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Coset construction from functional integrals
Nuclear Physics B320 (1989) 625-668 North-Holland, Amsterdam
COSET CONSTRUCTION FROM FUNCTIONAL INTEGRALS K. GAWt~DZKI
CNRS, IHES, 91440 Bures-sur-Yuette, France A. KUPIAINEN
Research Institute for Theoretical Physics, Helsinki University, 00170 Helsinki, Finland Received 26 September 1988
A detailed study of the gauged Wess-Zumino-Witten models is presented. These models are shown to be conformal field theories realizing the G o d d a r d - K e n t Olive coset construction. Partition functions are computed for an arbitrary group G with a subgroup H gauged. Correlation functions are shown to be computable in terms of WZW ones. Explicit cases of the minimal models and parafermionic theories are worked out.
1. Introduction The coset construction was first introduced [2,3] as the method to obtain new representations of Virasoro algebras by decomposing the representations of the group G K a c - M o o d y algebra into factor representations of K a c - M o o d y algebra based on a subgroup H c G. The factors carry the action of the Virasoro algebra given by the difference of two Sugawara representations corresponding to G and H. This way all the unitary representations of the Virasoro algebras with the central charge c < 1 as well as those of super-Virasoro algebras for c < ~ were obtained by taking G = SU(2) × SU(2) and H = diagonal SU(2). In ref. [4] it was realized that the coset construction allows relating modular invariant combinations of SU(2) K a c - M o o d y characters and those of unitary Virasoro characters. This established a correspondence between toroidal partition functions of SU(2) W e s s - Z u m i n o - W i t t e n models and of the unitary minimal conformal theories [5-7]. Since similar relations were also discovered in the monodromies of chiral fourpoint functions of the models [8,9], it was natural to expect that the coset construction can be extended to the level of complete conformal field theories. In ref. [10] an attempt was made to associate to a general pair G ~ H a coset conformal field theory whose relation to the WZW models based on groups G and H was studied algebraically. Several discrete series of conformal field theories [11-26] 0550-3213/89/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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constitute examples of such coset models and the method seems to provide a quite general tool to construct rational conformal field theories. It was suggested quite early [2] that on the lagrangian level the coset construction should correspond to a coupling of the group G WZW model to group H gauge field without the tr F 2 term so that the integration over the gauge field would freeze the H degrees of freedom. This is the procedure whose variant works for symmetric space G / H asymptotically free models without the Wess-Zumino term [27,28]. In the conformal-invariant context however, this idea was making its way rather slowly. Various couplings to (and without) the gauge field were considered [29-32] but the arguments advanced did not seem very convincing. In the present paper, we attempt to clarify the situation by a detailed study of the functional integral for the WZW theory coupled to group H gauge field (for a preliminary announcement, see ref. [1]). Our analysis, although not strictly rigorous in the handling of infinite-dimensional integrals, proves nevertheless very instructive and allows a deeper understanding of the coset construction. It may be viewed as a more systematic version of the idea employed in ref. [31] to compute the central charge of the fermions coupled to a gauge field without the kinetic term. By the change of variables which parametrizes the gauge fields by chiral gauge transformations and moduli of holomorphic H C-bundles, we transform the functional integral of the gauged WZW model into an (almost) factorized form. The factors correspond to (1) a (non-gauged) group G WZW model, (2) a WZW-type sigma model with fields taking values in the non-compact coset space H C / H , (3) anti-commuting ghosts transforming in the adjoint representation of H c. The different factors are coupled only by the moduli parameters entering through twisted boundary conditions. The non-compact WZW model is a (non-unitary) theory exactly soluble to the same extent as the compact (unitary) WZW model. Its toroidal partition functions can be even computed by gaussian integration, as in the case of the models with values in the hyperbolic spaces considered in ref. [34]. Alternatively they can be deduced from the K a c - M o o d y symmetry content of the model, as for the compact case [35, 36]. The correlation functions of the non-compact model obey (the analytically continued version of) the KnizhnikZamolodchikov equations [37] arising in the compact case and can be solved in the same special cases. As a result, the gauged WZW model can be in principle solved exactly, similarly to the non-gauged model. The paper is organized as follows. Sect. 2, the longest, illustrates the above ideas on the example of toroidal partition functions. Here, the calculation may be done quite effectively, the integration over the moduli parameters (twists) included. The result is the G / H toroidal partition function given as a sesquilinear combination of the G D H branching functions [38], as postulated in ref. [10]. In sect. 3 we discuss a set of planar correlation functions for the coset models which factorize into (sums
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of) products of compact and non-compact WZW correlations. Sect. 4 discusses the specific example of the coset construction: the minimal unitary series. In particular we show how the "non-diagonal" series of the partition functions appear and we discuss the factorization of the simplest correlation functions. Sect. 5 contains a brief discussion of parafermionic theories [15, 39, 40]. This is the simplest illustration of the general case. H is here the Cartan subgroup of G and the non-compact H C / H WZW model reduces to free fields. The paper is closed by appendices containing some technical details omitted in its main body. After the submission of the paper, we have obtained a revised version of ref. [33] which studied the canonical quantization of the gauged WZW models and ref. [76] which examined the energy-momentum tensor algebra in these models using path-integral arguments. Both papers have also concluded that the coset theories can be realized by gauging the WZW models.
2. G / H toroidal partition functions 2.1. THE GAUGED WZW MODEL Let us consider a general WZW model [35-37, 41, 42] on the Riemann surface Z with fields g taking values in the compact Lie group G (or, more generally, its complexification G C). We shall assume that
G=
t i__I~l n Gi
t/
Z
where G i are connected simple and simply connected compact groups and Z is a subgroup of the center of IJG~. On the Lie algebra ~ c of G c, ~ c = ~ ~ic, consider the Killing form ( .,. )k normalized so that half the length squared of long roots of leg is equal to k~ 1, k~ ~ Z, k - (ki). The action of group G WZW model of level k is given by
so,k(g)-
i
i
(2.11 whenever there exists an extension of the field g to a map ~ of a three-dimensional manifold B with boundary Z, g[aB = g- Restricting eventually possible values of k [36], we may reduce the ambiguity in the second (Wess-Zumino) term of eq. (2.1), due to the freedom of choice of g, to 27riZ so that the amplitudes exp[-S~,k(g)] are well defined. If Hz(G ) ~ 0, then additionally to eq. (2.1), we should choose consistently the amplitudes of field configurations generating H2(G ). Different
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choices lead to different "0-vacua", see refs. [42] and [36] for a more systematic discussion. The WZW field theory is given formally by functional integrals
f - e - s'~.~(g)Dg.
(2.2)
where Dg stands for the local product F I ~ z d g ( ~ ) of Haar measures on G. Although formal, integrals (2.2) may be in many cases exactly computed by using the conformal and current-algebra symmetries of the WZW theory [35-37,43-46]. We shall couple group G WZW model to a gauge field A with values in the complexification o~ c of a Lie algebra J¢'. Let us assume that
.]= 1
where ~ are simple factors (abelian factors can nevertheless be easily taken into account, see sect. 5). Below, we shall denote by H - - H i Hj the simply connected group corresponding to J'~'. Field A is an imaginary one-form, A = A I ° + A °1,
A 1°= - ( A ° l ) *,
where A a° and A °1 are its dz and d£ parts and " *" is the complex conjugation in ~c leaving af' invariant (we use the convention H = ei•). Suppose that af' is realized as Lie sub-algebras ~'~L and of"R of N, i.e. embedded into N in two ways. In particular, we may have -;~L =-'z~R = ~ with the canonical embedding, which we shall call the (left-right) symmetric case. In the asymmetric case, we shall additionally require that the pull-backs of the Killing form ( . , . ) k to ~ be equal. The pull-back form is characterized by half the length squared ksa of the long roots of the simple components af~j of ~ and we shall denote it by ( .,- )T,, k = (kg). The coupling of the gauge field A to the WZW model is described by the action i
So, k(g, A) = SG.k(g ) + 2~ f [(A~, g - 1 0 g ) k + (gOg -1, AO1)k + (gAng 1, Am)k _ (AlO, A0l)/,] ,
(2.3)
where the subscript L or R indicates the image under the imbedding of H into ~. Action (2.3) is conformal invariant (only the complex structure of 2J is used in its definition). It has an important covariance property under the chiral gauge transformations. Let h: ~ ~ H c. Put
hg = hLgh~
(2.4)
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and (hA)01 = hAmh -~ + h -Oh i
(2.5)
Then (as follows from the so-called "Polyakov-Wiegmann" formula for the action S, see appendix B)
SG,k ( g, A) = Sc,,k (hg,hA ) -- Sm f,( hh*,hA )
(2.6)
where the action SH, f, is gauged with respect to the identity imbedding of o~. Notice that in the special case when h is H-valued, eq. (2.6) implies the gauge invariance of the action since the second term on the right-hand side of (2.6) expressing the explicit chiral anomaly, vanishes then. We shall consider the G / H theory formally given by the functional integral
f- e x p [ - S G , k ( g ,
A)] D g D A .
(2.7)
We shall call it, somewhat abusively, the G / H WZW model. DA in (2.7) stands for the flat "measure" on the space of a'f'C-valued connections equipped with the L 2 scalar product. Notice that, since A enters the action at most quadratically, the A integration is gaussian. It fixes A to its classical value l o _- (1 - EL AdgER~-1"] ELg t 3g- 1 Acl AcC~= (1 - E~ A d g - 1 E L ) - I E ~ g
l~g,
where EL, E R are the inclusions of ~ c into Nc and E L, t E~ their transpositions with respect to the Killing forms (.,-)T, and ( . , - ) k - The integral (2.7) becomes formally
f-exp[-S~,mk(g)lI-ldet(l-E[Adg(~)ER) l d g ( ~ )
(2.8)
where SG, H, k is like in (2.1) but with the first term on the right replaced by
i
1 + ERE ~ Adg
£< 1 -ERE
1
Og,g-l g>k.
Integral (2.8) corresponds to another G-valued sigma model with a different metric on G and unchanged topological term. That it can be solved exactly is by no means
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clear from the representation (2.8). Notice that this is in contrast to the situation discussed in refs. [29, 31, 32], where the coupling to the gauge field is of the form f A ~ J ~ and the A-integral freezes the H-parts of the currents Jg to zero. To compute integrals of type (2.7), we shall choose another, four-step strategy.
2.2. C H I R A L R E P R E S E N T A T I O N OF A
The first step consists of a convenient parametrization of the space of the gauge fields A. Let us first gather our conventions on the Lie algebras. We denote the Cartan subgroup of H by T and the corresponding Cartan subalgebra of off by Y. The roots of off (lying in the dual space of Y ) will be denoted by or, the corresponding step operators in offc by e~ and the coroots by a v, a v~ y . offc is spanned by the e~'s and the elements t 1. . . . . t r forming the basis of J-. We have
[ti, e~] = e~, e* = e_a,
[e,, e_~] = a v ,
ti* = t i .
The root lattice of off is denoted by Q, its coroot lattice by Q v, its Weyl group by W. Now consider the space of smooth connections A. The group of chiral gauge transformations h: 2 ~ H c acts on this space by (2.5). We shall decompose the integration over A into integrals over the orbits of this action and over the quotient space: the moduli space of holomorphic He-bundles [47]. For this a convenient parametrization of the latter space will be needed. For A from an open dense set, we may find a flat connection in the chiral gauge orbit of A [47-49] i.e. such an h: 27 ~ H c that hA = A is flat. Consider the parallel transport for Y(~) = Pexp f~i-~ which is a map from the universal covering space of 27 to H. Under the action of the fundamental group ~q(27) =
where 7q(27) ~ p ~ q ( p ) ~ H is the holonomy of A. Notice that A 01 = ( ~ / h ) - 1 0 ( ' y h )
and this representation is unique up to the left multiplication of 3'h by a constant element in H c. The conjugacy classes of the holonomy representations ~ of ~q(2) may be used to parametrize the moduli of connections A.
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Let us concentrate on the case of torus T ( r ) -= C / ( Z + rZ), r = r~ + ir2, r 2 > O. Here the holonomy ? is given by commuting elements ?a, "~2~ H such that 7 ( z + 1) = ?aT(Z), 7 ( z + T) = 727(z). By conjugation, we may achieve that ?, ~ T, the Cartan subgroup: 5'1 = e
2,~i,~,
72 = e -
2~'iO,
(2.9)
~, 0 ~ Y.
q~ and @ are determined by "Yi modulo the coroot lattice Q v. ?i may be still conjugated (simultaneously) by a Weyl group element. To fix this freedom we should pick one q~ from each orbit of the affine Weyl group (generated by W and Q v) and one {9 in each orbit of Q v. Restricting the considered set of A's to an even smaller open dense set, we shall take q) from the open Weyl alcove cg0 c 3-- and O from U ~ wWCg0. We remind that
U
(W~o+~)=°.Y - and
(w%+~)n%.
wEW,~Q v
only if w = 1 and ~ = 0. Choosing
y,o(z)
=exp
,
we shall parametrize
A 01= (Tq,oh) lo(Tq~oh ).
(2.10)
If h and h' correspond through (2.10) to the same A then 7~oh' = hoT, oh,
where h 0 ~ H c. In particular h o commutes with "~i- In appendix A we show that necessarily h o ~ T c (this could be not the case if we admitted ~b and O from the b o u n d a r y of (go). Consequently, the ambiguity of h in (2.10) is the left multiplication by T c. In what follows, it will be more convenient to use a complex modular parameter u = O - r • and to introduce yu(z) = y , o e2~ize = e-{~r/,2)u(:-z).
(2.11)
A m = (yuh) -1 0 (Tuh)
(2.12)
Still
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632
with the difference that y,h is periodic in one direction and has a TC-twist in the other. Relation (2.12) (as well as (2.10)) parametrizes an open dense set of connections by multivalued chiral gauge transformations. Formal use of the transformation property (2.6) would give
s o . k ( g , A) = So,k( uLhLgh
) --
(2.13)
In appendix B, we show that the definition (2.1) of the WZW action can be naturally extended to twisted field configurations and that (2.13) does hold indeed. Thus the toroidal partition function of the gauged WZW model can be written as
Z~/H( r) - f e- s, .k
Dg DA
= f e x p [ - Sc, k (y~L hLgh~Y~*R)+ SH, T,(yuhh*y*)] Dg DA
= fexp[--S~,k('y,,Lg'y~*R)+ SH,~(Yuhh*y*)]DgDA,
(2.14)
where in the last step we have used the invariance
f F(glgg ~ ) d g = fc,F(g)dg of the Haar integrals of functions analytic on G c under the action of gl, g2 ~ G c, generalized formally to infinite dimensions.
2.3. CHANGE OF VARIABLESTO HC/H The next step in the computation of (2.14) is to make a change of variables from A to u and h and, eventually, to reduce the H c variables h to H C / H as the right-hand side of (2.14) involves only hh*. To get rid of the residual ambiguity of h, we shall fix h (0) in a suitable way. For this, and for a later purpose, let us discuss parameters in the homogeneous space H C / H . We may uniquely decompose an h ~ H c as
h = bU with U ~ H and
exp[ voeo]e.J2 neJ2
(2.15)
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where ¢p ~ 3- and v, e C. Eqs. (2.15, 2.16) are the generalization of the upper-triangular times unitary decomposition in GI(N, C). We fix the freedom of h (0) by fixing ¢p(0) and the phases of v~ for a simple roots. Since under h ~ e2=%, t e j - c , q0 ~ cp - 4~r Im t and v~ ~ eZ~i(t"~)v~, it is easy to see that this fixes h(0) (and h) up to the multiplication by the center (= e 2~iQ*,Q* being the dual lattice to Q). Let us first introduce into (2.14) the dummy variables % ~ J - and ~, ~ [0,1] for a simple roots:
ZG/H(r) = f ( exp[- SG. k(7.LgT,,~.) + Sn, ~(7.hh*"/* )])
8 ( % ) Dg d % dq~ DA. (2.17)
Now, we would like to change the variables from (%, +, A) to (¢b, {9, h), with % becoming ~0(0) and 2~rq, the phase of c~(0). The computation of the Jacobian
j(u, h) d % d + DA
=j(u, h ) d ~ d O D h
(2.18)
(Dh = I ] ~ d h ( ~ ) , d h being the left invariant measure on H c) is standard and has been done in detail in appendix C. We have not kept track there of various G or H dependent constants which may be absorbed into the normalization of the invariant volume elements in (2.17). Denoting such a constant by C, we have
det'(D~DA) j(h, u) = Cr~ det(ei, ej)det(ai, aj)
(2.19)
where DA = ~ + AdA01 is the operator on JgC-valued functions and the denominator contains the L 2 scalar products of ei = Ad e ,t, and a i = ~* d£ spanning respectively kerD A and kerD ]. r is the rank of H. The h-dependence of (2.19) is given by the chiral anomaly [50, 51]: det'(DA*Da ) det'(DA*Da) det(e,, ej)det(ai, aj) = det(ei, ej)det(ai,
h=l
aj)
exp[SH'2hv(hh*'A ] h = l ) ]
'
where h v_= (h j ) is the vector of the dual Coxeter numbers of the simple components of Jt ° and the determinants are ~'-function regularized. Since for A, = Alh_l, A°u1 = ")tu-1 0 ~u,
(2.20)
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634
we get by using again the twisted version of (2.6): det'(D~DA) det(e i, ej)det(a,, a j)
= Cr22~det'(L)*aS)&)exp[SH,2h~(y, hh*y* ) - Sn,2h~(g,g* )]. (2.21)
rz 2~ has been contributed by (det(ei, ej)det(ai, a j)) 1 at h = 1. Combining (2.17, 2.19) and (2.21), we obtain
Z o / n ( r ) = Cr2 ~f Z c . k ( ' r , u ) ~ m ~ ( r, u ) F ( u ) d e b d O
(2.22)
where
ZG, k(', ") ----fe &;,k(y~Lgy*R)Dg
(2.23)
is the group G level k WZW partition function with left-right twists e 2~iuL and e 2~iuR in the r-direction,
~ . ~ ( , , u) = fe*-.~'~o~*~e~a(m(o))Dh
(2.24)
is the twisted partition function of a WZW-type H C / H sigma model at level ~: = ~: + 2h v
(2.25)
F ( u ) = det'(Da* D&) e s,,.2,,v(v,vu*)
(2.26)
and
is a ghost-type contribution. The group G level k WZW model can be quantized [35] in a Hilbert space carrying representation of the pair of K a c - M o o d y algebras J,,(x), ]n(x), x ~ fgc,
[J.(x), J~(y)] =L+m([x, y]) + (x, y)~8.+.,.o
(2.27)
and the same for Jn - ' s. The conformal symmetries of the model are realized by the Sugawara construction Virasoro algebras:
L n = ~ Y~. K~b:J,,(x~)J,,_,,(xb): rtt, a , b
(2.28)
K. Gawgdzki, A. Kupiainen / Coset construction
and the same for L, with J ~ J
635
where (x~) is a basis of ~ and
(K-X)ab = {xa, Xb)k+gv.
(2.29)
g v = (gV) is the vector of the dual Coxeter numbers for ~. The Wick ordering puts annihilators arm, m > 0, to the right of creators arm,m < 0. The central charge of the Virasoro algebra is
CG,k
k i dim Ni k dim f¢ = E -ki + - gi v - -k + - g. V
(2.30)
i
The Hilbert space of states is obtained by applying the creators Jm, J,,, m < 0, to build the descendants of the primary states anihilated by Jm, f~, m > 0. For simply connected G, the space of primary states (constant modes of the model) can be identified with a subspace of L2(G, dg) on which J0 and ,~ act as left and right regular representations of fcc. Under this action, L2(G, dg) may be decomposed into the sum of irreducible highest weight representations labeled by left-right highest weights A L = A R. Only a finite (k-dependent) number of the so-called integrable weights [52, 35, 53, 36] appear in the subspace of primary states. The irreducible subspaces of the primary vectors together with their descendants build irreducible representations of the Kac-Moody algebras labeled by highest left-right weights A L, A R. These representations may be integrated to the K a c - M o o d y groups [53] (hence the name of the corresponding weights). In the operator language, the twisted partition function is equal to
Zc,,k('r, u) = ( q~)-c~"JZaTree~iJo("L)qCoe2"iYo(",)?t z°
(2.31)
f~ a( '7", UL,R) of the which becomes a sesquilinear combination of the characters Xk, integrable highest weight representations of the K a c - M o o d y algebra. These characters may be computed by a careful analysis of the null descendant vectors [52, 53].
Zo,k(~', u) = (q{t) -c°'jz4
E
NALARX~k,AL(~, ~/L)Xk~,AR(T, ~/R)"
(2.32)
AL,AR
For simply connected G, NALAR = ~A1AR. In general, the "mass matrices" N are symmetric, positive matrices of non-negative integers, N0o = 1. They may be computed by a careful analysis of the space of primary states [36].
2.4. COMPUTATION OF THE H C / H PARTITION FUNCTION
Under decomposition (2.15), H c -= H C / H × H and the left-invariant measure dh on H c becomes db dU where db is the ~C-invariant measure on H C / H and dU is
K. Gawgdzki, A. Kupiainen / Coset construction
636
the Haar measure on H. The H C / H model is formally given by functional integrals
f - e s",~(bb*) db
(2.33)
(with Db = FI~ db(~)). Notice that in (2.33) with H c invariant insertions, the global H c symmetry of the action should be fixed leading to the factorization of the (infinite) volume of HC/H. Although the action in (2.33) contains the Wess-Zumino term, it can be globally written as a local integral since H C / H is topologically trivial, see below. In particular, it can be defined for any ~: (no quantization of the level!). Classical stability requires that the components kj of k be positive (this is clear for the first term of the action S/~,k; it can be shown that the inclusion of the Wess-Zumino term, now real, does not change the result. The quantization leads to strengthening of the stability requirement to lc > h v component-wise, see below. Although stable, the H C / H model is not unitary. Unlike the compact group WZW model, it lacks physical positivity. Similarly to the compact case however, it has the conformal and K a c - M o o d y symmetry. It may be quantized in an indefinite-metric space of states carrying a representation of the pair of Kac Moody algebras
c,
[L(x),
y]) - (x,
(2.34)
(the same for Jn - ' s). Notice the minus sign in the second term. The Virasoro algebra is now
Ln=-½
Y'. Kca:J,,(xc)J, m(xa):
(2.35)
m,c,d
and similarly for Ln with x c spanning J~ and
(K 1), a = (xc, xa)?_hv
(2.36)
(again notice the sign). Wick ordering is as in (2.28) and the central charge becomes kj dim @
k dim ,)g -
(2.37)
As before the space of states may be generated by applying the creators
J,,(x), J,(x), n < 0, to primary states annihilated by Jn(x), J~(x), n > 0. The space of primary states can be identified with the space of constant modes L2(HC/H, db).
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637
In this identification,
(Jo(x)f)(b)= 71 ~d
~:o f ( e
i~b)
and = J0(x*)
.
Lo = L 0 on the space of zero modes and is proportional to the laplacian on H C / H with a continuous, bounded below spectrum. LZ(HC/H,db) may be decomposed into a direct integral of irreducible (principal series) representations of H c which leads to the decomposition of the space of states into a direct integral of the irreducible representations of the K a c - M o o d y algebras. Unlike in the compact case the integrability of the algebras to the K a c - M o o d y group does not lead to the elimination of states from L2(HC/H, db). Also, for almost all irreducible components of the latter space, the towers of the decendent states built over them do not contain null states. The last property allows to immediately reduce the computation of the toroidal partition functions of the theory to the pure constant-mode problem, i.e. to the quantum mechanics on H C / H . We shall not however follow this strategy, pursuing instead a direct functional integral approach (see nevertheless the comments below, after formula (2.53)). Upon factoring out of the DU integral, the twisted toroidal partition function (2.24) becomes
zf.,x(.,
.) :
Db
(2.38)
(the 6-function fixes the residual global symmetry). We shall see that in the variables (2.16), expression (2.38) becomes an iterative gaussian integral. First, we need a formula for the left-invariant measure on HC/H in terms of the parameters (2.16). For this purpose, it is convenient to introduce variables % via
G=e(*'"5/2w,,.
(2.39)
db = dq~ ] 7 dw~dw~ - dqodw.
(2.40)
Then ~>0
Indeed, (2.40) is given by a maximal degree differential form on H C / H , as is the invariant measure. Since the subgroup (2.16) of H c acts transitively on HC/H, it is enough to check that (2.40) is invariant under it. First, for ~b~.Y-,
e¢/2b=exp[ ~[] e(¢'~')/2u,~e~,]e (w+¢)/2 L~>O
]
638
K. Gawgdzki, A. Kupiainen /
i.e., by (2.39), c p ~ q o + t ) , w e ~ we which exp[)2, > oU,ea], we get b o b = exp[ ~
Coset ~vnstruction
leaves
(2.40)
invariant.
For
bo =
v'eele ~/2
La>O
J
with v" a p o l y n o m i a l in ue, , ve, such that F~c~i = c~. Order now the positive roots as
cq, a 2 .... , a d
(2,41)
n i 1Ol~/where a~/are simple. Then in the following way. We m a y uniquely write a~ = 52/= in (2.41) we require that n~ >/nj for i >~j. N o w
v" = v,, + u~, + ge~ with ge, depending only on ve, with j < i. Thus Ova,lOve, is a triangular matrix with ones on the diagonal. The invariance of (2.40) follows. Thus we need to study ~eH,i (~', U) = f 6 ( r p (0)) e s".~(Y-bb'Y*, Dqp D w .
(2.42)
First write (denoting - (¢r/'r2)u( z - ~) = X)
Yubb*y * = eXn eq)n* e x* = n x e ~ n x with
n × = e × n e - X = e x p [ ~ e' ( Xe' e > v ~o% ] q~x=eP+X+X*. T o c o m p u t e SH,7,('y, bb*y* ), we shall use the " P o l y a k o v - W i e g m a n n formula", p r o v e d in a p p e n d i x B for twisted fields. It states that if h I and h 2 are HC-valued fields periodic under z ~ z + 1 and satisfying
hl(z+¢)=ahl(z)b,
h2(z+~)=b
'h2(z)c
then
S( hlh2) = S( hi) + S( h2) - r ( h~, h2) with
r ( h l , h2) =
i
f
COSET CONSTRUCTION FROM FUNCTIONAL INTEGRALS K. GAWt~DZKI
CNRS, IHES, 91440 Bures-sur-Yuette, France A. KUPIAINEN
Research Institute for Theoretical Physics, Helsinki University, 00170 Helsinki, Finland Received 26 September 1988
A detailed study of the gauged Wess-Zumino-Witten models is presented. These models are shown to be conformal field theories realizing the G o d d a r d - K e n t Olive coset construction. Partition functions are computed for an arbitrary group G with a subgroup H gauged. Correlation functions are shown to be computable in terms of WZW ones. Explicit cases of the minimal models and parafermionic theories are worked out.
1. Introduction The coset construction was first introduced [2,3] as the method to obtain new representations of Virasoro algebras by decomposing the representations of the group G K a c - M o o d y algebra into factor representations of K a c - M o o d y algebra based on a subgroup H c G. The factors carry the action of the Virasoro algebra given by the difference of two Sugawara representations corresponding to G and H. This way all the unitary representations of the Virasoro algebras with the central charge c < 1 as well as those of super-Virasoro algebras for c < ~ were obtained by taking G = SU(2) × SU(2) and H = diagonal SU(2). In ref. [4] it was realized that the coset construction allows relating modular invariant combinations of SU(2) K a c - M o o d y characters and those of unitary Virasoro characters. This established a correspondence between toroidal partition functions of SU(2) W e s s - Z u m i n o - W i t t e n models and of the unitary minimal conformal theories [5-7]. Since similar relations were also discovered in the monodromies of chiral fourpoint functions of the models [8,9], it was natural to expect that the coset construction can be extended to the level of complete conformal field theories. In ref. [10] an attempt was made to associate to a general pair G ~ H a coset conformal field theory whose relation to the WZW models based on groups G and H was studied algebraically. Several discrete series of conformal field theories [11-26] 0550-3213/89/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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constitute examples of such coset models and the method seems to provide a quite general tool to construct rational conformal field theories. It was suggested quite early [2] that on the lagrangian level the coset construction should correspond to a coupling of the group G WZW model to group H gauge field without the tr F 2 term so that the integration over the gauge field would freeze the H degrees of freedom. This is the procedure whose variant works for symmetric space G / H asymptotically free models without the Wess-Zumino term [27,28]. In the conformal-invariant context however, this idea was making its way rather slowly. Various couplings to (and without) the gauge field were considered [29-32] but the arguments advanced did not seem very convincing. In the present paper, we attempt to clarify the situation by a detailed study of the functional integral for the WZW theory coupled to group H gauge field (for a preliminary announcement, see ref. [1]). Our analysis, although not strictly rigorous in the handling of infinite-dimensional integrals, proves nevertheless very instructive and allows a deeper understanding of the coset construction. It may be viewed as a more systematic version of the idea employed in ref. [31] to compute the central charge of the fermions coupled to a gauge field without the kinetic term. By the change of variables which parametrizes the gauge fields by chiral gauge transformations and moduli of holomorphic H C-bundles, we transform the functional integral of the gauged WZW model into an (almost) factorized form. The factors correspond to (1) a (non-gauged) group G WZW model, (2) a WZW-type sigma model with fields taking values in the non-compact coset space H C / H , (3) anti-commuting ghosts transforming in the adjoint representation of H c. The different factors are coupled only by the moduli parameters entering through twisted boundary conditions. The non-compact WZW model is a (non-unitary) theory exactly soluble to the same extent as the compact (unitary) WZW model. Its toroidal partition functions can be even computed by gaussian integration, as in the case of the models with values in the hyperbolic spaces considered in ref. [34]. Alternatively they can be deduced from the K a c - M o o d y symmetry content of the model, as for the compact case [35, 36]. The correlation functions of the non-compact model obey (the analytically continued version of) the KnizhnikZamolodchikov equations [37] arising in the compact case and can be solved in the same special cases. As a result, the gauged WZW model can be in principle solved exactly, similarly to the non-gauged model. The paper is organized as follows. Sect. 2, the longest, illustrates the above ideas on the example of toroidal partition functions. Here, the calculation may be done quite effectively, the integration over the moduli parameters (twists) included. The result is the G / H toroidal partition function given as a sesquilinear combination of the G D H branching functions [38], as postulated in ref. [10]. In sect. 3 we discuss a set of planar correlation functions for the coset models which factorize into (sums
K. Gaw¢dzki, A. Kupiainen / Coset construction
627
of) products of compact and non-compact WZW correlations. Sect. 4 discusses the specific example of the coset construction: the minimal unitary series. In particular we show how the "non-diagonal" series of the partition functions appear and we discuss the factorization of the simplest correlation functions. Sect. 5 contains a brief discussion of parafermionic theories [15, 39, 40]. This is the simplest illustration of the general case. H is here the Cartan subgroup of G and the non-compact H C / H WZW model reduces to free fields. The paper is closed by appendices containing some technical details omitted in its main body. After the submission of the paper, we have obtained a revised version of ref. [33] which studied the canonical quantization of the gauged WZW models and ref. [76] which examined the energy-momentum tensor algebra in these models using path-integral arguments. Both papers have also concluded that the coset theories can be realized by gauging the WZW models.
2. G / H toroidal partition functions 2.1. THE GAUGED WZW MODEL Let us consider a general WZW model [35-37, 41, 42] on the Riemann surface Z with fields g taking values in the compact Lie group G (or, more generally, its complexification G C). We shall assume that
G=
t i__I~l n Gi
t/
Z
where G i are connected simple and simply connected compact groups and Z is a subgroup of the center of IJG~. On the Lie algebra ~ c of G c, ~ c = ~ ~ic, consider the Killing form ( .,. )k normalized so that half the length squared of long roots of leg is equal to k~ 1, k~ ~ Z, k - (ki). The action of group G WZW model of level k is given by
so,k(g)-
i
i
(2.11 whenever there exists an extension of the field g to a map ~ of a three-dimensional manifold B with boundary Z, g[aB = g- Restricting eventually possible values of k [36], we may reduce the ambiguity in the second (Wess-Zumino) term of eq. (2.1), due to the freedom of choice of g, to 27riZ so that the amplitudes exp[-S~,k(g)] are well defined. If Hz(G ) ~ 0, then additionally to eq. (2.1), we should choose consistently the amplitudes of field configurations generating H2(G ). Different
628
K. Gawgdzki, A. Kupiainen / Coset construction
choices lead to different "0-vacua", see refs. [42] and [36] for a more systematic discussion. The WZW field theory is given formally by functional integrals
f - e - s'~.~(g)Dg.
(2.2)
where Dg stands for the local product F I ~ z d g ( ~ ) of Haar measures on G. Although formal, integrals (2.2) may be in many cases exactly computed by using the conformal and current-algebra symmetries of the WZW theory [35-37,43-46]. We shall couple group G WZW model to a gauge field A with values in the complexification o~ c of a Lie algebra J¢'. Let us assume that
.]= 1
where ~ are simple factors (abelian factors can nevertheless be easily taken into account, see sect. 5). Below, we shall denote by H - - H i Hj the simply connected group corresponding to J'~'. Field A is an imaginary one-form, A = A I ° + A °1,
A 1°= - ( A ° l ) *,
where A a° and A °1 are its dz and d£ parts and " *" is the complex conjugation in ~c leaving af' invariant (we use the convention H = ei•). Suppose that af' is realized as Lie sub-algebras ~'~L and of"R of N, i.e. embedded into N in two ways. In particular, we may have -;~L =-'z~R = ~ with the canonical embedding, which we shall call the (left-right) symmetric case. In the asymmetric case, we shall additionally require that the pull-backs of the Killing form ( . , . ) k to ~ be equal. The pull-back form is characterized by half the length squared ksa of the long roots of the simple components af~j of ~ and we shall denote it by ( .,- )T,, k = (kg). The coupling of the gauge field A to the WZW model is described by the action i
So, k(g, A) = SG.k(g ) + 2~ f [(A~, g - 1 0 g ) k + (gOg -1, AO1)k + (gAng 1, Am)k _ (AlO, A0l)/,] ,
(2.3)
where the subscript L or R indicates the image under the imbedding of H into ~. Action (2.3) is conformal invariant (only the complex structure of 2J is used in its definition). It has an important covariance property under the chiral gauge transformations. Let h: ~ ~ H c. Put
hg = hLgh~
(2.4)
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629
and (hA)01 = hAmh -~ + h -Oh i
(2.5)
Then (as follows from the so-called "Polyakov-Wiegmann" formula for the action S, see appendix B)
SG,k ( g, A) = Sc,,k (hg,hA ) -- Sm f,( hh*,hA )
(2.6)
where the action SH, f, is gauged with respect to the identity imbedding of o~. Notice that in the special case when h is H-valued, eq. (2.6) implies the gauge invariance of the action since the second term on the right-hand side of (2.6) expressing the explicit chiral anomaly, vanishes then. We shall consider the G / H theory formally given by the functional integral
f- e x p [ - S G , k ( g ,
A)] D g D A .
(2.7)
We shall call it, somewhat abusively, the G / H WZW model. DA in (2.7) stands for the flat "measure" on the space of a'f'C-valued connections equipped with the L 2 scalar product. Notice that, since A enters the action at most quadratically, the A integration is gaussian. It fixes A to its classical value l o _- (1 - EL AdgER~-1"] ELg t 3g- 1 Acl AcC~= (1 - E~ A d g - 1 E L ) - I E ~ g
l~g,
where EL, E R are the inclusions of ~ c into Nc and E L, t E~ their transpositions with respect to the Killing forms (.,-)T, and ( . , - ) k - The integral (2.7) becomes formally
f-exp[-S~,mk(g)lI-ldet(l-E[Adg(~)ER) l d g ( ~ )
(2.8)
where SG, H, k is like in (2.1) but with the first term on the right replaced by
i
1 + ERE ~ Adg
£< 1 -ERE
1
Og,g-l g>k.
Integral (2.8) corresponds to another G-valued sigma model with a different metric on G and unchanged topological term. That it can be solved exactly is by no means
630
K. Gaw(dzki, A. Kupiainen /
Coset construction
clear from the representation (2.8). Notice that this is in contrast to the situation discussed in refs. [29, 31, 32], where the coupling to the gauge field is of the form f A ~ J ~ and the A-integral freezes the H-parts of the currents Jg to zero. To compute integrals of type (2.7), we shall choose another, four-step strategy.
2.2. C H I R A L R E P R E S E N T A T I O N OF A
The first step consists of a convenient parametrization of the space of the gauge fields A. Let us first gather our conventions on the Lie algebras. We denote the Cartan subgroup of H by T and the corresponding Cartan subalgebra of off by Y. The roots of off (lying in the dual space of Y ) will be denoted by or, the corresponding step operators in offc by e~ and the coroots by a v, a v~ y . offc is spanned by the e~'s and the elements t 1. . . . . t r forming the basis of J-. We have
[ti, e~] =
[e,, e_~] = a v ,
ti* = t i .
The root lattice of off is denoted by Q, its coroot lattice by Q v, its Weyl group by W. Now consider the space of smooth connections A. The group of chiral gauge transformations h: 2 ~ H c acts on this space by (2.5). We shall decompose the integration over A into integrals over the orbits of this action and over the quotient space: the moduli space of holomorphic He-bundles [47]. For this a convenient parametrization of the latter space will be needed. For A from an open dense set, we may find a flat connection in the chiral gauge orbit of A [47-49] i.e. such an h: 27 ~ H c that hA = A is flat. Consider the parallel transport for Y(~) = Pexp f~i-~ which is a map from the universal covering space of 27 to H. Under the action of the fundamental group ~q(27) =
where 7q(27) ~ p ~ q ( p ) ~ H is the holonomy of A. Notice that A 01 = ( ~ / h ) - 1 0 ( ' y h )
and this representation is unique up to the left multiplication of 3'h by a constant element in H c. The conjugacy classes of the holonomy representations ~ of ~q(2) may be used to parametrize the moduli of connections A.
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K. Gaw~dzki, A. Kupiainen / Coset construction
Let us concentrate on the case of torus T ( r ) -= C / ( Z + rZ), r = r~ + ir2, r 2 > O. Here the holonomy ? is given by commuting elements ?a, "~2~ H such that 7 ( z + 1) = ?aT(Z), 7 ( z + T) = 727(z). By conjugation, we may achieve that ?, ~ T, the Cartan subgroup: 5'1 = e
2,~i,~,
72 = e -
2~'iO,
(2.9)
~, 0 ~ Y.
q~ and @ are determined by "Yi modulo the coroot lattice Q v. ?i may be still conjugated (simultaneously) by a Weyl group element. To fix this freedom we should pick one q~ from each orbit of the affine Weyl group (generated by W and Q v) and one {9 in each orbit of Q v. Restricting the considered set of A's to an even smaller open dense set, we shall take q) from the open Weyl alcove cg0 c 3-- and O from U ~ wWCg0. We remind that
U
(W~o+~)=°.Y - and
(w%+~)n%.
wEW,~Q v
only if w = 1 and ~ = 0. Choosing
y,o(z)
=exp
,
we shall parametrize
A 01= (Tq,oh) lo(Tq~oh ).
(2.10)
If h and h' correspond through (2.10) to the same A then 7~oh' = hoT, oh,
where h 0 ~ H c. In particular h o commutes with "~i- In appendix A we show that necessarily h o ~ T c (this could be not the case if we admitted ~b and O from the b o u n d a r y of (go). Consequently, the ambiguity of h in (2.10) is the left multiplication by T c. In what follows, it will be more convenient to use a complex modular parameter u = O - r • and to introduce yu(z) = y , o e2~ize = e-{~r/,2)u(:-z).
(2.11)
A m = (yuh) -1 0 (Tuh)
(2.12)
Still
K. Gaw~dzki, A. Kupiainen / Coset construction
632
with the difference that y,h is periodic in one direction and has a TC-twist in the other. Relation (2.12) (as well as (2.10)) parametrizes an open dense set of connections by multivalued chiral gauge transformations. Formal use of the transformation property (2.6) would give
s o . k ( g , A) = So,k( uLhLgh
) --
(2.13)
In appendix B, we show that the definition (2.1) of the WZW action can be naturally extended to twisted field configurations and that (2.13) does hold indeed. Thus the toroidal partition function of the gauged WZW model can be written as
Z~/H( r) - f e- s, .k
Dg DA
= f e x p [ - Sc, k (y~L hLgh~Y~*R)+ SH, T,(yuhh*y*)] Dg DA
= fexp[--S~,k('y,,Lg'y~*R)+ SH,~(Yuhh*y*)]DgDA,
(2.14)
where in the last step we have used the invariance
f F(glgg ~ ) d g = fc,F(g)dg of the Haar integrals of functions analytic on G c under the action of gl, g2 ~ G c, generalized formally to infinite dimensions.
2.3. CHANGE OF VARIABLESTO HC/H The next step in the computation of (2.14) is to make a change of variables from A to u and h and, eventually, to reduce the H c variables h to H C / H as the right-hand side of (2.14) involves only hh*. To get rid of the residual ambiguity of h, we shall fix h (0) in a suitable way. For this, and for a later purpose, let us discuss parameters in the homogeneous space H C / H . We may uniquely decompose an h ~ H c as
h = bU with U ~ H and
exp[ voeo]e.J2 neJ2
(2.15)
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K. Gawgdzki, A. Kupiainen / Coset construction
where ¢p ~ 3- and v, e C. Eqs. (2.15, 2.16) are the generalization of the upper-triangular times unitary decomposition in GI(N, C). We fix the freedom of h (0) by fixing ¢p(0) and the phases of v~ for a simple roots. Since under h ~ e2=%, t e j - c , q0 ~ cp - 4~r Im t and v~ ~ eZ~i(t"~)v~, it is easy to see that this fixes h(0) (and h) up to the multiplication by the center (= e 2~iQ*,Q* being the dual lattice to Q). Let us first introduce into (2.14) the dummy variables % ~ J - and ~, ~ [0,1] for a simple roots:
ZG/H(r) = f ( exp[- SG. k(7.LgT,,~.) + Sn, ~(7.hh*"/* )])
8 ( % ) Dg d % dq~ DA. (2.17)
Now, we would like to change the variables from (%, +, A) to (¢b, {9, h), with % becoming ~0(0) and 2~rq, the phase of c~(0). The computation of the Jacobian
j(u, h) d % d + DA
=j(u, h ) d ~ d O D h
(2.18)
(Dh = I ] ~ d h ( ~ ) , d h being the left invariant measure on H c) is standard and has been done in detail in appendix C. We have not kept track there of various G or H dependent constants which may be absorbed into the normalization of the invariant volume elements in (2.17). Denoting such a constant by C, we have
det'(D~DA) j(h, u) = Cr~ det(ei, ej)det(ai, aj)
(2.19)
where DA = ~ + AdA01 is the operator on JgC-valued functions and the denominator contains the L 2 scalar products of ei = Ad e ,t, and a i = ~* d£ spanning respectively kerD A and kerD ]. r is the rank of H. The h-dependence of (2.19) is given by the chiral anomaly [50, 51]: det'(DA*Da ) det'(DA*Da) det(e,, ej)det(ai, aj) = det(ei, ej)det(ai,
h=l
aj)
exp[SH'2hv(hh*'A ] h = l ) ]
'
where h v_= (h j ) is the vector of the dual Coxeter numbers of the simple components of Jt ° and the determinants are ~'-function regularized. Since for A, = Alh_l, A°u1 = ")tu-1 0 ~u,
(2.20)
K. Gawgdzki, A. Kupiainen / Coset construction
634
we get by using again the twisted version of (2.6): det'(D~DA) det(e i, ej)det(a,, a j)
= Cr22~det'(L)*aS)&)exp[SH,2h~(y, hh*y* ) - Sn,2h~(g,g* )]. (2.21)
rz 2~ has been contributed by (det(ei, ej)det(ai, a j)) 1 at h = 1. Combining (2.17, 2.19) and (2.21), we obtain
Z o / n ( r ) = Cr2 ~f Z c . k ( ' r , u ) ~ m ~ ( r, u ) F ( u ) d e b d O
(2.22)
where
ZG, k(', ") ----fe &;,k(y~Lgy*R)Dg
(2.23)
is the group G level k WZW partition function with left-right twists e 2~iuL and e 2~iuR in the r-direction,
~ . ~ ( , , u) = fe*-.~'~o~*~e~a(m(o))Dh
(2.24)
is the twisted partition function of a WZW-type H C / H sigma model at level ~: = ~: + 2h v
(2.25)
F ( u ) = det'(Da* D&) e s,,.2,,v(v,vu*)
(2.26)
and
is a ghost-type contribution. The group G level k WZW model can be quantized [35] in a Hilbert space carrying representation of the pair of K a c - M o o d y algebras J,,(x), ]n(x), x ~ fgc,
[J.(x), J~(y)] =L+m([x, y]) + (x, y)~8.+.,.o
(2.27)
and the same for Jn - ' s. The conformal symmetries of the model are realized by the Sugawara construction Virasoro algebras:
L n = ~ Y~. K~b:J,,(x~)J,,_,,(xb): rtt, a , b
(2.28)
K. Gawgdzki, A. Kupiainen / Coset construction
and the same for L, with J ~ J
635
where (x~) is a basis of ~ and
(K-X)ab = {xa, Xb)k+gv.
(2.29)
g v = (gV) is the vector of the dual Coxeter numbers for ~. The Wick ordering puts annihilators arm, m > 0, to the right of creators arm,m < 0. The central charge of the Virasoro algebra is
CG,k
k i dim Ni k dim f¢ = E -ki + - gi v - -k + - g. V
(2.30)
i
The Hilbert space of states is obtained by applying the creators Jm, J,,, m < 0, to build the descendants of the primary states anihilated by Jm, f~, m > 0. For simply connected G, the space of primary states (constant modes of the model) can be identified with a subspace of L2(G, dg) on which J0 and ,~ act as left and right regular representations of fcc. Under this action, L2(G, dg) may be decomposed into the sum of irreducible highest weight representations labeled by left-right highest weights A L = A R. Only a finite (k-dependent) number of the so-called integrable weights [52, 35, 53, 36] appear in the subspace of primary states. The irreducible subspaces of the primary vectors together with their descendants build irreducible representations of the Kac-Moody algebras labeled by highest left-right weights A L, A R. These representations may be integrated to the K a c - M o o d y groups [53] (hence the name of the corresponding weights). In the operator language, the twisted partition function is equal to
Zc,,k('r, u) = ( q~)-c~"JZaTree~iJo("L)qCoe2"iYo(",)?t z°
(2.31)
f~ a( '7", UL,R) of the which becomes a sesquilinear combination of the characters Xk, integrable highest weight representations of the K a c - M o o d y algebra. These characters may be computed by a careful analysis of the null descendant vectors [52, 53].
Zo,k(~', u) = (q{t) -c°'jz4
E
NALARX~k,AL(~, ~/L)Xk~,AR(T, ~/R)"
(2.32)
AL,AR
For simply connected G, NALAR = ~A1AR. In general, the "mass matrices" N are symmetric, positive matrices of non-negative integers, N0o = 1. They may be computed by a careful analysis of the space of primary states [36].
2.4. COMPUTATION OF THE H C / H PARTITION FUNCTION
Under decomposition (2.15), H c -= H C / H × H and the left-invariant measure dh on H c becomes db dU where db is the ~C-invariant measure on H C / H and dU is
K. Gawgdzki, A. Kupiainen / Coset construction
636
the Haar measure on H. The H C / H model is formally given by functional integrals
f - e s",~(bb*) db
(2.33)
(with Db = FI~ db(~)). Notice that in (2.33) with H c invariant insertions, the global H c symmetry of the action should be fixed leading to the factorization of the (infinite) volume of HC/H. Although the action in (2.33) contains the Wess-Zumino term, it can be globally written as a local integral since H C / H is topologically trivial, see below. In particular, it can be defined for any ~: (no quantization of the level!). Classical stability requires that the components kj of k be positive (this is clear for the first term of the action S/~,k; it can be shown that the inclusion of the Wess-Zumino term, now real, does not change the result. The quantization leads to strengthening of the stability requirement to lc > h v component-wise, see below. Although stable, the H C / H model is not unitary. Unlike the compact group WZW model, it lacks physical positivity. Similarly to the compact case however, it has the conformal and K a c - M o o d y symmetry. It may be quantized in an indefinite-metric space of states carrying a representation of the pair of Kac Moody algebras
c,
[L(x),
y]) - (x,
(2.34)
(the same for Jn - ' s). Notice the minus sign in the second term. The Virasoro algebra is now
Ln=-½
Y'. Kca:J,,(xc)J, m(xa):
(2.35)
m,c,d
and similarly for Ln with x c spanning J~ and
(K 1), a = (xc, xa)?_hv
(2.36)
(again notice the sign). Wick ordering is as in (2.28) and the central charge becomes kj dim @
k dim ,)g -
(2.37)
As before the space of states may be generated by applying the creators
J,,(x), J,(x), n < 0, to primary states annihilated by Jn(x), J~(x), n > 0. The space of primary states can be identified with the space of constant modes L2(HC/H, db).
K. Gaw~dzki, A. Kupiamen / Coset construction
637
In this identification,
(Jo(x)f)(b)= 71 ~d
~:o f ( e
i~b)
and = J0(x*)
.
Lo = L 0 on the space of zero modes and is proportional to the laplacian on H C / H with a continuous, bounded below spectrum. LZ(HC/H,db) may be decomposed into a direct integral of irreducible (principal series) representations of H c which leads to the decomposition of the space of states into a direct integral of the irreducible representations of the K a c - M o o d y algebras. Unlike in the compact case the integrability of the algebras to the K a c - M o o d y group does not lead to the elimination of states from L2(HC/H, db). Also, for almost all irreducible components of the latter space, the towers of the decendent states built over them do not contain null states. The last property allows to immediately reduce the computation of the toroidal partition functions of the theory to the pure constant-mode problem, i.e. to the quantum mechanics on H C / H . We shall not however follow this strategy, pursuing instead a direct functional integral approach (see nevertheless the comments below, after formula (2.53)). Upon factoring out of the DU integral, the twisted toroidal partition function (2.24) becomes
zf.,x(.,
.) :
Db
(2.38)
(the 6-function fixes the residual global symmetry). We shall see that in the variables (2.16), expression (2.38) becomes an iterative gaussian integral. First, we need a formula for the left-invariant measure on HC/H in terms of the parameters (2.16). For this purpose, it is convenient to introduce variables % via
G=e(*'"5/2w,,.
(2.39)
db = dq~ ] 7 dw~dw~ - dqodw.
(2.40)
Then ~>0
Indeed, (2.40) is given by a maximal degree differential form on H C / H , as is the invariant measure. Since the subgroup (2.16) of H c acts transitively on HC/H, it is enough to check that (2.40) is invariant under it. First, for ~b~.Y-,
e¢/2b=exp[ ~[] e(¢'~')/2u,~e~,]e (w+¢)/2 L~>O
]
638
K. Gawgdzki, A. Kupiainen /
i.e., by (2.39), c p ~ q o + t ) , w e ~ we which exp[)2, > oU,ea], we get b o b = exp[ ~
Coset ~vnstruction
leaves
(2.40)
invariant.
For
bo =
v'eele ~/2
La>O
J
with v" a p o l y n o m i a l in ue, , ve, such that F~c~i = c~. Order now the positive roots as
cq, a 2 .... , a d
(2,41)
n i 1Ol~/where a~/are simple. Then in the following way. We m a y uniquely write a~ = 52/= in (2.41) we require that n~ >/nj for i >~j. N o w
v" = v,, + u~, + ge~ with ge, depending only on ve, with j < i. Thus Ova,lOve, is a triangular matrix with ones on the diagonal. The invariance of (2.40) follows. Thus we need to study ~eH,i (~', U) = f 6 ( r p (0)) e s".~(Y-bb'Y*, Dqp D w .
(2.42)
First write (denoting - (¢r/'r2)u( z - ~) = X)
Yubb*y * = eXn eq)n* e x* = n x e ~ n x with
n × = e × n e - X = e x p [ ~ e' ( Xe' e > v ~o% ] q~x=eP+X+X*. T o c o m p u t e SH,7,('y, bb*y* ), we shall use the " P o l y a k o v - W i e g m a n n formula", p r o v e d in a p p e n d i x B for twisted fields. It states that if h I and h 2 are HC-valued fields periodic under z ~ z + 1 and satisfying
hl(z+¢)=ahl(z)b,
h2(z+~)=b
'h2(z)c
then
S( hlh2) = S( hi) + S( h2) - r ( h~, h2) with
r ( h l , h2) =
i
f
-
(2.43)
K. Gawgdzki, A. Kupiainen / Coset construction
639
and S's of twisted fields defined in appendix B. We have
nx(Z+.r)=e e~(z + r)
2~i~n~(z) e2~i~,
= e-2~i"e~(z)e
2~i~*.
The repeated application of (2.43) yields
S(n xe~nx)=S(nx)+S(n*)+S(e~x)
-F(nx,e~nx)-F(e~,nx).
(2.44)
Since e~ and t~ are orthogonal, the last term in (2.44) vanishes. Also S(nx) =
S(nx)
= 0
(2.45)
and SH'~(e~x)
=
-
i -4~r -;
see appendix B. Thus, we obtain (d2z =
s.,~(v.bb*v: ) -
1
where X + X* = Expand now
2~ri¢(z - ~) was
I
(2.46)
(1/2i)d~dz)
~ f
(71(0¢px, 0cpx) ? ,
0~9))~d22
-
2~"r2llq)ll-~
Oenx,e~xnx(Oze*x ~*
lXe-~°~)id2z,
(2.47)
used.
nx 10~nx=foldsexp [ = ELeo a>O
where wd = e(~/2+x,~)w~. Developing the under-integral expression into the (finite) Taylor series around s = 0 we see that
Io= O~(e(~J2+x'°~wo)+ L(w, ~)e ~ja+X'°~ where f~ depends only on w,, (and
3~w~,)with
a' preceding a in the sequence (2.41).
K. Gaw¢dzki,A. Kupiainen / Cosetconstruction
640
Moreover, f~ is uni-valued on the torus. Now, the last term of (2.47) becomes 1 -
-
Y', e <~'">ILI2= - -
'/7" a > 0
1
~ ID~w~+~1',
(2.48)
'/7" a > 0
where b~ = e - (~/2 +x, ~5 37 e (~/2 +x, ~5 .
Notice that kerD~=cokerD~ = 0. Indeed, if f ~ k e r D ~ then f = e(~/2+x'~)f is holomorphic with f ( z + 1) =f-(z) and f ( z + r ) = e - 2 ~ i ( u " > f ( z ) . Since for our choice of u = {9 - r@, (u, a) ~ Z + rZ, these have no solutions. CokerD~ is analogous. With (2.48), eq. (2.47) becomes
SH-k(yubb*y* ) -
2-~rl /(OzCp, ¢9~cP)~ d2z - 2rrr2ll~ll 2 - 71 Z ID.Wa -{-LI 2" ~>o
That f~ depends only on earlier % and D~ is invertible enables us to perform the w-integration in (2.42) iteratively in order w~ ..... wm to get ~H,~(T,
U)=
f(azW, 3#~P)7< d2z -
cfs(w(0))exp(-2~ X I-I det(D~D,~)-XDcp
2rrr:[]~][ { ]
•
(2.49)
a>0
The computation of det(D~D,0 is again standard. By the chiral anomaly
det(D,D.): exp[-2~ f"(~'a)':(%")d2e]det(D'D.) i.-o" Now, D.I~= o is also equal to
e x p [ - ('/.,o, a)] ~ exp[('/.o, a ) ] , see (2.11), so that det D2D.[~o=0 = det 0 ~ where 0 acts of functions f satisfying f ( z + 1) = e >"(*"Of(z),
f ( z + r) = e-2~'g(°"~>f(z).
(2.50)
Thus [54,55] (q-= e 2~'i~') det(D~tD.) L=o = qq;~"5'/2 + 1/12]e~ri(u,a) __ e-'i<" .>[2 x ~=:fi( 1 - e 2 " i ( " " ) q " ) ( 1 - e
2rri(u,a)qn)2.
(2.51)
K. Gawgdzki, A. Kupiainen / Coset construction
641
The remaining q0 integral in (2.49) is also gaussian,
[ Jt (o~w,o~w)~- Z
fs(w(O))exp - ~g
t]
Oz(w,~)o~(w, ~) d~z DW
ct>0
=
fs(w(0))exp[- ~1 f ( Ozw,O~w)~ h~d2z] Dq~
= C~.2r/2det,(0,~) x/2 =
C.r~/2(qgt)
~/24 n=lI~(1
--
qn)
-2r
(2.52)
where we have used the relation
Y' (t,@(t',C~)=~t,t')h~
for
t,t', ~.~-c
or>0
and 0 above operated on periodic functions. Gathering (2.49, 2.51 and 2.52) we obtain finally
,~H,~('r, U) = C'r2r/2(qgl)llq't12-h~/2-dimH/24
I-I Ie'<~''~> - e-"~<~'~>l-2 ~>0
X J~I=1 (1-q")~J~I>o(1-e2"i(~'")q")(1-e-2~il"'~)q")
2.
(2.53)
It is instructive for interpret (2.53) in the operator language in which
&L~'mT,('r,u) = ( q~)-c"~/~'~/24Tre2~iJ°(")qL°e2~iJo("*)~Zo
(2.54)
(in fact Tr in (2.54) should factor out the infinite volume of the group T c of residual symmetries). The infinite product in (2.53) is the contribution of the descendants of the primary states of the K a c - M o o d y symmetry which (unlike in the compact case) simply factors out. The rest of the expression is the constant-mode contribution given by the heat-kernel integral
(qq)-C"c'"~/2"fe""2~ "V(e 2=i"b,b)8(W) db where AT, h v denotes the laplacian of H C / H corresponding to the HC-invariant metric generated by the Killing form ( .,. )? hr. It will be convenient to rewrite (2.53) using the (Kac-Moody) affine roots c~ = (c~, n) where c~ is a root of .,vt°or zero and n ~ Z, ~ ~ 0. Recall that ff > 0 means n > 0 or n = 0 and c~ > 0. The affine roots (0, n) have multiplicity r. Using the
K. Gaw~dzki, A. Kupiainen / Coset construction
642
notation ea( q, e2~rit) = q, e2~ri(t,a) for t ~,Y-c, we obtain from (2.53) ~ " H , 7 ~ ( ' r , U ) = C'r2r/2(qq)llq'll~+hv/2-(cb'o)--dimH/2411~(,r,U)[
2
(2.55)
1 where O = ~E~ > oa is the Weyl vector,
n ( ~ ' , u ) = 1-I ( 1 - ea(q,e:"i")) 5>0
and we have also used (2.25). 2.5. T W I S T I N T E G R A L
Coming back to (2.22), we compute first SH,2h ~(7~'*) = - 2rrT21lOIIa2h v see (2.46). For det'(DJ D&) note that satisfying
DJuDAu is just the laplacian on functions f
f ( z + l)=e-2~i~f(z)e 2~i~,
f(z +,:)=e-2~iOf(z)e 2~i°,
i.e. expanding f = E~f~e~ + S,ifit i, f~ satisfy (2.50) whereas fi are periodic. Thus (compare (2.51) and (2.52)) d e t ' ( D ] D & ) = C~-22~(qq) IIq~ll~~+ 2(q~'°) + dimH/121/][( 'F, U )14 and (22) becomes
zo/.(':) = C':;/2f d~dO (qFt)'(*)ZG,k('c, u)lFl('r, u)12
(2.56)
c ( V ) = ' ~ll~ll~+h 2 v + (~, o) + d i m H / 2 4 .
(2.57)
where
Before we insert into (2.56) the expression (2.32) for Zo, k(~-, u), we shall rewrite the f¢ A( T , UL,R) in terms of the characters X ,K# characters Xk, L x (T, u ) of the group H K a c - M o o d y algebra: X k~t, A ( T , U L , R ) = z_.,VbL'R['r)X-~k,X('r,A,X\ U) . X
(2.58)
K. Gawgdzki,A. Kupiainen / Coset construction
643
Eq. (2.58) is obtained [38] by the decomposition of level k highest weight A representation of the group G Kac-Moody algebra into the highest weight representations of the group H Kac-Moody algebra embedded in the previous one by means of EL, R: jgc,,._.),~. X in (2.58) runs through the highest weights of ~ integrable at level ,~. These are the weights such that the image t(~) of ~ - ?~+ p under the identification of the dual space to ~ with de~ induced by the Killing form ( . , . ) k + h ~ lies in the Weyl alcove cg0. The coefficients bA,x(r) are called the (G ~ H) branching functions. We shall also use the Weyl-Kac formula [53] for the characters X~: X~,A(r,u) =
Y'~ E (--1)'(W)q a(x'*) w~W ~cQ ~
×exp[--2~ri(u,w(t(X) + ~)--t(O))i+h~]l-I(r,u) -1, (2.59) where l(w) is the length of w ( ~ of elementary reflections) and
d(x,
=
+ llL v
Inserting (2.32), (2.58) and (2.59) into (2.57), we obtain Z G / H ( ' r ) = C'r~/2(qq) (dimH-cG'k)/24 E E NALARbLL,XL(T)bRR, XR('r)mXL,Xe.("r)' AL, AR ~L, ~R
where
mxcX.=
E
fd~dO(--1)'(Wc)+'(w"~(qq)llellL*~/2qd(X~'~L~gl
~
a(x~'*~
WL'WR~W (L,~R~Q v
x exp[-2~ri(u, WL(t(XL)}- ~L))~.+hVq- 2rri(u*, WR(t(XR)+ ~R))k+hv]. (2.60) We shall first integrate over O. If t* is a weight then
dOe2rri(°'~') = CSu, 0 .
fuw~ww%
Since (0, i~)= (0, t(l~))~+hv and each ~ ~ Q v is of the form t(/~) for some weight #, the 0 integral in (2.60) will impose WL(t(XL) q- ~ L ) = WR(/'(XR) -I- ~R)"
(2.61)
644
K. Gawcdzkt A. Kupiainen / Coset construction
But L(~.L), L(XR) ~ cgo and w(Cg0+ ~) C3 cg0 :# ~ if and only if w = 1 and ( = 0. Hence (2.61) implies that wL = WR, & = & , )t L = )t a and m)tLX R = CaXLX R
= Ca~kL~kR
2 . , ~ W , ~=QV
E .,~W,IEQ v
f%dq) (qq),~il},,~/2+ll,(Y,~.)*aiL~/2
It,{o}ll~-+,~/2+(.*..x)~ ;)£,,,
.
f%dq}(qFl)pl.~+;+,{;,}llL,~/2
,,(o~lr},,,~/2.
The sum over w and ~ restores 3- as the full domain of integration for q). Since by the Freudenthal strange formula [56]
II (o)
II +,v =
v 1 hV d i m ~ E hj dim S j = - 12 ~ ' + h v ' / k/+ h/
we finally get mXLXR
=
C~XL)~RT2r/2
and
ZG/H(r)=C(qFI)
(~.{;., cH.~)/24 E
N,I,ARb,LI,x(T)b~RR,x(T)
•
(2.62)
AL. "~R. )t
The power of (q~) 1/24 gives the central charge CG,H = CG, * -- CH, ~ of the theory which is that of the coset construction [2, 3]. Notice also [31] that CG/H = CG, k -Jr-CHC/H" ~ q- Cghost
(2.63)
where CHC/H,~ is given by (2.37) and Cgho~t= -- 2 dim o~ is the central charge of the free anticommuting ghosts transforming in the adjoint representation of H c. Eq. (2.63) corresponds to the decomposition (2.22) of the theory into three factors: the compact WZW model, the H C / H model and the free ghosts. The partition functions (2.62) are modular invariant ZG/H(q')
=
ZG/H(T q- 1) = Z G / H ( - l / r ) .
This follows from the modular invariance of the partition functions Zo, a (r, u) as in ref. [32].
K. Gawgdzki, A. Kupiainen / Coset construction
645
3. Planar correlation functions The same strategy as applied in previous section for the toroidal partition functions may be used to study the correlation functions of the gauged WZW model. Below, we shall limit ourselves to the planar case. Let us first consider for the symmetric model (with Jf~--Jt'~L=YRC ~ ) the simplest correlations given formally as N
f,I~I_l (Retr
ga,,(z,,))e S'~,~(g'A'Dg DA = E,
(3.1)
where gA, stands for the matrix of g in the irreducible representation of G c of (integrable) highest weight A , and the functional integral is over fields g: P C ~ = C u {oc} ---, G and group H connections A on P C ~. Notice that the integrand of (1) has the local H-symmetry. On the Riemann sphere, the gauge fields A with A m = h -1 0h, h: P C 1 ~ H c, form already an open dense set so that the situation is much simpler than on the torus. Moreover the ambiguity in h may be removed if we fix h(0). Multiplying (3.1) by the dummy integral f S ( b o ) d b odUo = 1, where bo ~ H C / H , Uo ~ H, and changing variables from (b 0, Uo, A) to h = bo = b(0), Uo = U(0), we obtain immediately
E=/ ]-I
bU with
N
n
=
(RetrgA,,(z.))e
S(~'~(hgh*)+S"~(hh*)J(h)6(b(O)) Dh
(3.2)
1
where the jacobian is
j(h) = C
det'(DA*~4 )
(3.3)
det( ec, ea)
Here ~c = Adh 'xc, with x c spanning ~ , form a basis of kerD A and cokerD A = 0, compare appendix C. As in (2.22)
det'(D~DA) - C(Area) dim~'det'(0~)e s".2'v'hh*'. det(ec,
%)
Inserting (3.3, 3.4) into (3.2) and changing variables
(3.4)
g ~ h lgh*- 1, we obtain N
E=C(Area)
dim'A~'det'(0*0)
Y'~
2 - u f I--[ tr(g(bb*)
a,,= +1
×
,','
Dg Db
')~£',(z,,)
n=l
(3.5)
K. Gawgdzki, A. Kupiainen / Coset construction
646
i.e. an essentially factorized expression. Hence the computation of the correlation function (3.1) reduces to the computation of the correlations separately in the group G W Z W model and in the non-compact H C / H theory. The compact WZW model correlations satisfy the Knizhnik-Zamolodchikov equations [37] obtained from mixed conformal and current-algebra Ward identities and the Sugawara relation
0 OZm Here
Kab
N
1
~_, Kab(X~)m(Xb).-- f n@_lg~:,(z,)e-S°.k(g)Dg=O.
a,b, nq:m
Zm-- Zn
(3.6)
is as in (1.29) and d e=0 (ei~X"g(Z))A x~gn(z)°= 71 d----~
are generators of the left global G c symmetry, with subscripts m or n indicating on which field they act. Similarly,
[ 3
OZm
~2
a,b, n4~rn
K<~b( "7<') " (Xb) " .~m-1
Zn
If I~) g~"(z,)e s'~.k(g)Dg=O,
(3.7)
n=l
where _ a 1 d xaga(z) - i de
~=0
(g(z)
e-i~x°)~
generate right G c. The solutions of (3.6) and (3.7) are of the form
Z EaGp(z)Xpp'ED'(z) , p,p'
(3.8)
where a and fl label the global G c invariants made of tensor product of A n-representations and p, p' run through the set of independent solutions of (3.6) [37, 57, 8]. (X ep') is a hermitian matrix. The monodromy of the " W Z W blocks" E ~ ( z ) is given
K. Gaw~dzki, A. Kupiainen / Cosetconstruction
647
by the braid group representations arising as the holonomy of the connection
Y'.
a,b,m~n
dz m
K,,b(X~)m(Xb)n--
Zm-- Zn
,
see ref. [58]. In several cases (for G = SU(2) or G = SU(N) and fields in the fundamental representation) [37,59,57], one can obtain expressions (3.8) for the four point functions in a closed form. The H C / H correlations N
f ®
(3.9)
e s",~(bb*, D b
n=l
are similar. A slight complication is the presence of the delta function in (3.9) which breaks the global H c × H c covariance. This is however not a problem if we notice that
f ~n tr( g( bb* ) -l X"° " Xe-Sa .k(g) Dg )a°tz.) = fl-~I tr(g(Yobb*yd)
1 °n
)A,(z,)e-S(;'k'g)DbdYodYd
where 7o, Y(~~ H. The integration over 7o, 7d will select only H C × H c invariant combinations of correlations (3.9). For such combinations, the conformal and current-algebra Ward identities may be obtained exactly as for the compact case, leading to the KZ equations [~z0m
+
1
E
c,d,n~rn
Zm-- Zn
If
®N n=l
°" )Ao(Z.)
× 8(b(0)) e s".~(bb') Db dTodY~ = 0.
(3.10)
K~.d is as in eq. (1.36) and
xc(bb , o)A(z) : 71 -~e ~=o(ei~xcbb*(z))°a" An analogous equation holds for 3/35. Again, the solutions are of the form
E gaHqC/H(z)Yqq'g~qC/H(z) •
(3.11)
q,q'
which can be made explicit for the four point functions in the same cases as for the
648
K. Gaw~dzki, A. Kupiainen / Coset construction
compact model. In fact the solutions are given by the same expressions [37,57] analytically continued to negative levels (notice the sign difference between (3.6) and (3.10)). In the special case where we take G = H, k = lc, we infer from the KZ equations that N
f,I]I__,
tr(h(bb*)-~)a°,(z,)6(b(0))e-S-.~(h)+s.,*+~(h6*'Dh Db = const.
More exactly this holds separately for the WZW blocks:
E M~BEH(z)E~C/H(z) = Qpq
=
const.,
(3.12)
a,fl
where M~;3 gives the scalar product in the subspace of H c invariants. Relation (3.12) may be sometimes inverted to express the level k H C / H correlations by the ones of group H level ~: ones. In general however, the matrices ( E ~ ( z ) ) are not square. Eq. (3.12), in conjunction with (3.5), provides a precise version of the relation, postulated in ref. [10], between the group G level k and group H level /( WZW correlations and the coset model correlations for the simplest fields Retr gA of the gauged WZW models. The conditions under which Retr gA is a primary field are easily visible in the factorized form (3.5). g A ( z ) +1 is a primary field of the group G WZW model with the conformal weights equal 4 o
~i - - A, i • k i + g~i - v
k + gV '
where the sum is over the simple components of ~ and c£~ i stands for the (quatratic) Casimir of the component ~i in representation A. Similarly, the Ward identities for the H C / H model show that for each ~--algebra highest weight fi~,(bb*)~ 1 (in H C × H c invariant correlations) is a primary field with the conformal weights equal
where the sum runs this time over the simple factors of J'F. Thus field Retra ga is primary if and only if these numbers for the highest weights A labeling the irreducible components in the restriction to ~ of the representation A of (~ are the
K. Gawgdzki,A. Kupiainen / Cosetconstruction
649
same. The conformal weights of Retr ga are in such case equal to
k+g v
(3.13)
~:+h v
Fields R e t r g a ( z ) are by no means the only candidates for the primaries of the gauged WZW model. Suppose that in the general (not necessary left-right symmetric) case we start from same local (composite) field F(g, A 1°, Am), with F analytic, which is gauge invariant i.e.
F(g,
A 10, A ° l ) ( z )
=
F(ULgU~ 1, UAI°u-1
-F UOU 1, UAmU-I
+ U3U 1)(z)
for U H-valued. In the functional integral of the gauged WZW model (ignore for the moment the renormalization problem), upon the change of variables factorizing the model to the G times H C / H ones, such fields become the insertions of
if(g, b, b*)(z) = F(bLlgb~ 1, b*Ob* 1, b - l a b ) ( 2 ) .
(3.14)
Local functions i are invariant under the simultaneous chiral rotations g~'+~lL 6t2R°"*,
b~Ylb,
b* ~ b.7.2 ,
(3.15)
with y, H c valued, if Yl is (locally) analytic and ~'2 anti-analytic. Conversely, any local functional ff with such invariance defines by (3.14) F(g, A 1°, Am). Indeed, we can find b, b* such that A 1° = b*Ob*-1, A m = b lob (see ref. [47], p. 555) and put
F(g,
A 1°, A m) =
i(bLgb~, b, b*)
which does not depend on the choice of b, b*. A simple argument shows that F is still local if i was (in the sense of dependence on a finite number of derivatives at a point). Let us now perform a general (not necessarily analytic-anti-analytic) infinitesimal transformation (3.15) in the (factorized) functional integral with insertions of fields (3.14). The changes of the total action under the infinitesimal transformations (3.15) are given by the total a'gsymmetry currents of the factorized theory. Under (3.15) with ~'1 = e~&, ~2 = 1, 3 ( S o . k ( g ) --
SH,k(bb*))
and for Ya = 1, Y2
a(So,k(g)
-
=
-
i 2~r f[(0aAIL,
gag 1)k - (OaA,, bb*O(bb*)-')~]
eSa2,
=
i f[(g
lOg, aBA~R)k_((bb, ) lO(bb.),OSA~)~].
650
K. Gawcdzki,A. Kupiainen / Cosetconstruction
The variations of the insertions (3.14) gives rise to terms proportional to the values of 0~3A1, 8~3A~ and their derivatives at the insertion points. From the invariance of the functional integral, we infer in the standard way (stripping the variation of 3A's) the short distance behavior
ff(z)J~'(w) = O(1),
f f ( z ) J ~ ( w ) = O(1)
(3.16)
when z ~ w, z ¢ w (the variations of the insertions contribute only contact terms to (3.16)). The ~--symmetry currents are
J ~ ( x , w) = (XL, g Og-l(W))k -- (x, bb*O(bb*)-l(z))7~, Y~(X,W)=(XR, g 13g(w)) k - ( x , ( b b * )
lO(bb*)(z))~,
x ~ J ~ . Relation (3.16) should hold when inserted in the correlations of the combined G and H C / H WZW models. In fact it is equivalent to the invariance of ff under (3.15) with ~'1 analytic and ~'2 anti-analytic modulo contributions with vanishing correlation functions. Summarizing: the local gauge invariant fields of the gauged WZW model correspond in the factorized picture to joint composite fields ff of the tensor product of group G and H C / H WZW models, with the short distance behavior (3.16). In particular, they may be G-model composites with regular short distance behavior with G-model J~-symmetry currents ( ~ scalars in the language of ref. [26]). Since the short distance expansions of ff's with the Sugawara energy-momentum tensor of the factor theories are computable, it is in principle possible to determine in the factorized picture which ff's correspond to the primary fields of the gauged WZW model. In particular, the G-model composites should be just (Virasoro) primaries of this model. We shall see how this works in the simplest cases in the next section.
4. Unitary series Let us illustrate the general discussion of previous sections in the case of the minimal unitary series of conformal field theories [60,11] the, by now, standard test-ground for the coset construction ideas. We shall consider three cases (1)
G = SU(2) × SU(2)
(2)
G = SO(3) × SU(2)
(3)
G = SU(2) × S U ( 2 ) / d i a g Z 2
with the same Lie algebra su(2) • su(2) and the group G WZW model at level (k, 1) with the gauge-field coupled to the (left = right) diagonal su(2). In the first case k
K. Gawgdzki, A. Kupiainen / Coset construction
651
may be any positive integer, in the second one it has to be even and in the third one odd for the WZW action (2.1) to be defined up to 2~riZ, see ref. [36]. According to sect. 2, two ingredients are needed in order to find the toroidal partition functions of the gauged WZW models: (1)
partition functions for the twisted non-gauged WZW models (2.32) or the "mass matrices" N,
(2)
the branching functions of the ~ to J{' K a c - M o o d y characters (2.58).
The highest weights of su(2)~ su(2) integrable at level (k, 1) correspond to spins (j, e) j = 0, ~,11. . . . . ~k,1 e = 0,~.~ The mass matrices for cases 1, 2 and 3 are N(jL, eL)(JR ' eR) = ~jLJR~eLgR ,
(4.1)
N(jL,eL)(jR, eR) = (~jLJRl{jL=Omodl} q- ~jL(k/2_jR)l{jL=k/4modl} )~eLeR
(4.2)
and N(jt., eL)(JR, eR) = ~JLJ'R(~¢LeR1 { JL + t'L = 0 mod 1 }
q- (~jL(k/2_jR)~eL(1/2 ~.)1{JL +EL=(kq 1)/4rood1}
(4.3)
respectively, where 1 A denotes the characteristic function of set A. (4.1) and (4.2) have been worked out in ref. [35]. (4.6) follows by an immediate generalization of the discussion of ref. [36] to the semisimple case. As for the branching functions, the crucial realization of refs. [2, 3] was that they are the characters of the irreducible unitary representations of the Virasoro algebra for central charge ck = 1 - 6/(k + 2)(k + 3). More exactly, X su(2)/" ~,j , 'r ,
u)x~U~)(.~, u) =
E
su(2) l V Xk+I,ll, r, /,/) Xc,,hpq(T)
(4.4)
/ = 0 , 1 / 2 ..... (k + 1)/2,
l=j + e mod l
with p - 2 j + 1, q---2l+ 1 and x V G ( ~ ") standing for the Virasoro character of central charge c k and highest weight
[(k+ 3 ) p - (k + 2)q] 2- 1 hpq =
= h(k+2-p)(k+3 q)"
4 ( k q- 2)(k Jr- 3)
(4.5)
Thus
b(j.~,.,( ~) = ( x"V.~.~(~)
~0,
if
l=j+emodl,
otherwise.
(4.6)
K. Gawcdzki, A. Kupiainen / Coset construction
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Combining (4.1)-(4.3) and (4.6) with the result (2.62) gives for case 1: ZG/H(T)
=
C ( q q ) -ck/24
E
v
T
v
.
(4.7)
p=l ..... k+l q=l ..... k+2
for case 2:
Zc/H = C( q~)-,.,/24
E
k T app, XV~, ,,e~( r )XVc,, h,,.,,(),
(4.8)
p , p ' = l ..... k + l q = l ..... k + 2
where k
app, =
~pp,l{podd} q- ~p,k+2 p'l{k/2-podd},
for case 3 (with the use of symmetry (4.5)):
ZG/H(¢ ) =
C(qq)ck/24
...,k + l
E
v
v
( T ).
(4.9)
p=l ..... k+l q,q'=l ..... k + 2
These are, correspondingly, the (A, A), (D,A) and (A, D) modular invariant partition functions of the A - D - E classification of refs. [5, 6] (usually normalized so that C = 2, ~" it can be shown that the natural normalization of the volume elements in (2.14) gives this value of C). Here we have shown that (4.7-4.9), are the partition functions of three considered versions of the gauged WZW model. For the other modular invariants of refs. [5, 6], the (E, A)'s can be easily obtained by replacing first SU(2) by a bigger group G (at level 1) [61-64] into which SU(2) or SU(2) × G 1 for some group G 1 may be embedded in a conformal way [65] i.e. so that the corresponding branching functions are constant. Again one has to gauge left-right symmetrically diagSU(2)x G 1. To obtain the (A, E) partition functions one has to resort to a left right asymmetric gauge coupling [32] gauging in (SU(2) × SU(2)) × G (at level (k,l,1)) the subgroup diagSU(2) × (SU(2) × G1) with the left right embeddings differing by the permutation of the SU(2) factors. If in terms of the su(2) Kac-Moody characters
Zo/%(r, u) = C(q~) -d/z4)c~Ua'.... y'lV~,u,ak+l,,,~-(2),(~., -,ak+l,'t',"~"su(2)in u) I.l'
then we obtain from (2.62) ZSU(2)2xG/SU(2)2×G,('r)=C(qq) -cJ24
E N(q p,q,q'
1)/2(q'
1)/2XcV,hpq(~')XcV,hpq.(q') "
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653
Notice, that we could also use the asymmetric gauge coupling to rederive (by taking above G = SO(3)) the (A, D) partition functions. Let us discuss now the planar correlations for the unitary series models. For simplicity, we shall deal with the G = SU(2)x SU(2) case. The simplest primary fields are of the form Re(trgj(z)) with spins j = 0, ~2. . . . . ½k. Their correlations are given as N
fn
Re(tr gj,,(z,,)) e sk(g.A) &,g'.A}Dg Dg' DA
n= 1
f,,[I=,tr(g(bb*) _ N
= const. ×
E ~.=
1]°"(Zn)e-,/,,&(g)+s~-~{t"*)DgDb (4.10)
.' I
see (3.1) and (3.5). The conformal weights of Re(tr gi(z)) are equal to j(j + 1)/(k + 2)(k + 3), see (3.13), so that Re(trgj) is immediately identified with the 0pp field of refs. [60,66] and (4.10) gives the correlations of these fields in the form factorized into the SU(2) and H 3 ( = 3-dimensional hyperbolic space = SL(2, C)/SU(2)) correlations. The four-point functions of the eoqp fields were worked out in ref. [66] from the free field Feigin-Fuchs representation [67] of minimal models. The ((ppp(O)dppp(Z)dppp(l)(ppp(OC)) correlation is a sesquilinear combination of conformal blocks [60] p-1
FR(z)=f~ I-[ dtsdt.(f(t,z)~I(t~-t~(')-2f'(t',z) RS=I
(4.11)
s,s'
where functions f, f ' have singularities at 0, z, 1, oo and contours ~ a run between the singular points, see refs. [66, 8]. In particular for p = 2,
f(t,z)=
[z(1-- z)]l/2(k+2}[t(1-- t)(t -- z)]
f'(t',z)=[z(1-z)]
1/,k+2)
1 / 2 ( k + 3 ) [ t ' ( 1 - t ' ) ( t ' - z ) ] 1/~k+3)
Eq. (4.10) allows to express the conformal blocks (4.11) as a sum of products of SU(2) and H 3 WZW blocks: L~SU(2)[ ~ ] ~r'H3 FR(Z) = E M~,#'-'~r ,",~#r'(z) •
o~,[.I
The WZW blocks are again given by contour integrals [37, 57] p
£2(z) =
1
H dt .e2(t, z), oX s = l
(4.12)
654
K. Gaw~dzkL A. Kupiainen /
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where X stands for SU(2) or H 3. It is clearly not obvious that integrals (4.11) may be expressed in the factorized form (4.12). For p = 2 the right-hand side of relation (4.12) becomes
1
2
dtdt'f(t,z)
const.×~
1
2
]
~t' + (1 - t)t' + t(1 - t'~ + (1 - t)(1 - t') ]f'(t" z).
R
(4.13) The contour integrals in (4.13) give the hypergeometric functions leading to the expression for the ~22 four-point function already cited in ref. [68]. Other easily computable correlations of the gauged WZW model are those of the field Retr gj (g~/2 ® l j_ l/2) l(z). They become N
f
Retr gJ(g{/2 ® lj
1/2) l( zn)e- Sk(g'A)-Sl(g"A)Og Og'OA N
= const. ×
E
f n~=1 tr g],( g;/2 ® ( bb,) j 1/2)-°"(z.)e sk(g) Sl(g')
%=_+1 X e Sk+5(6b*) Dg
Dg' Db
so that the field is clearly primary. Its conformal weights are equal to
j ( j + 1) -
-
k+2
1 +
4
j2_ 1/4 -
-
k+3
_= h p ( p _ l )
leading to the identification d~(p_l) p =
Retrgj(g{/2 ® 1:.
1/2) - 1
(4.14)
In particular for q~12the H C / H contribution drops out and we obtain an expression in terms of the correlations of the SU(2) WZW models of level k and 1. The k = 1 correlations are given through non-abelian bosonization [41] by elementary functions [37]. For the four-point function of ~12 both sides of (4.12) become then expressions in hypergeometric functions whose equality was already noticed in ref. [10]. More complicated composite primary fields of the gauged WZW model are easier identified directly in their factorized picture as the primary composites with a regular short-distance behavior with respect to the total su(2) current, see sect. 3.
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The simplest of them are O
tr:(J Y"
(k+2+p)J')gj(J-(k+2+p)J')"
.(bb , - 1
+ c.c.
e~o~,eaoa~:(J-(k+2+P)J')~w2(gJ)~o~2 .... 2j. . . . . a2j
¢20,..., a2j ao . . . . . a z j
7t
.
,
-1
× ( J - ( k + Z + P ) J ) a 2 a o ' ( ( bb )J-1)ag..aij~3 .... 2 j ( z ) + c . c . (4.15) (with the convention that c.c. of gj = gg 1 and c.c. of (bb*)j 1 = ( b b * ) j l l ) where J = kgOzg -1, ] = kg -10~g, J' = g'Ozg '-l, ]' = g' 10~g' are the currents and the normal ordered expression is defined as the regular part of the field products, e.g.
:JgjJ:(z) = lim cgz,cg~,(z'- z ) ( U - Z)J(z')gj(z)J(z'). z,-.--~ z
In the operator language, taking the normal product with J corresponds to the commutator with J 1- To check that qs(p_2)p given by (4.15) corresponds to a primary field of the gauged WZW model, one has to verify that [L 0 + L~ + L0', 0
[Ln+L'n+L'~',q~
2)p(0)] = 0
for
n>0,
for
n>~0
as well as the right-handed counterparts of the above ( L " and J " correspond to the H 3 factor in the theory). An easy algebra is left to the reader. It should be clear now that, for the price of algebraic complication, one can build in the factorized picture any of the primary operators Oqp of the unitary series models out of the current-algebra descendants of level ¼(p - q)2 of gs and ( b b * ) i l for p - q even and of the descendants of level ~(p - q)2 _ ¼ of gigS~2 and (bb*)~ 1 for p - q odd. This way, any correlation functions of the unitary models may be expressed in terms of those of the SU(2) and H 3 WZW models.
5. Parafermionie theories
In previous sections, we have excluded abelian factors in the gauge field coupled to the WZW theory. This restriction was by no means essential. To show how to treat the abelian case, we shall consider a group G WZW theory, G as before, with the gauge field coupled to the Cartan subalgebra Jd' of ft. We shall denote by H the Cartan subgroup of the universal covering C of G.
656
K. Gawgdzki, A. Kupiainen / Coset construction
First consider the toroidal partition functions ZG/H(7")of the coupled theory. The parametrization (2.10) or (2.12) for the gauge fields becomes now a version of the Hodge decomposition and holds globally. We shall take • and O from a fundamental domain for JY'/Q v. The remaining freedom for h in (2.10) or (2.12) is removed by fixing h(0). If we parametrize
h = em/2U where qo ~
and U ~ H, relation (2.13) will become
S6,k(g, A) = So,k(y.hgh*7* ) - Sc.,k(e~), where
• ~=~+x+x*,
x = -(~/~).(~-
e)
and by (2.46) Sc" k (e+X) -
-
i f ( 3Wx'~,~5~ 4-~r 1 27
f
35q@kd2z - 2~rr21lq~l[2 ~
The jacobian of the change of variables from (%, U0, A) to (q~, O, h) where the dummy variables %, U0 becomes equal q~(0) and U(0) is
j( u, h) = Cr2 r det'( O*O )~, where 0 acts on the periodic functions on the toms. The change of variables leads then to the identity
Z(I/H( • ) = f e
s~ ~,g, A~a(W0) Dg DA d % dUo
=c~2rf{exp[-SG.k(r.gr~*)-(1/2~)f(o:~.Oem)kd2z-2~r2He~,l~] × det'( a*0 ) r8 (q0 (0)) dq) dO Dq~ DU = c r 2 r / 2 d e t , ( O t ~ ~r/2 g u )(q~/) (1/2)114~112dq~dO. j j[ Z c,,~tr,
(5.1)
Hence the result is that both the H C / H and the ghost contributions become free field expressions which factor completely out for group H abelian.
K. Gaw(dzki, A. Kupiainen / Coset construction
657
The partition function Zo, k(~', u) is given in terms of the characters of the group G K a c - M o o d y algebra by (2.32). In order to transform (5.1) further, we shall write the characters as the Fourier series x,~.~(,.,)=
e '"<"'> X ,~, a , x t ~r ) . '
E w e i g h t s )t
This gives
ZG/H("r ) = C'r2 r/2 det'( O~ ) r/2
NA~A.f ( q~) -c~;~/~""~'1~/~
E A L , Z R , ~k L , X R
The O integral is proportional to 8x~' XR. Thus
Zc, u(~)=C'r2~/2det'(O~) r/2
E
NALAR
A L , A R ,
= C'r;~/2det'(O~) r/2
E
NA~A~
A L, AR, X
× f(qO)
', ,/24+(1/2)11q~-,(X)]I2~N ~,,,,~.x('r)c~. a~,x('r)d~b
cc
(5.2)
where t is the isomorphism between the dual space of o~ and ~ induced by the Killing form ( . , . ) , and we have introduced the so-called "string functions" [69]
C*,A,X( "r -- q-(1/2)ll'~x)ll~XT, A, X( "r).
(5.3)
We shall need certain symmetries of the string functions which follow from a relation between Zo, k(r, u) and ZG, k(r, U - r() for ( ~ Q v. Notice that for h,(z) = e 2~i~:, by (2.13),
So., ( vuh,gegv: ) - So.,( v.h~h~v: ) = So.k ( Vugv* ) - So.,( v~v: ), since ( T u h t / ) - 1 0('yuh~) = "1,, -1 0Yu- Computing the WZW actions of the terms without g as in (2.46), we obtain
K. Gaw¢dzki, A. Kupiainen / Coset construction
658
from which we infer that ZG,k('r , U -- "r~) = (qgt)ll~ll~/Z e z'i(',~)*
2¢H(~t'~)kZG,k("l" , bl).
(5.4)
After Fourier transformation, eq. (5.4) implies that E NALARXSff, AL,~L(T)Xk,AR,~R f9 ('t") At-, A R
=
NaLARXk,AL,XL+,-,(t)('r)Xk,AF,.X.+-I(t)('r
)
At., AR
or that NALARCk, AL , )kL( 'r )Ck, AR, )~R( T ) AL, A R
=
NALARCk,At-,XL+t 1(~)( ~)Ck,AR,X~+, ~(t)( T) •
(5.5)
AL, AR
In fact one can show that Ck, A, ~ ~- Ck,A,?~+~-I(~)
but (5.5) will be sufficient for us. It allows to perform in (5.2) the sum over the t-~(Q v) translates of X as it simply restores the full domain of integration for the gaussian integral over ~b. Since d e t ' ( 0 ~ ) = ~-~l~/(~-)[4 where ~/(~-) is the Dedekind function, we obtain finally
Z ~ / n ( Z ) = C(qFt) c~,,/24[~/(~.)12r
E
NALA.ck, At-,
)c* A.,
) (5.6)
AL, AR ~. mod t l(QV)
i.e. the partition functions of the (generalized) parafermionic theories of [40] with central charge c~, k - rank G. The standard parafermions [15,39] correspond to G = SU(2). In this case the string functions are known explicitly [70]. Partition functions of (5.6) are modular invariant. Other modular invariant combinations of the string functions [39] may be obtained, as in the preceding section, by the asymmetric gauging of the product of the group G WZW model and (compactified) r free bosonic fields. For the parafermionic theories, one can easily control certain non-local correlations. Consider fields (f~,g~l(z)fp.)
(5.7)
K. Gawgdzki, A. Kupiainen / Coset construction
659
where fx and f, are ~-eigenvectors in the highest (integrable) weight A representation of G corresponding to weights ), and g. Fields (5.7) are not gauge invariant (for v~ g). However in the expression N
F - l--I (f~,,g~",(z,)f~°) ej(A'w> n=l
where w is a one-dimensional chain, a combination of non-intersecting paths, with values in the dual of j ~ c , the gauge invariance is restored provided that
17
Let us consider the correlation functions
(5.s)
f Fe s,i,~(g,m DgDA - E
for simplicity in the planar case. Upon parametrization A m = h- ~ 0h, h = e~/2U, we obtain N
E = C (Area) ~det'( 0 ~ ) ~ f f~I=1 ( fx., gz]:(z.) f..,) e -(°o/2)(~(~,,)" x,, + ~'-> e- so.~(g)
×exp
[
i
18(q~(O))DgDcp.
(5.9)
The ¢p integral in (5.9) is gaussian and can be easily computed after (multiplicative) Wick ordering. The result is E = C ( A r e a ) - ' / 2 ( d e t ' 0 ~ ) ~/2
I[
(Zn I -- ZtI 2)
k/2
?/1 :/~ ,v/2
×(Q1-Q2) <'(;')"(;2)>~/2 fl-I(fx°,g,°~".(z.)f..) e s';~(~)Dg, (5.10) n
where
X'=
Xn -g,
if if
on=l, o,=-1,
( gn /~"=t-X-
if if
on=l, %=-1"
The fight-hand side of (5.10) is multi-valued. The choice of its branch depends on the (homotopy class) of system of paths w. Notice that (5.9) can be inverted. Taking
K. Gawgdzki, ,4. Kupiainen / Coset construction
660
WE,R such that 8w L = ~2,,X'~z,,, 8w R = Z,tt' z,,, we may write
= cE(z~ ..... ~ . ) f I - I e ~j<°~'~> '/<~'W">e-('/~='J<°~"°+~D~
(5.11)
t/
expressing this way the group G WZW primary correlation functions through those of the parafermionic fields [40] (5.21) and of the (free) field e i~ taking values in the Cartan subgroup of G. These relations may be extended to the higher genus surface. Other easily identifiable parafermionic correlation functions are those of the parafermionic currents +~, ~7~ for a a root of N. Taking w with 8w = - E,~,/,,-
E/3.,~m,
tl
01
we have
-
f H ( g o g - ~ + gAl°g-l,%°)k(z~) × l-I(g l~g+g
1AOlg, e~o,),(Wm)ef(A,w)-s(~,,
m
= C (Area)
~det'(b*o)rfl-[(gOg 1, e,~o),(z.,)e(~<~,,),.,,)/2 n
X ]--I(g -a Og, eB,,,}k(w,, ) e - (~(w-,)'B,,,)/2 e f(~+-~+,w)/2 m
× e
(Znl--7--n2)-(t(Olnl)'~(Otn~))k/2
I11 ~ 112
× H
(Wml--w.,z)("Bm')"(Bm~-)Sk/2fI~n (gcgg 1,e,~,,)k(z,~)
× l-I ( g - 1 Og, e•,,,) k ( w m) e-So.~
(5.1 2)
K. Gawgdzki, A. Kupiainen / Coset construction
661
Again (5.12) may be inverted and slightly generalized to give the expression for the planar correlations of the WZW currents in terms of the parafermionic currents and free fields:
f~, J(e,..)(z,)ViJ(,j)(~))e s(~.~(g'Dg J
with (~W L = --~nOlnZ n and similarly for the right currents. Relation (5.13) generalizes the vertex construction of the current correlations to arbitrary group G and arbitrary level k of the WZW model [71, 72]. For G simple, simply laced and k = 1, the parafermionic contribution is (locally) constant and the WZW currents become free field expressions [73, 74]. The authors would like to acknowledge the hospitality of the IAS at Princeton where this work was begun. A.K. thanks the IHES at Bures-sur-Yvette for the invitation which enabled completing the paper.
Appendix A We shall prove a simple group-theoretic fact, used in sect. 2, that if h o ~ H c commutes with ~ = e 2~i° where O is in the Cartan subalgebra 3- and ( 0 , c~) ~ Z for roots a then h 0 must belong to the complexified Cartan subgroup T c. By the Bruhat decomposition [75], h 0 can be uniquely written as
E >O,eO] e,P[o,elwexp[ OAd
,A1,
where v,~ ~ C and w is in the normalizer of T c in H c. But then ?h°'7 l = e x p [ Ee2~<°'">v~e"l.<0
×
exp[27ri(O--AdwO)lwexp[ >O,Ad E ....
(A.2)
so that h 0=~ho'~ 1 implies that v ~ = 0 and O = A d w O + ~ for some ~ from the coroot lattice Q v. But the affine Weyl group acts without fixed points on the set of O such that (O, @ ¢ Z for roots a so that ~ = 0 and w e T c.
K. Gaw(dzkL A. Kupiainen / Coset construction
662
Appendix B We shall define and study here the WZW action for field configurations on the torus, twisted along the T-direction. Consider first the field configuration g satisfying g ( z + 1) = g ( z ) ,
(B.1)
g(z +'r) = ag(z)b,
a, b, g ( z ) ~ G c, on the cylinder Z(I") = ( zlO <~ l m z ~ . r 2 } / Z .
The boundary of Z is composed of two loops - 0 Z 0 and OZ 1 corresponding to Im z = 0 and Im z =~'2 respectively. As explained in [36,42], one can naturally define the amplitude (B.2)
e - S~"(g) ~£Pg*o ®~ag,
where .LP is a holomorphic line bundle over the loop group L G c, whose fibers over gi = glaz, i = 0,1, appear in (B.2). Group G ¢ acts on ~ by lifts of its actions on L G c by'left and right multiplications. Clearly a ' e-Sz(.(g)b -1 G,L,('g* ®~q~go~ C
(B.3)
where a-1 and b-1 act on the £,°g1 factor of the amplitude. We shall define the toroidal amplitude of the twisted field g, e x p [ - S r ( , ) ( g ) ] , as the number that one obtains in (B.3). Consider now a 1-parameter family gt of fields satisfying (1) with t-dependent a and b. Bundle £P is provided with a (holomorphic) connection such that
at
G
(O g -
ot l ,
G for g
~-,g
Og)k t dg)k e sz(g)
(B.4)
see formula (22) of appendix 2 in ref. [36]. Now d D _ _ e - S~(g) = __ ( a - 1 . e - &(g) . b - X) dt dt
(°
= a - 1 • - ~ e-Sz(g)
t "b-1
4~r
- ( 7 db 7b
Zo ( a
g-~
~-, (dg)g-1)k ida
dg) j] a - t . e - S z ( g ) . b
t
(B.5)
K. Gaw~dzki,A. Kupiainen / Cosetconstruction
663
where the second contribution can be easily extracted from formulae (24) and (25) of appendix 2 in ref. [36]. Substituting (B.4) to (B.5), we obtain
13g),g -1
d
__e-St(g)= dt
,
do
]
2~.fOzo(a 1 - - ~ - , ( d g ) g - ' ) k
e s~(g).
(B.6)
Using (B.6), it is easy to show that the Polyakov-Wiegmann formula ST(glg2) = ST(gl) "~ ST(g2) -- FZ(gl, g2) mod2eri
(B.7)
with
i F z ( g , , g2) = ~
fzk
holds for fields satisfying g,(z + 1 ) = g , ( z ) ,
gl(z + ¢) = agl(z)b,
(B.8)
g2(z + "r) = b-lg2(z)c,
(B.9)
whenever there exists a one-parameter family of fields satisfying (B.8) and (B.9) (with variable a, b, c) connecting fields gi to fields gO for which (B.7) can be checked directly. This occurs, for example, if b = 1 and either gl or g2 may be deformed to 1 with preservation of this condition or, without restrictions, for the simply connected group. The first case allows to prove (2.13) and the second one (2.43). To prove (2.45), one considers the deformation
e-X. Clearly
d / d t e-Sr(n× ') = 0 since (B.6) in this case involves only (e,, e~) k vanishing for a, fl > 0. Finally, by (B.6) again, d which proves (2.46).
i
2
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K. Gaw¢dzki, A. Kupiainen / Coset construction
Appendix C We shall compute here the formal jacobian j(u, h) of the change of variables
(~,O,h)
F,(~o,+,A)
performed in sect. 2, j ( u , h) d ~ dO Dh = F*(dq% dff DA).
(C.1)
First, we shall choose convenient bases in the tangent spaces. Consider the operator D A : e - - ~ where e is the space of YgC-valued functions and .~¢ of ovfC-valued (0, 1)-forms on the toms, both with the L 2 norm. L)AE=~E+
[A01, E] =]~-l~(y/E~ / 1)~,
(C.2)
where h - ~uh in the parametrization of (2.12). Its adjoint is given by DAt(ad Y) = - h * 0 z ( h *
lah)h*-i
(C.3)
Split e = kerD A • (ker D A ) ± = e 0 + e l , z~¢= kerDAt ~ (ker D J ) ± - z a ¢ 0 + z¢ 1 . By (C.2), a basis of kerD A is given by
ei= h ltih,
i = 1,..., r
(C.4)
where (t~) is a basis for the Cartan algebra 3-. Indeed, if 0(YueYu-1) = 0 then one easily sees that our restrictions on • and O force e to be a constant vector in .y-c so that k e r D a does not contain vectors independent of (C.4) Let (e,) be an orthornormal basis of e a. Obviously,
a i = e* d2,
a. = e ' d 2
form bases for d 0 and ~¢1 respectively. Let us introduce new coordinates
A m = Y'~yiai + ~ y . a . . i
n
K. Gaw¢dzki, A. Kupiainen / Coset construction
665
Putting
Dx =
dxidxi A dx, dY,,
A i
Dy =
n
A dy, dy, A d y . d y . ,
we have F*(d% d+ Dy) =jd~/'
(c.5)
d@ Dx
with the same j as in (C.1). Let us compute first F ' d %
and F*d~b. For
¢PO = ~ i 9 3 0 i t i '
F* dq0oi =
dxj +
and similarly for F* d~b~.
O%/Ox
OY----f
,, \ ax,
8 CPoi . +
- -
dx.
)
may be obtained by computing
hexp(~xi~o~)=,[exp(~xd,)lh=[exp(~xd~)lbu =exp[~>o(exP(Y'~xiti+~°P,a))w~G][exp(~ixiti+cp/2)]U
where
r ' = ~ + E(x,+ x,)t,,
(C.6)
i
(C.7)
U'=expI~(xi-xi)tslU. From (C.7), we see that
¢" = ~ + (l/4vri) ~ ( x i- Y,i)(ti, a). i
(c.8)
K. Gawcdzki,A. Kupiainen / Cosetconstruction
666 Hence
(C.9)
F* dcp0i = dx~ + dY~ + (dxn, dYn terms), 1
F * d ~ - 4qri E(dx,- dxi)(ti, a> + (dx,,dY, terms).
(C.10)
l
Next compute
oqy i d~ i + ~
F* d y i = Y'~
J
)
dOj
(C.11)
i
(y~ clearly does not depend on h). We have 0q
'fit
--h O~i
1),~t 3(y~h) = - - - h t , h ¢2
1 d_7,
--h 30i
tyulO(Tu h ) = --htih ld-7.
'7/"
¢2
(C.12)
(C.13)
We have to project the right hand side of (C.11), (C.12) to ~'o. The projection P0 to
~'o is
Poa = ~ a i ( m - 1 ) i j ( a j , i,j
a)
(C.14)
with
Mii (a,, a+) =
(c.15)
=
We have
(aj, h-XtihdY,)= fT(¢)((h*tyh* 1)*,h-ltih)d2z=CT28ij
(C.16)
where, for simplicity, we have assumed t i to be orthonormal in the considered scalar product on J-. From (C.11)-(C.16), we obtain
F* d y i = CY'~( M - 1 ) i j ( d O i - r d ~ i ) . ./ Finally, 0
--h ~Xm
1y-10(y~h)=Dne.,= Ea,,(a~,f)Aem), n
(C.17)
K. Gawgdzki, A. Kupiainen / Coset construction
667
which combined with (C.11) and (C.12) gives F* dy, = 2(a,,DAem)dxm+(dcb, dOterm). m
Eqs. (C.9)-(C.12) yield: F*(d% dq~Dy) = C,rfdet(MM*) 1det(D~D,4)dcbdODx. '----
(C.19)
Eq. (C.19), by virtue of (C.5) and (C.15), gives formula (2.19) for the jacobian
j(u, h). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
K. Gaw~dzki and A. Kupiainen, IHES/P/88/35, Helsinki Univ. HU-TFT-88-29 preprint P. Goddard, A. Kent and D. Olive, Phys. Lett, B152 (1985) 88 P. Goddard, A. Kent and D. Olive, Commun. Math. Phys. 113 (1987) 1 D. Gepner, Nucl. Phys. B287 (1987) 111 A. Cappelli, C. Itzykson and J.B. Zuber, Nucl. Phys. B280 [FS18] (1987) 445 A. Cappelli, C. Itzykson and J.B. Zuber, Commun. Math. Phys. 113 (1987) 1 A. Kato, Mod. Phys. Lett. A2 (1987) 585 P. Christe and R. Flume, Phys. Lett. B188 (1987) 219 P. Christe, preprint M.R. Douglas, CALT-68-1453 preprint D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575 H. Eichenherr, Phys. Lett. B151 (1985) 26 M.A. Bershadski, V.G. Knizhnik and M.G. Teitelmam Phys. Lett. B151 (1985) 31 D. Friedan, Z. Qiu and S. Shenker, Phys. Lett. B151 (1985) 37 V.A. Fateev and A.B. Zamolodchikov, Zh. Eksp. Teor. Fiz. 89 (1985) 380; 90 (1986) 1553 V.A. Fateev and A.B. Zamolodchikov, Teor. Mat. Fiz. 65 (1985) 1205 V.A. Fateev and A.B. Zamolodchikov, Nucl. Phys. B280 [FS18] (1987) 644 F. Ravanini and S.K. Yang, Phys. Lett. B195 (1987) 202 F.A. Bais, P. Bouwknegt and M. Surridge, ITFA 87-12 and 87-18 preprints V.A. Fateev and S.L. Lykyanov, Int. J. Mod. Phys. A3 (1988) 507 D. Kastor, E. Martinec and Z. Qiu, Phys. Lett. B200 (1988) 434 J. Bagger, D. Nemeschansky and S. Yankielowicz, Phys. Rev. Lett. 60 (1988) 389 F. Ravanini, Mod. Phys. Lett. A3 (1988) 397 D. Ahschuler, LBL-TH-25125 preprint P. Christe and F. Ravanini, Nordita-88/17 P preprint P. Bowcock and P. Goddard, DAMTP 88-18 preprint H. Eichenherr and M. Forger, Nucl. Phys. B155 (1979) 381; B164 (1980) 528 H. Eichenherr and M. Forger, Commun. Math. Phys. 82 (1981) 227 A. Kent, Phys. Lett. B173 (1986) 413 E. Guadagnini, M. Martellini and M. Mintchev, Phys. Lett. B194 (1987) 69 K. Bardacki, E. Rabinovici and B. S~iring, Nucl. Phys. B299 (1988) 151 W. Nahm, UCD-88-02 preprint P. Bowcock, DAMTP 87-31 (revised) preprint Z. Haba, BiBoS 309/88 preprint D. Gepner and E. Witten, Nucl. Phys. B278 (1986) 493 G. Felder, K. Gaw~;dzki and A. Kupiainen, Commun. Math. Phys. 117 (1988) 127 V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B243 (1984) 83
668
K. Gawgdzki, A. Kupiainen / Coset construction
[38] V.G. Kac and M. Wakimoto, Adv. Math., to be published [39] D. Gepner and Z. Qiu, Nucl. Phys. B285 [FS19] (1987) 423 [40] D. Gepner, Nucl. Phys. B290 [FS20] (1987) 10 [41] E. Witten, Commun. Math. Phys. 92 (1984) 455 [42] K. Gaw~dzki, I H E S / P / 8 7 / 4 7 preprint [43] D. Bernard, Nucl. Phys. B303 (1988) 77 [44] D. Bernard, PUPT-1087 preprint [45] S.D. Mathur and S. MukhL TIFR/TH/87-7 preprint [46] S.D. Mathur, S. Mukhi and A. Sen, TIFR/TH/88-22 preprint [47] M.F. Atiyah, R. Bott and Phil. Trans. R. Soc. London A308 (1982) 523 [48] M.S. Narasimhan and C.S. Seshadri, Ann. Math. 82 (1965) 540 [49] S.K. Donaldson, J. Diff. Geom. 18 (1983) 269 [50] A.M. Polyakov and P.B. Wiegmann, Phys. Lett. B131 (1983) 121 [51] J.-B. Bost, Sdminaire Bourbaki, Expos~ (1987) 676 [52] V.G. Kac, Infinite dimensional Lie algebras (Cambridge University Press, Cambridgc 1985) [53] A. Pressley and G. Segal, Loop groups (Clarendon Press, Oxford 1986) [54] D.B. Ray and I.M. Singer, Ann. Math. 98 (1973) 154 [55] L. Alvarez-Gaum~, G. Moore and C. Vafa, Commun. Math. Phys. 106 (1986) 1 [56] H. Freudenthal and H. de Vries, Linear Lie groups (Academic Press, New York, 1969) [57] P. Christe and R. Flume, Nucl. Phys. B282 (1987) 466 [58] T. Kohno, in Contemporary mathematics: Antin's braid group (Santa Cruz, 1987) [59] V.A. Fateev and A.B. Zamolodchikov, Yad. Fiz. 43 (1986) 1031 [60] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241 (1984) 333 [61] P. Bouwknegt and W. Nahm, Phys. Lett. B184 (1987) 359 [62] P. Bouwknegt, Nucl. Phys. B290 [FS20] (1987) 507 [63] W. Nahm, Duke Math. J. 54 (1987) 579 [64] W. Nahm, UCD-88-01 preprint [65] P. Bowcock and P. Goddard, Nucl. Phys. B285 [FS19] (1987) 651 [66] VI.S. Dotsenko and V.A. Fateev, Nucl. Phys. B240 [FS12] (1984) 312; B251 [FS13] (1985) 691 [67] B.L. Feigin and D.B. Fuchs, Funct. Anal. Appl. 16 (1982) 114:17 (1983) 241 [68] A.B. Zamolodchikov, Yad. Fiz. 44 (1986) 821 [69] V.G. Kac, Adv. Math. 35 (1980) 264 [70] V.G. Kac and D. Peterson, Adv. Math. 53 (1984) 125 [71] P. Goddard, W. Nahm, D. Olive and A. Schwimmer, Commun. Math. Phys. 102 (1986) 179 [72] D. Bernard and J. Thierry-Mieg, Commun. Math. Phys. 111 (1987) 181 [73] I.B. Frenkel and V.G. Kac, Invent. Math. 62 (1980) 23 [741 G. Segal, Commun. Math. Phys, 80 (1981) 301 [75] N. Bourbaki, Groupes et alg~bres de Lie, Ch. 6 (Hermann, Paris, 1968) [76] D. Karabali, Q-Han Park, H.J. Schnitzer and Zhu Yang, BRX TH-247 [revised] preprint
(2.43)
K. Gawgdzki, A. Kupiainen / Coset construction
639
and S's of twisted fields defined in appendix B. We have
nx(Z+.r)=e e~(z + r)
2~i~n~(z) e2~i~,
= e-2~i"e~(z)e
2~i~*.
The repeated application of (2.43) yields
S(n xe~nx)=S(nx)+S(n*)+S(e~x)
-F(nx,e~nx)-F(e~,nx).
(2.44)
Since e~ and t~ are orthogonal, the last term in (2.44) vanishes. Also S(nx) =
S(nx)
= 0
(2.45)
and SH'~(e~x)
=
-
i -4~r -;
see appendix B. Thus, we obtain (d2z =
s.,~(v.bb*v: ) -
1
where X + X* = Expand now
2~ri¢(z - ~) was
I
(2.46)
(1/2i)d~dz)
~ f
(71(0¢px, 0cpx) ? ,
0~9))~d22
-
2~"r2llq)ll-~
Oenx,e~xnx(Oze*x ~*
lXe-~°~)id2z,
(2.47)
used.
nx 10~nx=foldsexp [ = ELeo a>O
where wd = e(~/2+x,~)w~. Developing the under-integral expression into the (finite) Taylor series around s = 0 we see that
Io= O~(e(~J2+x'°~wo)+ L(w, ~)e ~ja+X'°~ where f~ depends only on w,, (and
3~w~,)with
a' preceding a in the sequence (2.41).
K. Gaw¢dzki,A. Kupiainen / Cosetconstruction
640
Moreover, f~ is uni-valued on the torus. Now, the last term of (2.47) becomes 1 -
-
Y', e <~'">ILI2= - -
'/7" a > 0
1
~ ID~w~+~1',
(2.48)
'/7" a > 0
where b~ = e - (~/2 +x, ~5 37 e (~/2 +x, ~5 .
Notice that kerD~=cokerD~ = 0. Indeed, if f ~ k e r D ~ then f = e(~/2+x'~)f is holomorphic with f ( z + 1) =f-(z) and f ( z + r ) = e - 2 ~ i ( u " > f ( z ) . Since for our choice of u = {9 - r@, (u, a) ~ Z + rZ, these have no solutions. CokerD~ is analogous. With (2.48), eq. (2.47) becomes
SH-k(yubb*y* ) -
2-~rl /(OzCp, ¢9~cP)~ d2z - 2rrr2ll~ll 2 - 71 Z ID.Wa -{-LI 2" ~>o
That f~ depends only on earlier % and D~ is invertible enables us to perform the w-integration in (2.42) iteratively in order w~ ..... wm to get ~H,~(T,
U)=
f(azW, 3#~P)7< d2z -
cfs(w(0))exp(-2~ X I-I det(D~D,~)-XDcp
2rrr:[]~][ { ]
•
(2.49)
a>0
The computation of det(D~D,0 is again standard. By the chiral anomaly
det(D,D.): exp[-2~ f"(~'a)':(%")d2e]det(D'D.) i.-o" Now, D.I~= o is also equal to
e x p [ - ('/.,o, a)] ~ exp[('/.o, a ) ] , see (2.11), so that det D2D.[~o=0 = det 0 ~ where 0 acts of functions f satisfying f ( z + 1) = e >"(*"Of(z),
f ( z + r) = e-2~'g(°"~>f(z).
(2.50)
Thus [54,55] (q-= e 2~'i~') det(D~tD.) L=o = qq;~"5'/2 + 1/12]e~ri(u,a) __ e-'i<" .>[2 x ~=:fi( 1 - e 2 " i ( " " ) q " ) ( 1 - e
2rri(u,a)qn)2.
(2.51)
K. Gawgdzki, A. Kupiainen / Coset construction
641
The remaining q0 integral in (2.49) is also gaussian,
[ Jt (o~w,o~w)~- Z
fs(w(O))exp - ~g
t]
Oz(w,~)o~(w, ~) d~z DW
ct>0
=
fs(w(0))exp[- ~1 f ( Ozw,O~w)~ h~d2z] Dq~
= C~.2r/2det,(0,~) x/2 =
C.r~/2(qgt)
~/24 n=lI~(1
--
qn)
-2r
(2.52)
where we have used the relation
Y' (t,@(t',C~)=~t,t')h~
for
t,t', ~.~-c
or>0
and 0 above operated on periodic functions. Gathering (2.49, 2.51 and 2.52) we obtain finally
,~H,~('r, U) = C'r2r/2(qgl)llq't12-h~/2-dimH/24
I-I Ie'<~''~> - e-"~<~'~>l-2 ~>0
X J~I=1 (1-q")~J~I>o(1-e2"i(~'")q")(1-e-2~il"'~)q")
2.
(2.53)
It is instructive for interpret (2.53) in the operator language in which
&L~'mT,('r,u) = ( q~)-c"~/~'~/24Tre2~iJ°(")qL°e2~iJo("*)~Zo
(2.54)
(in fact Tr in (2.54) should factor out the infinite volume of the group T c of residual symmetries). The infinite product in (2.53) is the contribution of the descendants of the primary states of the K a c - M o o d y symmetry which (unlike in the compact case) simply factors out. The rest of the expression is the constant-mode contribution given by the heat-kernel integral
(qq)-C"c'"~/2"fe""2~ "V(e 2=i"b,b)8(W) db where AT, h v denotes the laplacian of H C / H corresponding to the HC-invariant metric generated by the Killing form ( .,. )? hr. It will be convenient to rewrite (2.53) using the (Kac-Moody) affine roots c~ = (c~, n) where c~ is a root of .,vt°or zero and n ~ Z, ~ ~ 0. Recall that ff > 0 means n > 0 or n = 0 and c~ > 0. The affine roots (0, n) have multiplicity r. Using the
K. Gaw~dzki, A. Kupiainen / Coset construction
642
notation ea( q, e2~rit) = q, e2~ri(t,a) for t ~,Y-c, we obtain from (2.53) ~ " H , 7 ~ ( ' r , U ) = C'r2r/2(qq)llq'll~+hv/2-(cb'o)--dimH/2411~(,r,U)[
2
(2.55)
1 where O = ~E~ > oa is the Weyl vector,
n ( ~ ' , u ) = 1-I ( 1 - ea(q,e:"i")) 5>0
and we have also used (2.25). 2.5. T W I S T I N T E G R A L
Coming back to (2.22), we compute first SH,2h ~(7~'*) = - 2rrT21lOIIa2h v see (2.46). For det'(DJ D&) note that satisfying
DJuDAu is just the laplacian on functions f
f ( z + l)=e-2~i~f(z)e 2~i~,
f(z +,:)=e-2~iOf(z)e 2~i°,
i.e. expanding f = E~f~e~ + S,ifit i, f~ satisfy (2.50) whereas fi are periodic. Thus (compare (2.51) and (2.52)) d e t ' ( D ] D & ) = C~-22~(qq) IIq~ll~~+ 2(q~'°) + dimH/121/][( 'F, U )14 and (22) becomes
zo/.(':) = C':;/2f d~dO (qFt)'(*)ZG,k('c, u)lFl('r, u)12
(2.56)
c ( V ) = ' ~ll~ll~+h 2 v + (~, o) + d i m H / 2 4 .
(2.57)
where
Before we insert into (2.56) the expression (2.32) for Zo, k(~-, u), we shall rewrite the f¢ A( T , UL,R) in terms of the characters X ,K# characters Xk, L x (T, u ) of the group H K a c - M o o d y algebra: X k~t, A ( T , U L , R ) = z_.,VbL'R['r)X-~k,X('r,A,X\ U) . X
(2.58)
K. Gawgdzki,A. Kupiainen / Coset construction
643
Eq. (2.58) is obtained [38] by the decomposition of level k highest weight A representation of the group G Kac-Moody algebra into the highest weight representations of the group H Kac-Moody algebra embedded in the previous one by means of EL, R: jgc,,._.),~. X in (2.58) runs through the highest weights of ~ integrable at level ,~. These are the weights such that the image t(~) of ~ - ?~+ p under the identification of the dual space to ~ with de~ induced by the Killing form ( . , . ) k + h ~ lies in the Weyl alcove cg0. The coefficients bA,x(r) are called the (G ~ H) branching functions. We shall also use the Weyl-Kac formula [53] for the characters X~: X~,A(r,u) =
Y'~ E (--1)'(W)q a(x'*) w~W ~cQ ~
×exp[--2~ri(u,w(t(X) + ~)--t(O))i+h~]l-I(r,u) -1, (2.59) where l(w) is the length of w ( ~ of elementary reflections) and
d(x,
=
+ llL v
Inserting (2.32), (2.58) and (2.59) into (2.57), we obtain Z G / H ( ' r ) = C'r~/2(qq) (dimH-cG'k)/24 E E NALARbLL,XL(T)bRR, XR('r)mXL,Xe.("r)' AL, AR ~L, ~R
where
mxcX.=
E
fd~dO(--1)'(Wc)+'(w"~(qq)llellL*~/2qd(X~'~L~gl
~
a(x~'*~
WL'WR~W (L,~R~Q v
x exp[-2~ri(u, WL(t(XL)}- ~L))~.+hVq- 2rri(u*, WR(t(XR)+ ~R))k+hv]. (2.60) We shall first integrate over O. If t* is a weight then
dOe2rri(°'~') = CSu, 0 .
fuw~ww%
Since (0, i~)= (0, t(l~))~+hv and each ~ ~ Q v is of the form t(/~) for some weight #, the 0 integral in (2.60) will impose WL(t(XL) q- ~ L ) = WR(/'(XR) -I- ~R)"
(2.61)
644
K. Gawcdzkt A. Kupiainen / Coset construction
But L(~.L), L(XR) ~ cgo and w(Cg0+ ~) C3 cg0 :# ~ if and only if w = 1 and ( = 0. Hence (2.61) implies that wL = WR, & = & , )t L = )t a and m)tLX R = CaXLX R
= Ca~kL~kR
2 . , ~ W , ~=QV
E .,~W,IEQ v
f%dq) (qq),~il},,~/2+ll,(Y,~.)*aiL~/2
It,{o}ll~-+,~/2+(.*..x)~ ;)£,,,
.
f%dq}(qFl)pl.~+;+,{;,}llL,~/2
,,(o~lr},,,~/2.
The sum over w and ~ restores 3- as the full domain of integration for q). Since by the Freudenthal strange formula [56]
II (o)
II +,v =
v 1 hV d i m ~ E hj dim S j = - 12 ~ ' + h v ' / k/+ h/
we finally get mXLXR
=
C~XL)~RT2r/2
and
ZG/H(r)=C(qFI)
(~.{;., cH.~)/24 E
N,I,ARb,LI,x(T)b~RR,x(T)
•
(2.62)
AL. "~R. )t
The power of (q~) 1/24 gives the central charge CG,H = CG, * -- CH, ~ of the theory which is that of the coset construction [2, 3]. Notice also [31] that CG/H = CG, k -Jr-CHC/H" ~ q- Cghost
(2.63)
where CHC/H,~ is given by (2.37) and Cgho~t= -- 2 dim o~ is the central charge of the free anticommuting ghosts transforming in the adjoint representation of H c. Eq. (2.63) corresponds to the decomposition (2.22) of the theory into three factors: the compact WZW model, the H C / H model and the free ghosts. The partition functions (2.62) are modular invariant ZG/H(q')
=
ZG/H(T q- 1) = Z G / H ( - l / r ) .
This follows from the modular invariance of the partition functions Zo, a (r, u) as in ref. [32].
K. Gawgdzki, A. Kupiainen / Coset construction
645
3. Planar correlation functions The same strategy as applied in previous section for the toroidal partition functions may be used to study the correlation functions of the gauged WZW model. Below, we shall limit ourselves to the planar case. Let us first consider for the symmetric model (with Jf~--Jt'~L=YRC ~ ) the simplest correlations given formally as N
f,I~I_l (Retr
ga,,(z,,))e S'~,~(g'A'Dg DA = E,
(3.1)
where gA, stands for the matrix of g in the irreducible representation of G c of (integrable) highest weight A , and the functional integral is over fields g: P C ~ = C u {oc} ---, G and group H connections A on P C ~. Notice that the integrand of (1) has the local H-symmetry. On the Riemann sphere, the gauge fields A with A m = h -1 0h, h: P C 1 ~ H c, form already an open dense set so that the situation is much simpler than on the torus. Moreover the ambiguity in h may be removed if we fix h(0). Multiplying (3.1) by the dummy integral f S ( b o ) d b odUo = 1, where bo ~ H C / H , Uo ~ H, and changing variables from (b 0, Uo, A) to h = bo = b(0), Uo = U(0), we obtain immediately
E=/ ]-I
bU with
N
n
=
(RetrgA,,(z.))e
S(~'~(hgh*)+S"~(hh*)J(h)6(b(O)) Dh
(3.2)
1
where the jacobian is
j(h) = C
det'(DA*~4 )
(3.3)
det( ec, ea)
Here ~c = Adh 'xc, with x c spanning ~ , form a basis of kerD A and cokerD A = 0, compare appendix C. As in (2.22)
det'(D~DA) - C(Area) dim~'det'(0~)e s".2'v'hh*'. det(ec,
%)
Inserting (3.3, 3.4) into (3.2) and changing variables
(3.4)
g ~ h lgh*- 1, we obtain N
E=C(Area)
dim'A~'det'(0*0)
Y'~
2 - u f I--[ tr(g(bb*)
a,,= +1
×
,','
Dg Db
')~£',(z,,)
n=l
(3.5)
K. Gawgdzki, A. Kupiainen / Coset construction
646
i.e. an essentially factorized expression. Hence the computation of the correlation function (3.1) reduces to the computation of the correlations separately in the group G W Z W model and in the non-compact H C / H theory. The compact WZW model correlations satisfy the Knizhnik-Zamolodchikov equations [37] obtained from mixed conformal and current-algebra Ward identities and the Sugawara relation
0 OZm Here
Kab
N
1
~_, Kab(X~)m(Xb).-- f n@_lg~:,(z,)e-S°.k(g)Dg=O.
a,b, nq:m
Zm-- Zn
(3.6)
is as in (1.29) and d e=0 (ei~X"g(Z))A x~gn(z)°= 71 d----~
are generators of the left global G c symmetry, with subscripts m or n indicating on which field they act. Similarly,
[ 3
OZm
~2
a,b, n4~rn
K<~b( "7<') " (Xb) " .~m-1
Zn
If I~) g~"(z,)e s'~.k(g)Dg=O,
(3.7)
n=l
where _ a 1 d xaga(z) - i de
~=0
(g(z)
e-i~x°)~
generate right G c. The solutions of (3.6) and (3.7) are of the form
Z EaGp(z)Xpp'ED'(z) , p,p'
(3.8)
where a and fl label the global G c invariants made of tensor product of A n-representations and p, p' run through the set of independent solutions of (3.6) [37, 57, 8]. (X ep') is a hermitian matrix. The monodromy of the " W Z W blocks" E ~ ( z ) is given
K. Gaw~dzki, A. Kupiainen / Cosetconstruction
647
by the braid group representations arising as the holonomy of the connection
Y'.
a,b,m~n
dz m
K,,b(X~)m(Xb)n--
Zm-- Zn
,
see ref. [58]. In several cases (for G = SU(2) or G = SU(N) and fields in the fundamental representation) [37,59,57], one can obtain expressions (3.8) for the four point functions in a closed form. The H C / H correlations N
f ®
(3.9)
e s",~(bb*, D b
n=l
are similar. A slight complication is the presence of the delta function in (3.9) which breaks the global H c × H c covariance. This is however not a problem if we notice that
f ~n tr( g( bb* ) -l X"° " Xe-Sa .k(g) Dg )a°tz.) = fl-~I tr(g(Yobb*yd)
1 °n
)A,(z,)e-S(;'k'g)DbdYodYd
where 7o, Y(~~ H. The integration over 7o, 7d will select only H C × H c invariant combinations of correlations (3.9). For such combinations, the conformal and current-algebra Ward identities may be obtained exactly as for the compact case, leading to the KZ equations [~z0m
+
1
E
c,d,n~rn
Zm-- Zn
If
®N n=l
°" )Ao(Z.)
× 8(b(0)) e s".~(bb') Db dTodY~ = 0.
(3.10)
K~.d is as in eq. (1.36) and
xc(bb , o)A(z) : 71 -~e ~=o(ei~xcbb*(z))°a" An analogous equation holds for 3/35. Again, the solutions are of the form
E gaHqC/H(z)Yqq'g~qC/H(z) •
(3.11)
q,q'
which can be made explicit for the four point functions in the same cases as for the
648
K. Gaw~dzki, A. Kupiainen / Coset construction
compact model. In fact the solutions are given by the same expressions [37,57] analytically continued to negative levels (notice the sign difference between (3.6) and (3.10)). In the special case where we take G = H, k = lc, we infer from the KZ equations that N
f,I]I__,
tr(h(bb*)-~)a°,(z,)6(b(0))e-S-.~(h)+s.,*+~(h6*'Dh Db = const.
More exactly this holds separately for the WZW blocks:
E M~BEH(z)E~C/H(z) = Qpq
=
const.,
(3.12)
a,fl
where M~;3 gives the scalar product in the subspace of H c invariants. Relation (3.12) may be sometimes inverted to express the level k H C / H correlations by the ones of group H level ~: ones. In general however, the matrices ( E ~ ( z ) ) are not square. Eq. (3.12), in conjunction with (3.5), provides a precise version of the relation, postulated in ref. [10], between the group G level k and group H level /( WZW correlations and the coset model correlations for the simplest fields Retr gA of the gauged WZW models. The conditions under which Retr gA is a primary field are easily visible in the factorized form (3.5). g A ( z ) +1 is a primary field of the group G WZW model with the conformal weights equal 4 o
~i - - A, i • k i + g~i - v
k + gV '
where the sum is over the simple components of ~ and c£~ i stands for the (quatratic) Casimir of the component ~i in representation A. Similarly, the Ward identities for the H C / H model show that for each ~--algebra highest weight fi~,(bb*)~ 1 (in H C × H c invariant correlations) is a primary field with the conformal weights equal
where the sum runs this time over the simple factors of J'F. Thus field Retra ga is primary if and only if these numbers for the highest weights A labeling the irreducible components in the restriction to ~ of the representation A of (~ are the
K. Gawgdzki,A. Kupiainen / Cosetconstruction
649
same. The conformal weights of Retr ga are in such case equal to
k+g v
(3.13)
~:+h v
Fields R e t r g a ( z ) are by no means the only candidates for the primaries of the gauged WZW model. Suppose that in the general (not necessary left-right symmetric) case we start from same local (composite) field F(g, A 1°, Am), with F analytic, which is gauge invariant i.e.
F(g,
A 10, A ° l ) ( z )
=
F(ULgU~ 1, UAI°u-1
-F UOU 1, UAmU-I
+ U3U 1)(z)
for U H-valued. In the functional integral of the gauged WZW model (ignore for the moment the renormalization problem), upon the change of variables factorizing the model to the G times H C / H ones, such fields become the insertions of
if(g, b, b*)(z) = F(bLlgb~ 1, b*Ob* 1, b - l a b ) ( 2 ) .
(3.14)
Local functions i are invariant under the simultaneous chiral rotations g~'+~lL 6t2R°"*,
b~Ylb,
b* ~ b.7.2 ,
(3.15)
with y, H c valued, if Yl is (locally) analytic and ~'2 anti-analytic. Conversely, any local functional ff with such invariance defines by (3.14) F(g, A 1°, Am). Indeed, we can find b, b* such that A 1° = b*Ob*-1, A m = b lob (see ref. [47], p. 555) and put
F(g,
A 1°, A m) =
i(bLgb~, b, b*)
which does not depend on the choice of b, b*. A simple argument shows that F is still local if i was (in the sense of dependence on a finite number of derivatives at a point). Let us now perform a general (not necessarily analytic-anti-analytic) infinitesimal transformation (3.15) in the (factorized) functional integral with insertions of fields (3.14). The changes of the total action under the infinitesimal transformations (3.15) are given by the total a'gsymmetry currents of the factorized theory. Under (3.15) with ~'1 = e~&, ~2 = 1, 3 ( S o . k ( g ) --
SH,k(bb*))
and for Ya = 1, Y2
a(So,k(g)
-
=
-
i 2~r f[(0aAIL,
gag 1)k - (OaA,, bb*O(bb*)-')~]
eSa2,
=
i f[(g
lOg, aBA~R)k_((bb, ) lO(bb.),OSA~)~].
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The variations of the insertions (3.14) gives rise to terms proportional to the values of 0~3A1, 8~3A~ and their derivatives at the insertion points. From the invariance of the functional integral, we infer in the standard way (stripping the variation of 3A's) the short distance behavior
ff(z)J~'(w) = O(1),
f f ( z ) J ~ ( w ) = O(1)
(3.16)
when z ~ w, z ¢ w (the variations of the insertions contribute only contact terms to (3.16)). The ~--symmetry currents are
J ~ ( x , w) = (XL, g Og-l(W))k -- (x, bb*O(bb*)-l(z))7~, Y~(X,W)=(XR, g 13g(w)) k - ( x , ( b b * )
lO(bb*)(z))~,
x ~ J ~ . Relation (3.16) should hold when inserted in the correlations of the combined G and H C / H WZW models. In fact it is equivalent to the invariance of ff under (3.15) with ~'1 analytic and ~'2 anti-analytic modulo contributions with vanishing correlation functions. Summarizing: the local gauge invariant fields of the gauged WZW model correspond in the factorized picture to joint composite fields ff of the tensor product of group G and H C / H WZW models, with the short distance behavior (3.16). In particular, they may be G-model composites with regular short distance behavior with G-model J~-symmetry currents ( ~ scalars in the language of ref. [26]). Since the short distance expansions of ff's with the Sugawara energy-momentum tensor of the factor theories are computable, it is in principle possible to determine in the factorized picture which ff's correspond to the primary fields of the gauged WZW model. In particular, the G-model composites should be just (Virasoro) primaries of this model. We shall see how this works in the simplest cases in the next section.
4. Unitary series Let us illustrate the general discussion of previous sections in the case of the minimal unitary series of conformal field theories [60,11] the, by now, standard test-ground for the coset construction ideas. We shall consider three cases (1)
G = SU(2) × SU(2)
(2)
G = SO(3) × SU(2)
(3)
G = SU(2) × S U ( 2 ) / d i a g Z 2
with the same Lie algebra su(2) • su(2) and the group G WZW model at level (k, 1) with the gauge-field coupled to the (left = right) diagonal su(2). In the first case k
K. Gawgdzki, A. Kupiainen / Coset construction
651
may be any positive integer, in the second one it has to be even and in the third one odd for the WZW action (2.1) to be defined up to 2~riZ, see ref. [36]. According to sect. 2, two ingredients are needed in order to find the toroidal partition functions of the gauged WZW models: (1)
partition functions for the twisted non-gauged WZW models (2.32) or the "mass matrices" N,
(2)
the branching functions of the ~ to J{' K a c - M o o d y characters (2.58).
The highest weights of su(2)~ su(2) integrable at level (k, 1) correspond to spins (j, e) j = 0, ~,11. . . . . ~k,1 e = 0,~.~ The mass matrices for cases 1, 2 and 3 are N(jL, eL)(JR ' eR) = ~jLJR~eLgR ,
(4.1)
N(jL,eL)(jR, eR) = (~jLJRl{jL=Omodl} q- ~jL(k/2_jR)l{jL=k/4modl} )~eLeR
(4.2)
and N(jt., eL)(JR, eR) = ~JLJ'R(~¢LeR1 { JL + t'L = 0 mod 1 }
q- (~jL(k/2_jR)~eL(1/2 ~.)1{JL +EL=(kq 1)/4rood1}
(4.3)
respectively, where 1 A denotes the characteristic function of set A. (4.1) and (4.2) have been worked out in ref. [35]. (4.6) follows by an immediate generalization of the discussion of ref. [36] to the semisimple case. As for the branching functions, the crucial realization of refs. [2, 3] was that they are the characters of the irreducible unitary representations of the Virasoro algebra for central charge ck = 1 - 6/(k + 2)(k + 3). More exactly, X su(2)/" ~,j , 'r ,
u)x~U~)(.~, u) =
E
su(2) l V Xk+I,ll, r, /,/) Xc,,hpq(T)
(4.4)
/ = 0 , 1 / 2 ..... (k + 1)/2,
l=j + e mod l
with p - 2 j + 1, q---2l+ 1 and x V G ( ~ ") standing for the Virasoro character of central charge c k and highest weight
[(k+ 3 ) p - (k + 2)q] 2- 1 hpq =
= h(k+2-p)(k+3 q)"
4 ( k q- 2)(k Jr- 3)
(4.5)
Thus
b(j.~,.,( ~) = ( x"V.~.~(~)
~0,
if
l=j+emodl,
otherwise.
(4.6)
K. Gawcdzki, A. Kupiainen / Coset construction
652
Combining (4.1)-(4.3) and (4.6) with the result (2.62) gives for case 1: ZG/H(T)
=
C ( q q ) -ck/24
E
v
T
v
.
(4.7)
p=l ..... k+l q=l ..... k+2
for case 2:
Zc/H = C( q~)-,.,/24
E
k T app, XV~, ,,e~( r )XVc,, h,,.,,(),
(4.8)
p , p ' = l ..... k + l q = l ..... k + 2
where k
app, =
~pp,l{podd} q- ~p,k+2 p'l{k/2-podd},
for case 3 (with the use of symmetry (4.5)):
ZG/H(¢ ) =
C(qq)ck/24
...,k + l
E
v
v
( T ).
(4.9)
p=l ..... k+l q,q'=l ..... k + 2
These are, correspondingly, the (A, A), (D,A) and (A, D) modular invariant partition functions of the A - D - E classification of refs. [5, 6] (usually normalized so that C = 2, ~" it can be shown that the natural normalization of the volume elements in (2.14) gives this value of C). Here we have shown that (4.7-4.9), are the partition functions of three considered versions of the gauged WZW model. For the other modular invariants of refs. [5, 6], the (E, A)'s can be easily obtained by replacing first SU(2) by a bigger group G (at level 1) [61-64] into which SU(2) or SU(2) × G 1 for some group G 1 may be embedded in a conformal way [65] i.e. so that the corresponding branching functions are constant. Again one has to gauge left-right symmetrically diagSU(2)x G 1. To obtain the (A, E) partition functions one has to resort to a left right asymmetric gauge coupling [32] gauging in (SU(2) × SU(2)) × G (at level (k,l,1)) the subgroup diagSU(2) × (SU(2) × G1) with the left right embeddings differing by the permutation of the SU(2) factors. If in terms of the su(2) Kac-Moody characters
Zo/%(r, u) = C(q~) -d/z4)c~Ua'.... y'lV~,u,ak+l,,,~-(2),(~., -,ak+l,'t',"~"su(2)in u) I.l'
then we obtain from (2.62) ZSU(2)2xG/SU(2)2×G,('r)=C(qq) -cJ24
E N(q p,q,q'
1)/2(q'
1)/2XcV,hpq(~')XcV,hpq.(q') "
K. Gawgdzki, A. Kupiainen / Coset construction
653
Notice, that we could also use the asymmetric gauge coupling to rederive (by taking above G = SO(3)) the (A, D) partition functions. Let us discuss now the planar correlations for the unitary series models. For simplicity, we shall deal with the G = SU(2)x SU(2) case. The simplest primary fields are of the form Re(trgj(z)) with spins j = 0, ~2. . . . . ½k. Their correlations are given as N
fn
Re(tr gj,,(z,,)) e sk(g.A) &,g'.A}Dg Dg' DA
n= 1
f,,[I=,tr(g(bb*) _ N
= const. ×
E ~.=
1]°"(Zn)e-,/,,&(g)+s~-~{t"*)DgDb (4.10)
.' I
see (3.1) and (3.5). The conformal weights of Re(tr gi(z)) are equal to j(j + 1)/(k + 2)(k + 3), see (3.13), so that Re(trgj) is immediately identified with the 0pp field of refs. [60,66] and (4.10) gives the correlations of these fields in the form factorized into the SU(2) and H 3 ( = 3-dimensional hyperbolic space = SL(2, C)/SU(2)) correlations. The four-point functions of the eoqp fields were worked out in ref. [66] from the free field Feigin-Fuchs representation [67] of minimal models. The ((ppp(O)dppp(Z)dppp(l)(ppp(OC)) correlation is a sesquilinear combination of conformal blocks [60] p-1
FR(z)=f~ I-[ dtsdt.(f(t,z)~I(t~-t~(')-2f'(t',z) RS=I
(4.11)
s,s'
where functions f, f ' have singularities at 0, z, 1, oo and contours ~ a run between the singular points, see refs. [66, 8]. In particular for p = 2,
f(t,z)=
[z(1-- z)]l/2(k+2}[t(1-- t)(t -- z)]
f'(t',z)=[z(1-z)]
1/,k+2)
1 / 2 ( k + 3 ) [ t ' ( 1 - t ' ) ( t ' - z ) ] 1/~k+3)
Eq. (4.10) allows to express the conformal blocks (4.11) as a sum of products of SU(2) and H 3 WZW blocks: L~SU(2)[ ~ ] ~r'H3 FR(Z) = E M~,#'-'~r ,",~#r'(z) •
o~,[.I
The WZW blocks are again given by contour integrals [37, 57] p
£2(z) =
1
H dt .e2(t, z), oX s = l
(4.12)
654
K. Gaw~dzkL A. Kupiainen /
Coset construction
where X stands for SU(2) or H 3. It is clearly not obvious that integrals (4.11) may be expressed in the factorized form (4.12). For p = 2 the right-hand side of relation (4.12) becomes
1
2
dtdt'f(t,z)
const.×~
1
2
]
~t' + (1 - t)t' + t(1 - t'~ + (1 - t)(1 - t') ]f'(t" z).
R
(4.13) The contour integrals in (4.13) give the hypergeometric functions leading to the expression for the ~22 four-point function already cited in ref. [68]. Other easily computable correlations of the gauged WZW model are those of the field Retr gj (g~/2 ® l j_ l/2) l(z). They become N
f
Retr gJ(g{/2 ® lj
1/2) l( zn)e- Sk(g'A)-Sl(g"A)Og Og'OA N
= const. ×
E
f n~=1 tr g],( g;/2 ® ( bb,) j 1/2)-°"(z.)e sk(g) Sl(g')
%=_+1 X e Sk+5(6b*) Dg
Dg' Db
so that the field is clearly primary. Its conformal weights are equal to
j ( j + 1) -
-
k+2
1 +
4
j2_ 1/4 -
-
k+3
_= h p ( p _ l )
leading to the identification d~(p_l) p =
Retrgj(g{/2 ® 1:.
1/2) - 1
(4.14)
In particular for q~12the H C / H contribution drops out and we obtain an expression in terms of the correlations of the SU(2) WZW models of level k and 1. The k = 1 correlations are given through non-abelian bosonization [41] by elementary functions [37]. For the four-point function of ~12 both sides of (4.12) become then expressions in hypergeometric functions whose equality was already noticed in ref. [10]. More complicated composite primary fields of the gauged WZW model are easier identified directly in their factorized picture as the primary composites with a regular short-distance behavior with respect to the total su(2) current, see sect. 3.
K. Gaw~dzki, A. Kupiainen / Coset construction
655
The simplest of them are O
tr:(J Y"
(k+2+p)J')gj(J-(k+2+p)J')"
.(bb , - 1
+ c.c.
e~o~,eaoa~:(J-(k+2+P)J')~w2(gJ)~o~2 .... 2j. . . . . a2j
¢20,..., a2j ao . . . . . a z j
7t
.
,
-1
× ( J - ( k + Z + P ) J ) a 2 a o ' ( ( bb )J-1)ag..aij~3 .... 2 j ( z ) + c . c . (4.15) (with the convention that c.c. of gj = gg 1 and c.c. of (bb*)j 1 = ( b b * ) j l l ) where J = kgOzg -1, ] = kg -10~g, J' = g'Ozg '-l, ]' = g' 10~g' are the currents and the normal ordered expression is defined as the regular part of the field products, e.g.
:JgjJ:(z) = lim cgz,cg~,(z'- z ) ( U - Z)J(z')gj(z)J(z'). z,-.--~ z
In the operator language, taking the normal product with J corresponds to the commutator with J 1- To check that qs(p_2)p given by (4.15) corresponds to a primary field of the gauged WZW model, one has to verify that [L 0 + L~ + L0', 0
[Ln+L'n+L'~',q~
2)p(0)] = 0
for
n>0,
for
n>~0
as well as the right-handed counterparts of the above ( L " and J " correspond to the H 3 factor in the theory). An easy algebra is left to the reader. It should be clear now that, for the price of algebraic complication, one can build in the factorized picture any of the primary operators Oqp of the unitary series models out of the current-algebra descendants of level ¼(p - q)2 of gs and ( b b * ) i l for p - q even and of the descendants of level ~(p - q)2 _ ¼ of gigS~2 and (bb*)~ 1 for p - q odd. This way, any correlation functions of the unitary models may be expressed in terms of those of the SU(2) and H 3 WZW models.
5. Parafermionie theories
In previous sections, we have excluded abelian factors in the gauge field coupled to the WZW theory. This restriction was by no means essential. To show how to treat the abelian case, we shall consider a group G WZW theory, G as before, with the gauge field coupled to the Cartan subalgebra Jd' of ft. We shall denote by H the Cartan subgroup of the universal covering C of G.
656
K. Gawgdzki, A. Kupiainen / Coset construction
First consider the toroidal partition functions ZG/H(7")of the coupled theory. The parametrization (2.10) or (2.12) for the gauge fields becomes now a version of the Hodge decomposition and holds globally. We shall take • and O from a fundamental domain for JY'/Q v. The remaining freedom for h in (2.10) or (2.12) is removed by fixing h(0). If we parametrize
h = em/2U where qo ~
and U ~ H, relation (2.13) will become
S6,k(g, A) = So,k(y.hgh*7* ) - Sc.,k(e~), where
• ~=~+x+x*,
x = -(~/~).(~-
e)
and by (2.46) Sc" k (e+X) -
-
i f ( 3Wx'~,~5~ 4-~r 1 27
f
35q@kd2z - 2~rr21lq~l[2 ~
The jacobian of the change of variables from (%, U0, A) to (q~, O, h) where the dummy variables %, U0 becomes equal q~(0) and U(0) is
j( u, h) = Cr2 r det'( O*O )~, where 0 acts on the periodic functions on the toms. The change of variables leads then to the identity
Z(I/H( • ) = f e
s~ ~,g, A~a(W0) Dg DA d % dUo
=c~2rf{exp[-SG.k(r.gr~*)-(1/2~)f(o:~.Oem)kd2z-2~r2He~,l~] × det'( a*0 ) r8 (q0 (0)) dq) dO Dq~ DU = c r 2 r / 2 d e t , ( O t ~ ~r/2 g u )(q~/) (1/2)114~112dq~dO. j j[ Z c,,~tr,
(5.1)
Hence the result is that both the H C / H and the ghost contributions become free field expressions which factor completely out for group H abelian.
K. Gaw(dzki, A. Kupiainen / Coset construction
657
The partition function Zo, k(~', u) is given in terms of the characters of the group G K a c - M o o d y algebra by (2.32). In order to transform (5.1) further, we shall write the characters as the Fourier series x,~.~(,.,)=
e '"<"'> X ,~, a , x t ~r ) . '
E w e i g h t s )t
This gives
ZG/H("r ) = C'r2 r/2 det'( O~ ) r/2
NA~A.f ( q~) -c~;~/~""~'1~/~
E A L , Z R , ~k L , X R
The O integral is proportional to 8x~' XR. Thus
Zc, u(~)=C'r2~/2det'(O~) r/2
E
NALAR
A L , A R ,
= C'r;~/2det'(O~) r/2
E
NA~A~
A L, AR, X
× f(qO)
', ,/24+(1/2)11q~-,(X)]I2~N ~,,,,~.x('r)c~. a~,x('r)d~b
cc
(5.2)
where t is the isomorphism between the dual space of o~ and ~ induced by the Killing form ( . , . ) , and we have introduced the so-called "string functions" [69]
C*,A,X( "r -- q-(1/2)ll'~x)ll~XT, A, X( "r).
(5.3)
We shall need certain symmetries of the string functions which follow from a relation between Zo, k(r, u) and ZG, k(r, U - r() for ( ~ Q v. Notice that for h,(z) = e 2~i~:, by (2.13),
So., ( vuh,gegv: ) - So.,( v.h~h~v: ) = So.k ( Vugv* ) - So.,( v~v: ), since ( T u h t / ) - 1 0('yuh~) = "1,, -1 0Yu- Computing the WZW actions of the terms without g as in (2.46), we obtain
K. Gaw¢dzki, A. Kupiainen / Coset construction
658
from which we infer that ZG,k('r , U -- "r~) = (qgt)ll~ll~/Z e z'i(',~)*
2¢H(~t'~)kZG,k("l" , bl).
(5.4)
After Fourier transformation, eq. (5.4) implies that E NALARXSff, AL,~L(T)Xk,AR,~R f9 ('t") At-, A R
=
NaLARXk,AL,XL+,-,(t)('r)Xk,AF,.X.+-I(t)('r
)
At., AR
or that NALARCk, AL , )kL( 'r )Ck, AR, )~R( T ) AL, A R
=
NALARCk,At-,XL+t 1(~)( ~)Ck,AR,X~+, ~(t)( T) •
(5.5)
AL, AR
In fact one can show that Ck, A, ~ ~- Ck,A,?~+~-I(~)
but (5.5) will be sufficient for us. It allows to perform in (5.2) the sum over the t-~(Q v) translates of X as it simply restores the full domain of integration for the gaussian integral over ~b. Since d e t ' ( 0 ~ ) = ~-~l~/(~-)[4 where ~/(~-) is the Dedekind function, we obtain finally
Z ~ / n ( Z ) = C(qFt) c~,,/24[~/(~.)12r
E
NALA.ck, At-,
)c* A.,
) (5.6)
AL, AR ~. mod t l(QV)
i.e. the partition functions of the (generalized) parafermionic theories of [40] with central charge c~, k - rank G. The standard parafermions [15,39] correspond to G = SU(2). In this case the string functions are known explicitly [70]. Partition functions of (5.6) are modular invariant. Other modular invariant combinations of the string functions [39] may be obtained, as in the preceding section, by the asymmetric gauging of the product of the group G WZW model and (compactified) r free bosonic fields. For the parafermionic theories, one can easily control certain non-local correlations. Consider fields (f~,g~l(z)fp.)
(5.7)
K. Gawgdzki, A. Kupiainen / Coset construction
659
where fx and f, are ~-eigenvectors in the highest (integrable) weight A representation of G corresponding to weights ), and g. Fields (5.7) are not gauge invariant (for v~ g). However in the expression N
F - l--I (f~,,g~",(z,)f~°) ej(A'w> n=l
where w is a one-dimensional chain, a combination of non-intersecting paths, with values in the dual of j ~ c , the gauge invariance is restored provided that
17
Let us consider the correlation functions
(5.s)
f Fe s,i,~(g,m DgDA - E
for simplicity in the planar case. Upon parametrization A m = h- ~ 0h, h = e~/2U, we obtain N
E = C (Area) ~det'( 0 ~ ) ~ f f~I=1 ( fx., gz]:(z.) f..,) e -(°o/2)(~(~,,)" x,, + ~'-> e- so.~(g)
×exp
[
i
18(q~(O))DgDcp.
(5.9)
The ¢p integral in (5.9) is gaussian and can be easily computed after (multiplicative) Wick ordering. The result is E = C ( A r e a ) - ' / 2 ( d e t ' 0 ~ ) ~/2
I[
(Zn I -- ZtI 2)
?/1 :/~ ,v/2
×(Q1-Q2) <'(;')"(;2)>~/2 fl-I(fx°,g,°~".(z.)f..) e s';~(~)Dg, (5.10) n
where
X'=
Xn -g,
if if
on=l, o,=-1,
( gn /~"=t-X-
if if
on=l, %=-1"
The fight-hand side of (5.10) is multi-valued. The choice of its branch depends on the (homotopy class) of system of paths w. Notice that (5.9) can be inverted. Taking
K. Gawgdzki, ,4. Kupiainen / Coset construction
660
WE,R such that 8w L = ~2,,X'~z,,, 8w R = Z,tt' z,,, we may write
= cE(z~ ..... ~ . ) f I - I e ~j<°~'~> '/<~'W">e-('/~='J<°~"°+~D~
(5.11)
t/
expressing this way the group G WZW primary correlation functions through those of the parafermionic fields [40] (5.21) and of the (free) field e i~ taking values in the Cartan subgroup of G. These relations may be extended to the higher genus surface. Other easily identifiable parafermionic correlation functions are those of the parafermionic currents +~, ~7~ for a a root of N. Taking w with 8w = - E,~,/,,-
E/3.,~m,
tl
01
we have
-
f H ( g o g - ~ + gAl°g-l,%°)k(z~) × l-I(g l~g+g
1AOlg, e~o,),(Wm)ef(A,w)-s(~,,
m
= C (Area)
~det'(b*o)rfl-[(gOg 1, e,~o),(z.,)e(~<~,,),.,,)/2 n
X ]--I(g -a Og, eB,,,}k(w,, ) e - (~(w-,)'B,,,)/2 e f(~+-~+,w)/2 m
× e
(Znl--7--n2)-(t(Olnl)'~(Otn~))k/2
I11 ~ 112
× H
(Wml--w.,z)("Bm')"(Bm~-)Sk/2fI~n (gcgg 1,e,~,,)k(z,~)
× l-I ( g - 1 Og, e•,,,) k ( w m) e-So.~
(5.1 2)
K. Gawgdzki, A. Kupiainen / Coset construction
661
Again (5.12) may be inverted and slightly generalized to give the expression for the planar correlations of the WZW currents in terms of the parafermionic currents and free fields:
f~, J(e,..)(z,)ViJ(,j)(~))e s(~.~(g'Dg J
with (~W L = --~nOlnZ n and similarly for the right currents. Relation (5.13) generalizes the vertex construction of the current correlations to arbitrary group G and arbitrary level k of the WZW model [71, 72]. For G simple, simply laced and k = 1, the parafermionic contribution is (locally) constant and the WZW currents become free field expressions [73, 74]. The authors would like to acknowledge the hospitality of the IAS at Princeton where this work was begun. A.K. thanks the IHES at Bures-sur-Yvette for the invitation which enabled completing the paper.
Appendix A We shall prove a simple group-theoretic fact, used in sect. 2, that if h o ~ H c commutes with ~ = e 2~i° where O is in the Cartan subalgebra 3- and ( 0 , c~) ~ Z for roots a then h 0 must belong to the complexified Cartan subgroup T c. By the Bruhat decomposition [75], h 0 can be uniquely written as
E >O,eO] e,P[o,elwexp[ OAd
,A1,
where v,~ ~ C and w is in the normalizer of T c in H c. But then ?h°'7 l = e x p [ Ee2~<°'">v~e"l.<0
×
exp[27ri(O--AdwO)lwexp[ >O,Ad E ....
(A.2)
so that h 0=~ho'~ 1 implies that v ~ = 0 and O = A d w O + ~ for some ~ from the coroot lattice Q v. But the affine Weyl group acts without fixed points on the set of O such that (O, @ ¢ Z for roots a so that ~ = 0 and w e T c.
K. Gaw(dzkL A. Kupiainen / Coset construction
662
Appendix B We shall define and study here the WZW action for field configurations on the torus, twisted along the T-direction. Consider first the field configuration g satisfying g ( z + 1) = g ( z ) ,
(B.1)
g(z +'r) = ag(z)b,
a, b, g ( z ) ~ G c, on the cylinder Z(I") = ( zlO <~ l m z ~ . r 2 } / Z .
The boundary of Z is composed of two loops - 0 Z 0 and OZ 1 corresponding to Im z = 0 and Im z =~'2 respectively. As explained in [36,42], one can naturally define the amplitude (B.2)
e - S~"(g) ~£Pg*o ®~ag,
where .LP is a holomorphic line bundle over the loop group L G c, whose fibers over gi = glaz, i = 0,1, appear in (B.2). Group G ¢ acts on ~ by lifts of its actions on L G c by'left and right multiplications. Clearly a ' e-Sz(.(g)b -1 G,L,('g* ®~q~go~ C
(B.3)
where a-1 and b-1 act on the £,°g1 factor of the amplitude. We shall define the toroidal amplitude of the twisted field g, e x p [ - S r ( , ) ( g ) ] , as the number that one obtains in (B.3). Consider now a 1-parameter family gt of fields satisfying (1) with t-dependent a and b. Bundle £P is provided with a (holomorphic) connection such that
at
G
(O g -
ot l ,
G for g
~-,g
Og)k t dg)k e sz(g)
(B.4)
see formula (22) of appendix 2 in ref. [36]. Now d D _ _ e - S~(g) = __ ( a - 1 . e - &(g) . b - X) dt dt
(°
= a - 1 • - ~ e-Sz(g)
t "b-1
4~r
- ( 7 db 7b
Zo ( a
g-~
~-, (dg)g-1)k ida
dg) j] a - t . e - S z ( g ) . b
t
(B.5)
K. Gaw~dzki,A. Kupiainen / Cosetconstruction
663
where the second contribution can be easily extracted from formulae (24) and (25) of appendix 2 in ref. [36]. Substituting (B.4) to (B.5), we obtain
13g),g -1
d
__e-St(g)= dt
,
do
]
2~.fOzo(a 1 - - ~ - , ( d g ) g - ' ) k
e s~(g).
(B.6)
Using (B.6), it is easy to show that the Polyakov-Wiegmann formula ST(glg2) = ST(gl) "~ ST(g2) -- FZ(gl, g2) mod2eri
(B.7)
with
i F z ( g , , g2) = ~
fz
holds for fields satisfying g,(z + 1 ) = g , ( z ) ,
gl(z + ¢) = agl(z)b,
(B.8)
g2(z + "r) = b-lg2(z)c,
(B.9)
whenever there exists a one-parameter family of fields satisfying (B.8) and (B.9) (with variable a, b, c) connecting fields gi to fields gO for which (B.7) can be checked directly. This occurs, for example, if b = 1 and either gl or g2 may be deformed to 1 with preservation of this condition or, without restrictions, for the simply connected group. The first case allows to prove (2.13) and the second one (2.43). To prove (2.45), one considers the deformation
e-X. Clearly
d / d t e-Sr(n× ') = 0 since (B.6) in this case involves only (e,, e~) k vanishing for a, fl > 0. Finally, by (B.6) again, d which proves (2.46).
i
2
664
K. Gaw¢dzki, A. Kupiainen / Coset construction
Appendix C We shall compute here the formal jacobian j(u, h) of the change of variables
(~,O,h)
F,(~o,+,A)
performed in sect. 2, j ( u , h) d ~ dO Dh = F*(dq% dff DA).
(C.1)
First, we shall choose convenient bases in the tangent spaces. Consider the operator D A : e - - ~ where e is the space of YgC-valued functions and .~¢ of ovfC-valued (0, 1)-forms on the toms, both with the L 2 norm. L)AE=~E+
[A01, E] =]~-l~(y/E~ / 1)~,
(C.2)
where h - ~uh in the parametrization of (2.12). Its adjoint is given by DAt(ad Y) = - h * 0 z ( h *
lah)h*-i
(C.3)
Split e = kerD A • (ker D A ) ± = e 0 + e l , z~¢= kerDAt ~ (ker D J ) ± - z a ¢ 0 + z¢ 1 . By (C.2), a basis of kerD A is given by
ei= h ltih,
i = 1,..., r
(C.4)
where (t~) is a basis for the Cartan algebra 3-. Indeed, if 0(YueYu-1) = 0 then one easily sees that our restrictions on • and O force e to be a constant vector in .y-c so that k e r D a does not contain vectors independent of (C.4) Let (e,) be an orthornormal basis of e a. Obviously,
a i = e* d2,
a. = e ' d 2
form bases for d 0 and ~¢1 respectively. Let us introduce new coordinates
A m = Y'~yiai + ~ y . a . . i
n
K. Gaw¢dzki, A. Kupiainen / Coset construction
665
Putting
Dx =
dxidxi A dx, dY,,
A i
Dy =
n
A dy, dy, A d y . d y . ,
we have F*(d% d+ Dy) =jd~/'
(c.5)
d@ Dx
with the same j as in (C.1). Let us compute first F ' d %
and F*d~b. For
¢PO = ~ i 9 3 0 i t i '
F* dq0oi =
dxj +
and similarly for F* d~b~.
O%/Ox
OY----f
,, \ ax,
8 CPoi . +
- -
dx.
)
may be obtained by computing
hexp(~xi~o~)=,[exp(~xd,)lh=[exp(~xd~)lbu =exp[~>o(exP(Y'~xiti+~°P,a))w~G][exp(~ixiti+cp/2)]U
where
r ' = ~ + E(x,+ x,)t,,
(C.6)
i
(C.7)
U'=expI~(xi-xi)tslU. From (C.7), we see that
¢" = ~ + (l/4vri) ~ ( x i- Y,i)(ti, a). i
(c.8)
K. Gawcdzki,A. Kupiainen / Cosetconstruction
666 Hence
(C.9)
F* dcp0i = dx~ + dY~ + (dxn, dYn terms), 1
F * d ~ - 4qri E(dx,- dxi)(ti, a> + (dx,,dY, terms).
(C.10)
l
Next compute
oqy i d~ i + ~
F* d y i = Y'~
J
)
dOj
(C.11)
i
(y~ clearly does not depend on h). We have 0q
'fit
--h O~i
1),~t 3(y~h) = - - - h t , h ¢2
1 d_7,
--h 30i
tyulO(Tu h ) = --htih ld-7.
'7/"
¢2
(C.12)
(C.13)
We have to project the right hand side of (C.11), (C.12) to ~'o. The projection P0 to
~'o is
Poa = ~ a i ( m - 1 ) i j ( a j , i,j
a)
(C.14)
with
Mii (a,, a+) =
(c.15)
=
We have
(aj, h-XtihdY,)= fT(¢)((h*tyh* 1)*,h-ltih)d2z=CT28ij
(C.16)
where, for simplicity, we have assumed t i to be orthonormal in the considered scalar product on J-. From (C.11)-(C.16), we obtain
F* d y i = CY'~( M - 1 ) i j ( d O i - r d ~ i ) . ./ Finally, 0
--h ~Xm
1y-10(y~h)=Dne.,= Ea,,(a~,f)Aem), n
(C.17)
K. Gawgdzki, A. Kupiainen / Coset construction
667
which combined with (C.11) and (C.12) gives F* dy, = 2(a,,DAem)dxm+(dcb, dOterm). m
Eqs. (C.9)-(C.12) yield: F*(d% dq~Dy) = C,rfdet(MM*) 1det(D~D,4)dcbdODx. '----
(C.19)
Eq. (C.19), by virtue of (C.5) and (C.15), gives formula (2.19) for the jacobian
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668
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