( 2 ) , SO(3) X S U ( 2 ) ~ diagSU(2) and ( S U ( 2 ) xSU(2))/772 = diagSO(3) at level k, 1 gives the (A, A), (D, A) and (A, D) series of modular invariant partition functions [ 27 ]. The realization of the closet theories as gauged WZW models allows to express certain (primary) coset correlations by the WZW ones. For example on the sphere, we immediately get
f l-[ trA,(g(z,) ) exp[ -kS(g, A) ]DgDA t
=const. • j" 17 '
trA,(g(zi) (hh*)-' (z,) ) exp [ -kS(g) ] Dg exp[~S(hh*) ] Dh,
(26 )
where tra denotes (the analytic continuation of) the real part of the trace in the highest weight representation A,. Thus (26) factorizes to WZW and HC/H correlations and the latter obey the Knizhnik-Zamolodchikov equations [ 13 ], the same as the group H WZW correlators but for the level -/7. For four-point functions, these can be solved in many cases explicitly [13,28,29 ] leading to the expressions for coset correlations factorized into (a sum of) products of WZW type ones. In application to the unitary series this gives the correlators of the diagonal operators (1)(p,p) in the factorized form, not evident in the Feigin-Fuchs representation [ 30 ] but known already for q~2.2~ four-point functions [31 ]. Also correlators of non-diagonal fields can be worked out, the simplest being that of (~)(2,1) represented as
I ~ trl/2(g'(zi)g2(z')-') exp[-kS(g~,A)-S(&,A)] Dg, Dg2 DA ~const.
•
I H tr~/2(g~(z,)&(z~) -~ ) exp[ -kS(g~,
A)] Dg~ exp[
~ S(g2
~
A)] Dg2,
(27)
and so factorized in terms of level k and level one SU (2) correlators, as noticed already in ref. [ 11 ]. If relation (26) is specialized to the case when A couples to the full group then, as easily follows from the Knizhnik-Zamolodchikov equations, one obtains z,-independent expressions. Thus the level/7 H C/H correla122
Volume 215, number 1
PHYSICS LETTERS B
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t o r s i n v e r t t h e level jk o n e s o f t h e g r o u p H W Z W m o d e l a n d ( 2 6 ) m a y b e r e g a r d e d as a p r e c i s e f o r m o f t h e r e l a t i o n s b e t w e e n t h e c o s e t a n d W Z W c o r r e l a t o r s c o n j e c t u r e d i n ref. [ 11 ]. F o r t h e p a r a f e r m i o n i c m o d e l s , b e s i d e s t h e p r i m a r y fields, also t h e p a r a f e r m i o n i c c u r r e n t s c a n b e r e p r e s e n t e d b y s i m p l e ( b u t in t h e l a t t e r case n o n - l o c a l ) f u n c t i o n a l - i n t e g r a l e x p r e s s i o n s [ 19].
References [ 1 ] P. Goddard, A. Kent and D. Olive, Phys. Len. B 152 (1985) 88. [2] D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Len. 52 (1984) 1575. [3] H. Eichenherr, Phys. Lett. B 151 (1985) 26. [ 4 ] M.A. Bershadsky, V.G. Knizhnik and M.G. Teitelman, Phys. Lett. B 151 (1985) 31. [ 5 ] D. Friedan, Z. Qiu and S. Shenker, Phys. Lett. B 151 ( 1985 ) 37. [6] A.B. Zamolodchikov and V.A. Fateev, Sov. Phys. JETP 62 (1985) 215. [7] V.A. Fateev and A.B. Zamolodchikov, Theor. Math. Phys. 65 (1985) 1205. [ 8 ] V.A. Fateev and A.B. Zamolodchikov, Nucl. Phys. B 280 (1987) 644. [ 9 ] D. Kastor, E. Martinec and Z. Qiu, Phys. Lett. B 200 (1988) 434. [ 10 ] J. Bagger, D. Nemeschansky and S. Yankielowicz, Phys. Rev. Lett. 60 ( 1988 ) 389. [ l 1 ] M.R. Douglas, G / H conformal field theory, preprint CALT-68-1453 ( 1987 ). [ 12 ] V. Kac and M. Wakimoto, Adv. Math., to be published.. [ 13 ] V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 243 (1984) 83. [ 14] D. Gepner and E. Winen, Nucl. Phys. B 278 (1986) 493. [ 15 ] E. Guadagnini, M. Martellini and M. Mincthev, Phys. Lett. B 191 (1987) 69. [ 16] K. Bardacki, E. Rabinovici and B. S~iring, Nucl. B 299 (1988) 151. [ 17 ] W. Nahm, preprint UCD-88-02 ( 1988 ). [ 18 ] P. Bowcock, preprint Imperial TH/87/88 ( 1988 ). [ 19 ] K. Gawcdzki and A. Kupianen, in preparation. [20] M.F. Atiyah and R. Bott, Phil. Trans. R. Soc. Lond. A 308 (1982) 523. [ 21 ] Z. Haba, Bochum University preprint BiBoS 309/88 ( 1988 ). [22 ] A, Pressley and G. Segal, Loop groups (Clarendon, Oxford, 1986 ). [23 ] D. Gepner and Z. Qiu, Nucl. Phys. B 285 ( 1987 ) 423. [24] V. Kac and D.H. Peterson, Adv. Math. 53 (1984) 125. [ 25 ] G. Felder, K. Gaw~dzki and A. Kupiainen, Commun. Math. Phys. 117 (1988) 127. [26] P. Goddard, A. Kent and D. Olive, Commun. Math. Phys. 103 (1986) 105. [27] A. Cappelli, C. Itzykson and J.-B. Zuber, Commun. Math. Phys. 113 (1987) 1. [28] P. Christie and R. Flume, Nucl. Phys. B 282 (1987) 466. [29] V.A. Fateev and A.B. Zamolodchikov, Yad. Fiz. 43 (1986) 75. [30] VI.S. Dotsenko and V.A. Fateev, Nucl. Phys. B 240 [FSI2] (1984) 312; B 251 [FSI3] (1985) 691. [31 ] A.B. Zamolodchikov, Yad. Fiz. 44 (1986) 821.
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G/H CONFORMAL FIELD THEORY FROM GAUGED WZW MODEL K. G A W I ~ D Z K I lnstitut des tlautes Etudes Scientifiques, CNRS, F-91440 Bures sur Yvette, France and A. K U P I A I N E N Research lnstitute jor Theoretical Physics, Helsinki University, SF-O0170 Helsinki 17, Finland Received 22 July 1988
We show that the coset construction for affine algebras ~ = h can be realized by coupling a group G WZW model to a .gauge field taking values in the Lie algebra h. The partition function of the coset models is computed exactly in terms of the branching functions of~ ~ ft. Correlation functions may be expressed in terms of those of the G-valued WZW model and of the He/H-valued one, also exactly soluble. The special cases include unitary, superconformal, parafermionic and other discrete series.
The coset construction [ 1 ] o f representations o f the Virasoro algebra has led to the identification o f several interesting families o f c o n f o r m a l field theories [ 2 - 1 0 ] . F o r a general c o m p a c t Lie group G and a subgroup H, it has been conjectured [ 11 ] that a G / H theory exists with a p a r t i t i o n function expressed in terms o f the branching functions [ 12 ] o f the representations o f the affine algebras ~ and fi and correlations in principle d e t e r m i n e d in terms o f the G and H W Z W m o d e l [ 13,14 ] correlations. In this letter we present a functional-integral a p p r o a c h to a G / H theory formulated as a gauged W Z W model. Previous attempts in this direction [ 15-17 ] were not quite conclusive as witnessed by ref. [ 18 ]. Here we achieve an exact solution o f the group G W Z W model, with subgroup H gauged, by transforming it to the essentially decoupled (non-gauged) G W Z W model, a new W Z W - t y p e H C / H n o n - c o m p a c t sigma model and free (ghost) fields. F o r simplicity, we shall consider G and H simple and simply connected. This can be easily relaxed. F o r the general analysis and m o r e details we refer to ref. [ 19 ]. Let us illustrate our argument on the example o f the G / H partition function on the torus. This will be given by the functional integral Z~;/H (Z) = J exp[ - k S ( g , A ) ] D g D A .
( 1)
Here g is a G - v a l u e d field on the torus T~ with m o d u l a r p a r a m e t e r r. A, the gauge field, is an i m a g i n a r y he-valued one-form where h is the Lie algebra of H. The gauged W Z W action is given by
if Tr
-~
B
[(Am, J)+(J,A°')+(gA'°g-',A°')-(A'°,A°')],
(2)
Tr
where B has T~ as the b o u n d a r y , ~ extends g to B a n d the currents are J = g O g -~, J = g - ~ Og. A m and A°~= 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
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Volume 215, number 1
PHYSICS LETTERS B
8 December 1988
- (A ~°)* are the dz parts of A. The Killing form is normalized so that (qb, q~) = 2 for the long roots of G. The level k is a positive integer. Notice that the coupling to A is not of the fJ~'A~,type discussed in ref. [ 17 ]. We shall perform the A integral in ( 1 ) in the following steps. First, on a dense open set of connections, A may be written [20] as a chiral gauge transform of a fiat connection, i.e.,
A°~=h-l(A°~+O)h,
(3)
with h: T<-.H c, the complexification of H. The flat A's are parametrized by their holonomies 7~, 72 around the two cycles to T~. We may take 7~ = exp (2niqb), 72=exp(2niO) in T, the Caftan subgroup of H, and write A ° ' = ( h . h ) - ' 0 ( h . h ) = h - 1 (A°ul + 0 ) h ,
(4)
with u = e x p [ 2 n i ( O - rq~) ] in T c and h,,=u ~-:)/~-:~ Inserting (4) to (2) and using the Polyakov-Wiegmann formula (S(g) =S(g, O) )
if
S(gt,g2)=S(g~)+S(g2)+ ~n
(g(-Z OgL, (Og2)g~ ~) ,
(5)
we get
S(g, A) =S(h.hgh*h]) -jS(h.hh*h*)
(6)
(it can be shown, that the WZW action is well defined for twisted fields as in ( 6 ), see ref. [ 19 ] ). j is the Dynkin index of the embedding h ~ g and occurs in the last S we use the h normalization of ( , ) . Next, we want to make a change of variables in ( 1 ) to h and O and q~. h~H c may be uniquely decomposed as h = e x p ( ~,~>oVc~e~)e~U-bU'
(7)
with o~ roots of h, e, the step operators ~et, the Cartan subalgebra of h, UeH. Eq. (7) is the analogue of the unitary-upper triangular decomposition of an h e GL (N, C). Given A o~ as in (4), with O, q~ generic, h is fixed (up to an element in the center) if we fix ~0(0) and e.g. phases o f v , ( 0 ) for o~ simple roots. With such gauge fixing, we may perform the change of variable and ( 1 ) becomes Z(;/H (r) = ~ " j d g d ~ d O D h d(~0(0) )F(A) exp[ - k S ( g , A) ] ,
(8)
with r = rank (H),
F(A) = d e t ' (13~D ,) [det(a~, eJ) det(a~, aj) ] -~ ,
(9)
e, and a, being the basis of ker IbA and coker 13A respectively transforming by Adh under (3) and orthonormal for A =A,. Computation of (9) (by chiral anomaly) is standard:
F(A ) =exp [2h"S( h,hh*h *) - 2h~S( h,,h * ) ] F ( A , ) ,
(10)
exp[ - 2 h ' S ( h . h * ) ]F(A.) = z~r (qc~) -2
( 11)
with
H(q,u)= II (1-ea)(q,u),
(12)
&>0
where ~ = ( o~, n ) are the affine roots of fi acting on C* × T c as ea (qe z) = q"exp ( - ( o~,Z) ). h" is the dual Coxeter number of H, and p half the sum over its positive roots. Combining now eqs. ( 8 ) - ( 11 ) and (6), we obtain ZG/H('C)='~ r
120
j dqbdOZG('C, u) ~eH(Z, U) exp[-2h~'S(h.h*) lF(A.) ,
(13)
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where Zc; is the WZW partition function [GW, FGK]
Zc,(r, u)= j exp[-kS(h.gh*.) ] Dg= (q~/)--(l/24)kdinaG/(k+g")~A IZA(r' U)
(14)
with g" denoting the dual Coxeter number of G and Z. being the characters of the level k integrable representations A of~. With E=jk+ 2h ~', Sen(z, u ) = j Dhexp[FcS(h.hh*h.)] .
(15)
Since only hh* appears in (15), the integral can be restricted to HC/H valued fields and becomes the partition function of an exactly soluble WZW-type model based on the coset space HC/H [ 19 ]. For example, for H = SU (2), we obtain the sigma model with values in the three-dimensional hyperboloid. As realized in ref. [21 ] for the latter case, Sen can be computed essentially by gaussian integrals. The H c invariant measure on HC/H becomes in the variables of (7) d~01~.> odu. dd~, where u. =exp( - (o~, ~0) ) v . .
(16)
The WZW action in (15) may be computed using (5) repeatedly. The result is
S(h,,hh*h*)= -
27~ f d2z(( 0-~,a-~>+2 a>O
I I~)a it. + f a I 2) _ 27cr 2 ( qO, ~iO) ,
with
(17)
I)~ =exp( - (c~, fD+Z) )~z exp( ( a , {0+Z) )
and f . certain polynomials in (derivatives of) u. It turns out, that there is an ordering of positive roots ~1 C~ds.t.f., only depends on u.j with j < i. Thus, since ker I ) . = 0 = c o k e r l ) . except for a discrete set ofq~, O, we may perform the u.-integrals in (15) getting ....
~n(r,u)=(qO) ~/2)<¢'¢'> fD~,~(~(O))exp( -
~ f
D~D.)
-~
.
,
(18)
The determinant ( 18 ) is again standard and we get finally ~fi)fH(Z, U) ='r2"/2(qq) {l/2)~'c-h')
(19)
We now combine ( 13 ), ( 12 ), (14) and ( 19 ) with
z,,(r, u)= E2 b~(r)Z~(r,u),
(20)
where 2 are level jk weights of fi and b~ (r) are the branching functions [ 12 ] from representation A to 2. Using also the Kac formula [22] for the character Z~.we obtain ZG/H(r'
bl)-~'T~/2(q4)(I/24)tdirnH--kdirnG/(l"+g")]E E bXA'(r)b~2(r)m~.,~.2,
(21)
with ~,~2 w~,w2
× f dcbdO exp{ (w, [2-~ + (jk+h")~ ] - p , Z) + (w2 [2-2 + 0 k + h")~2] - p , 2 ) } , where ~, runs over the co-root lattice of h, w~ over its Weyl group, c(2, ~)= ½1112-+(jk+h'~)~ll 2_ Ilpll 2](Jk + h'3 ~, and 2-=2 +p. Z= 2hi (~9- rqb). The integrals dO, dqb are over t/co-root lattice and Weyl alcove Co of t, 121
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respectively. The O integrals w~ = w2, (~ =~2 and 2L = / ~ 2 and the qMntegral combined with the sum over w~ and ~ yields a gaussian integral over the Cartan subalgebra of h. We get rn; ,>.2= c~a,a2 z ~-r/2( q c ] ) -,pll 2/2 (J/" + hv) Finally, since ILPt[2= Zc;/n(r) = with
~2 h z, dim
( 22 )
H, (21 ) and (22) yield
(q(l) -(*/24~C<'/~Y'. Ib~(r)12 A,2
C(~/Hthe
,
(23)
closet construction central charge [ 1 ]
kdimG
CG/H= k+g"
jk dim H jk+h"
(24)
Relation (23) generalizes to arbitrary semisimple G and H with (24) involving the sums over simple factors. U ( 1 ) factors in H are easily accommodated too, for example the level k parafermionic partition functions SU(2) = U ( 1 ) come out asin [23] k
= (qcT)-k/~gk+ 16) Y',
Zst,~2),/u~,~
/=0
2k-- I
Z
(25)
I Cg,(r) I21r/(T) 12 ,
m=0 m = / rood 2
where the C}I~(z) are the so-called functions [24] and q(T) is the Dedekind function. Also the assumption that G is simply connected can be easily relaxed, with the help of the results of ref. [ 25 ], leading to partition functions given by non-diagonal expressions in branching functions. For example the variants of the classical coset construction [26] of the unitary series, S U ( 2 ) × S U ( 2 ) ~ d i a g S U ( 2 ) , SO(3) X S U ( 2 ) ~ diagSU(2) and ( S U ( 2 ) xSU(2))/772 = diagSO(3) at level k, 1 gives the (A, A), (D, A) and (A, D) series of modular invariant partition functions [ 27 ]. The realization of the closet theories as gauged WZW models allows to express certain (primary) coset correlations by the WZW ones. For example on the sphere, we immediately get
f l-[ trA,(g(z,) ) exp[ -kS(g, A) ]DgDA t
=const. • j" 17 '
trA,(g(zi) (hh*)-' (z,) ) exp [ -kS(g) ] Dg exp[~S(hh*) ] Dh,
(26 )
where tra denotes (the analytic continuation of) the real part of the trace in the highest weight representation A,. Thus (26) factorizes to WZW and HC/H correlations and the latter obey the Knizhnik-Zamolodchikov equations [ 13 ], the same as the group H WZW correlators but for the level -/7. For four-point functions, these can be solved in many cases explicitly [13,28,29 ] leading to the expressions for coset correlations factorized into (a sum of) products of WZW type ones. In application to the unitary series this gives the correlators of the diagonal operators (1)(p,p) in the factorized form, not evident in the Feigin-Fuchs representation [ 30 ] but known already for q~2.2~ four-point functions [31 ]. Also correlators of non-diagonal fields can be worked out, the simplest being that of (~)(2,1) represented as
I ~ trl/2(g'(zi)g2(z')-') exp[-kS(g~,A)-S(&,A)] Dg, Dg2 DA ~const.
•
I H tr~/2(g~(z,)&(z~) -~ ) exp[ -kS(g~,
A)] Dg~ exp[
~ S(g2
~
A)] Dg2,
(27)
and so factorized in terms of level k and level one SU (2) correlators, as noticed already in ref. [ 11 ]. If relation (26) is specialized to the case when A couples to the full group then, as easily follows from the Knizhnik-Zamolodchikov equations, one obtains z,-independent expressions. Thus the level/7 H C/H correla122
Volume 215, number 1
PHYSICS LETTERS B
8 December 1988
t o r s i n v e r t t h e level jk o n e s o f t h e g r o u p H W Z W m o d e l a n d ( 2 6 ) m a y b e r e g a r d e d as a p r e c i s e f o r m o f t h e r e l a t i o n s b e t w e e n t h e c o s e t a n d W Z W c o r r e l a t o r s c o n j e c t u r e d i n ref. [ 11 ]. F o r t h e p a r a f e r m i o n i c m o d e l s , b e s i d e s t h e p r i m a r y fields, also t h e p a r a f e r m i o n i c c u r r e n t s c a n b e r e p r e s e n t e d b y s i m p l e ( b u t in t h e l a t t e r case n o n - l o c a l ) f u n c t i o n a l - i n t e g r a l e x p r e s s i o n s [ 19].
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