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Progress of Theoretical Physics Supplement No. 102, 1990 67 Free Field Approach to 2-Dimensional Conformal Field Theories Peter BOUWKNEGT,*'#l Jim MCCARTHY**·tl and Krzysztof PILCH***·t n *Center for Theoretical Physics, Massachusetts Institute of Technology Cambridge, MA 02139, U.S.A. and Institute for Theoretical Physics, University of California Santa Barbara, CA 93106, U. S. A. **Department of Physics, Brandeis University, Waltham, MA 02254, U.S.A. ***Department of Physics, University of Southern California Los Angeles, CA 90089-0484, U.S.A. We review various aspects of the free field approach to (rational) conformal field theories. In particular, we will discuss resolutions of irreducible modules in terms of free field Fock spaces for WZNW-models and their coset models, as well as the free field realization of chiral vertex operators. We provide a host of clarifying examples and detailed proofs of results that were announced elsewhere. Contents §L Ll § 2. 2.1 Introduction Notations Fock space realizations and resolutions for finite dimensional Lie algebras Fock space realizations 2.2. Intertwiners 2.3. The EGG-resolution 2.4. Twisted Verma modules, realizations and resolutions 2.5. Finite dimensional coset models 2.6. Finite dimensional vertex operators § 3. Free field approach to affine Kac-Moody algebras 3.L The Fock space realization 3.2. Intertwining operators 3.3. The resolution Supported in part by the U.S. Department of Energy under Contract #DE-AC02-76ER03069 and by NSF grant #PHY-89-04035, supplemented by funds from NASA. Address after Nov. 1; CERN-TH, CH-1211 Geneve 23, Switzerland. tJ Supported by the NSF Grant #PHY-88-04561. t t > Supported in part by the USC Faculty Research and Innovation Fund. #) 68 P. Bouwknegt, J. McCarthy and K. Pilch 3.4. Chiral vertex operators and fusion rules 3.5. Coset conformal field theories Appendices A. Some lemmas on the Weyl group B. Quantum group identities C. Cohomology of double complexes D. Proof of Theorems 2.12 and 2.12' E. Restricted quantum group Verma modules References § 1. Introduction Free field realizations, so widely used in the early works on string theory (see e.g. Ref. 1) and references therein), have naturally found their way into the study of two dimensional conformal field theories. 2>-s> Their power is illustrated in the work of Dotsenko and Fateev4 >,s> who, using ideas of Feigin and Fuchs, 2>managed to compute the correlators of the Virasoro minimal models. 6> A complete free field description of a conformal field theory has three major ingredients : i) a realization of the chiral algebra Jl by free fields, ii) a projection from the free field Fock spaces to the irreducible representations of Jl ("null-state decoupling"), iii) a realization of the chiral vertex operators. Although these ingredients are implicitly present in the work of Dotsenko and Fateev, the full underlying structure has only been realized and appreciated recently through the work of Felder. 7 > This work not only shows why the Dotsenko-Fateev prescription works, but also immediately suggests how to compute the higher genus conformal blocks. 8 >-IZ> The structure revealed by Felder in the case of the Virasoro minimal models is expected to exist for generic 2D conformal field theories ; i.e. the projection can be enforced by a "ERST-like" procedure, and the chiral vertex operators can be naturally realized as "ERST-invariant" operators. Specifically, the projection to the irreducible representations LA of Jl can be achieved by a "two-sided resolution" of LA in terms of Fock space modules FAu>, i.e. a complex (FA, d)={(FAu>, d<i>), iEZ}, with a differential d<i> that intertwines with Jl (1·1) and such that the cohomology of this complex is given by Hd(i)(FA)={LA 0 if i=O otherwise. 0 (1·2) Second, the chiral vertex operators are realized as equivalence classes of chain maps V(z):(FA, d)-+(FA', d'), i.e. a collection of maps v<i): FA<i>-+FW satisfying d'u>v<i) = v<•+l>du>, iEZ, Free Field Approach to 2-Dimensional Conformal Field Theories ···~ FA(- 1 ) .1- ... d(-1) ~ FA(O) .1- v<-1> v<•> d'(-1) ~ F1-; 1 > ~ d(O) ~ F1g> FA( 1 ) .1- v(l> d'(O) ~ d(1) ~ FA(Z) .1- ~··· v<2> d'(l) F1V ~ 69 F1~> ~ ... (1· 3) modulo trivial chain maps v<il=d'u- 1>j<il+ jU+ 1>du> where f={fu>: FAu>~ F1~- 1 >}. A particularly useful tool that will be used throughout the paper is the algebraic Lefschetz theorem (1·4) for a chain map {Cl<il: FA<il~FAUl} such that o<o>ILA=O. An important parallel development was the discovery of free field realizations for more general chiral algebras, such as affine Kac-Moody algebras/ 3 >- 16 >parafermion algebras/ 7 >-21 >CW-algebras. 22 > Proposals of free field parametrizations that could be relevant to other coset conformal field theories have also been made. 18 > For other related papers see Refs. 23)---....35). The corresponding resolutions in terms of those free field Fock spaces were constructed in Refs. 36)---....40) for affine Kac-Moody algebras, Ref. 37) for the CWalgebras (through the quantum Hamiltonian reduction 41 >-43 >), in Refs. 44) and 45) for parafermion algebras and in Ref. 44) for generic coset conformal field theories. The investigation of chiral vertex operators in this context, i.e. of the chain maps between two resolutions, was initiated in Refs. 36) and 46). The above formulation puts the free field approach to 2D conformal field theories in a convenient setting, namely that of homological algebra.47l' 48 > Here, in general, one tries to deal with a module over a ring fR by "replacing" the module by a resolution in terms of modules with certain suitable properties. A relevant example in our context is the Bernstein-Gel'fand-Gel'fand (BGG) 49 > resolution of g-modules in terms of Verma modules. Verma modules have the special property that they are free over CU(n-) on a single generator, the highest weight vector (in fact they are uniquely characterized by this property). However, the relevant Fock space modules of chiral algebras Jl are neither isomorphic to a Verma module nor to a dual Verma module, but rather "intermediate" between a Verma module and a dual Verma module. This has the important consequence that the corresponding resolutions are two-sided. The proofs of standard textbook theorems on homological algebra, which all assume one-sided resolutions, must then be suitably modified-in particular, the use of induction techniques has to be adapted. Such developments may be critical in supplying a rigorous formulation of the work described here. This paper is in large part a review of our work, 39 >' 40 >' 44 >' 46 >but there is a significant fraction which is new. In particular the proofs of several results, as promised in Refs. 44) and 46), are included in the relevant sections. Throughout, we will use an instructive finite dimensional analogue-the free oscillator approach to finite dimensional Lie algebras-as a guide. Therefore, we will start (in § 2) by developing the concept of free field realizations, resolutions, coset models and vertex operators in this finite dimensional context. We will not review the geometric interpretation of the 70 P. Bouwknegt, ]. McCarthy and K. Pilch finite dimensional algebra realization, but present instead a completely algebraic approach. We include a discussion of the "twisted Verma modules" of Feigin and Frenkel, 50>which are in some respects the closest finite dimensional analogues, since their corresponding resolutions turn out to be two-sided as well. In § 3 we will treat the infinite dimensional case along the same lines. Specifically, the results that we will discuss in § 3 include: i) For any Kac-Moody algebra fj, at arbitrary level k, a free field realization may be constructed from t (=rank g) scalar fields ¢;(z), i=1, ···, t and ILI+I pairs of conjugate spin (1, 0) bosonic first order fields (/3a(z), ra(z)), aE.d+. The Sugawara energy momentum tensor in this realization reduces to the free field tensor for !3r¢, with an additional background charge (Feigin-Fuchs) term for the scalar fields ¢;(z). ii) There are screening operators s;(z), of conformal dimension one, whose operator products with the Kac-Moody currents contain at most a total derivative singular term. Thus, to construct intertwiners with the full Kac-Moody algebra we look at the class of integrated products of screening currents. Since the operator product algebra of screening currents is nonlocal this is a nontrivial problem. In fact there exists a set of "basis contours" within which products of screenings obey the algebra of the positive root generators of the quantum group CfJ q(g ), q =exp(i7r/(k+hv)). It follows then quite easily that given any singular vector in an associated quantum group Verma module we can construct an intertwiner between two Fock space modules. iii) For A an integrable {j-weight, there is a complex of Fock modules (1·1), where each FA <iJ is the direct sum of an (in general) infinite set of Fock space modules F w*A characterized by affine W eyl group elements w of a certain "twisted length" lr(w)=i. It is conjectured that this complex (FA, d) provides a resolution of the irreducible highest weight module LA. iv) For integrable {j-weight A;,, i=1, 2, 3, the chiral vertex operators mapping LA, to LA., transforming according to LA,, may be realized as chain maps between the resolutions (FA,, d) and (FA., d'). Clearly each v<o is itself a collection { Vw',w: Fw*A•-+Fw'*A•, lr(w)=lr(w')=i}. The components Vw',w are built with integrated screenings, and can be identified with elements of the quantum group Verma modules. We have proven that a chain map exists iff there is v<o> and v< 1>such that d'(O) v<o)= vn>d(O). We conjecture-as can be proven in the finite dimensional analogue problem-that the dimension of the vector space of chain maps modulo trivial chain maps is precisely given by the fusion rules of the WZNWmodels. v) We have also begun to extend the above results i)"""'iv) to G/H coset models. 50 Of course, since the chiral algebra is not known in general for these models, we must take a more "extrinsic" approach. As a first step, we have shown that for any irreducible highest weight module LA of fj, and LA' of ii, there exists a subcomplex of the resolution (FA, d) which provides a resolution of the coset module whose character is the appropriate branching function. The subcomplex is just the restriction of (FA, d) to the subspaces of fi-singular vectors-i.e. those vectors annihilated by the generators in ii+ H-as may be enforced by a "usual" Free Field Approach to 2-Dimensional Conformal Field Theories 71 BRST procedure. That the quantum group CU q(g) plays an important role should perhaps not be surprising. In fact, it has been conjectured that the problem of classification of rational conformal field theories is intimately related to the theory of quantum groups. 52 >-ss> For example, the braiding matrices corresponding to exchange of chiral vertex operators56 >' 57 > are related to quantum group 6j-symbols. It has also been suggested that the fusion rules are related to truncated tensor product rules of irreducible quantum group representations, 55 > as is known to be true empirically for (2). 58> The apparent relation of conformal field theories to quantum group theory has not been easy to understand. We feel that the results outlined above indicate that the free field approach to 2D conformal field theory is a natural arena to address this issue. Unfortunately, lack of space and time has forced us to restrict the discussions in this paper mainly to our own work on this subject. For closely related interesting developments the reader may want to consult for example Refs. 59)""-'66). su 1.1. Notations Throughout the paper we will use the following notations (see, e.g. Ref. 67)): g a semi -simple Lie algebra t a Cartan subalgebra with dual t* CU( ·) the universal enveloping algebra functor g=n-ffitffin+ a Cartan (triangular) decomposition b±= n±ffit the two Borel sub algebras G, T, N±, B± the corresponding groups .t the rank of g L1± a system of positive/negative roots av=(2a)/(a, a) the co-roots M = Z · Lh v the co-root lattice of g W the Weyl group of g ra the reflection in the root aEL1+ r, the reflection in a simple root a;, i = 1, · · ·, .t (,) the bilinear form on t or t*, sometimes also denoted by · p the element of t* such that (p, a/)=1, i=1, ···, .t co*tl=w(tl+p)-p for wE W, t\Et* a shifted Weyl group action Z+={O, 1, 2, ···} h v the dual Coxeter number of g P, P+ the set of integral, and integral dominant weights, respectively C[t] the set of polynomials in the variable t with coefficients in C. Throughout this paper we will use the Chevalley basis of g, which we recall is defined by the following commutators : i, j=1, ···, .t (1·5) 72 P. Bouwknegt,]. McCarthy and K. Pilch as well as the Chevalley-Serre relations for ai.isO, (1·6) where ai.i=(ai, a;v) is the Cartan matrix of g. For convenience we will also sometimes use the notations ea for the generator corresponding to a root a ELl, and Ia= e-a for aE.L/+. We fix a normalization of (,) such that (8, 8)=2 for the highest root and put d;= -Ha;, a;) such that the matrix d;ai.i is symmetric. We will distinguish between quantities of the affine Kac-Moody algebra fj and its underlying finite dimensional Lie algebra g by putting hats on the former. Furthermore we will implicitly identify the affine Weyl group W of fj with its projection Waff onto t*. Also we will identify an affine weight AEP with its components (Jf, k) in tEe Cc, and use the same notation for quantum group weights through the correspondence q=exp{in/(k+ hv)}. e § 2. 2.1. Fock space realizations and resolutions for finite dimensional Lie algebras Fock space realizations A finite dimensional analogue of the free field realizations of affine Kac-Moody algebras that we are interested in, is provided by realizations of finite dimensional (semi-simple) Lie algebras in terms of differential operators on polynomial spaces. These arise naturally from the group action on (local holomorphic) sections of a line bundle over the flag manifold cc /B, where B is a Borel subgroup of the complexified group cc. This is known as Borel-Weil theory, 68 > and has previously entered physics in discussions of coherent states (see Ref. 69) and references therein). The algebra is then realized in terms of differential operators of degree one on the space of polynomials in Za, aE.LI+/0> (see also Ref. 71) and references therein), giving a description naturally isomorphic to a dual Verma module (see e.g. Ref. 40) for more details). In this section we will instead discuss these free field realizations in the closely related algebraic setting of Verma modules, thereby of course losing track of the global properties of the Borel-Weil theory. To make the transition to the infinite dimensional case as natural as possible we will give the realization in terms of a set of bosonic oscillators ya, {r, aE.L/+, satisfying [ra, pa']=oaa' on a Fock space built on a vacuum lA> satisfying .BaiA>=O. One may, of course, always keep the specific realization r=z, .8= -(o/oz) in mind. To keep track of the highest weight A we will use coordinate momentum pairs (pi, q i) with commutators [pi, qi] = - ioi.i such that P;IA>=A;IA>, where A' are the components of A with respect to some orthonormal basis. In particular we will have translation operators TAA' =exp(i(A'- A)· q) such that lA'>= TAA'IA>. Now suppose we want to obtain a realization of g on such a Fock space and such Free Field Approach to 2-Dimensional Conformal Field Theories 73 that the resulting module is isomorphic to a Verma module MA of highest weight A. We proceed as follows. By the Poincare-Birkhoff-Witt (PBW) theorem72 > we choose a certain basis of MA =CU(n-)vA. Then, through the replacement Ia~ ra we identify every basis element of MA with a Fock space monomial (i.e. we choose a map from CU(n-) to the symmetric algebra S (n_) on n-). The action of g on MA can be represented by an action of g on these Fock space monomials in terms of /3roscillators. This leads to the required free field realization, as is perhaps best illustrated in an example : Example 2.1. Let g=su(3). We choose the PEW-basis of MA as follows (/3= Ia.) (2·1) Using the commutators [A lz]=- /3, [/I, /3]=0, [/z, /3]=0, we find (2·2) and with a little more work, e.g. hiVAp,q,r=( -2p-q+ r+(A, ai))vAp,q,r, eiVAp,q,r =((r- q )p- p(p-1)+ p(A, ai))VAP-I,q,r- QVAp,q-I,r+I. (2·3) Upon identifying VAp,q,r with the monomial (ri)P(y 3)q(r2YIA>, we find the following realization of g ei = -(ai· p)f3I-( ri 13 I_ r2 132+ r3 /33)/3I + r2 133, e2= -(a2· P)/32_ rz /32 /32_ ri /33, e3= _ (ai. p)/33- (a2•p)(/33 +pi /32) _ ( ri pi+ r2 132+ r3 /33)/33 _ r2 pi 132132, hi= ai. p+(2ri /3I_ f /32+ r3 /33)' h2= az· p+(- ri pi +2r2 /32+ r3 /33)' !I=ri, /2= r 2 - r 3/3I, /3=r3 . (2·4) Obviously, for generic g, the generators of the Cartan subalgebra in this realization are independent of the choice of PEW-basis. Indeed, (2·5) The form of the other generators of course does depend on the specific choice of basis and although every choice of PEW-basis of MA would suffice, the following choice 74 P. Bouwknegt, ]. McCarthy and K. Pilch seems to be a convenient one. Choose a presentation of the longest Weyl group element Wo of g in terms of simple reflections (2·6) and take the basis (2·7) Let us illustrate this choice in several examples. Example 2.2. For su(n) we could take (2·8) such that (2·9) where li.i denotes Ia,; for the root ai.i= E;- EJ. This gives rise to the following realization (for the sake of simplicity we only give the simple root generators) e;=- (a;. p)pu+l_ "2J rji/3ji+l + "2J j<i ri+lj pi.i- ("'2, ri.i pi.i- "2J ri+lj!3i+lj)/3ii+l i>t i>i+l i>t+l ' (2·10) Example 2.3. Let g=so(5). Taking wo=r1r2r1r2, where a2 is the short root, leads to the PBW-basis (we use the shorthand notation /122= !a,+2a., etc.) !/!1~f{2d2 8 VA ~( r 1)P( 'Y12 )q( r 122 Y( r 2)8 \A>. (2·11) We find the following realization e1= _ (a1v. p)/31-( r1 13 1+ r12 1312_ r2 /32)/31 + f 1312_ ~ r122p12 1312 , e2= _ (a2v. p)/32- r2 132132_ 2 r1 1312_ 2 r12 13 122, h1 =a1 v. P+(2r1 /31 + r12 /312_ r2 /32)' h2=a2 v. P+2(- r1 /31 + r122 !3122+ r2 /32), /1=r1 , (2·12) For specific applications other choices of basis might be more convenient. For instance, in the discussion of coset models described by some embedding H c G it is convenient to have a realization of n- 8 in terms of oscillators pa, ra for aELh 8 only. This can be achieved by choosing a PBW-basis of CU(n- G) respecting the decomposition CU(n_ G)=CU(n~ 18 )®CU(n- 8 ). Free Field Approach to 2-Dimensional Conformal Field Theories 75 Example 2.4. Consider the diagonal embedding SU(3)cSU(3) X SU(3). We distinguish the groups by superscripts or subscripts in round brackets. The appropriate PEW-basis places the generators of the diagonal subgroup c<D> (1/D>= f/ 1>+ f/ 2>, I = 1, 2, 3) on the left, (2·13) Identifying this with the monomial ( r/)P•( ri)P•( ri)P•( r_/)q•( r.L3)q•( Y.L 2)q•IA (1>, A <2 >> , (2·14) one can find the explicit realization as in the other examples. The fact the f/D> are realized purely in terms of the r/ is manifest by construction since they act freely on (2·13) from the left, not disturbing the perpendicular coordinates. In contrast, note that the subgroup generators corresponding to positive roots, e/D>, will contain both parallel and perpendicular coordinates since their action must be pulled all the way through (2 ·13) onto the highest weight state. As a final comment, we note that the realization obtained by the direct product of that in Example 2.1 is related to the present one by a simple coordinate transformation. What characterizes the above realizations is that they are free over CU(n_) on one generator, the highest weight vector (vacuum) lA>. By taking the Fock space adjoint -i.e. rat=pa, pat=ra, aELI+-as well as relabelling e;~/; in the realizations described above, we obtain another highest weight realization of g on the same Fock space. Clearly this corresponds to a realization of g in terms of differential operators ([3~o/oz) of order one-exactly that on sections of a line bundle over cc /B alluded to at the beginning of this section. One easily observes that this realization is co-free over CU(n+) on one generator, hence it is isomorphic to the dual Verma module. In fact, there exists a whole spectrum of modules which are intermediate between a Verma module and a dual Verma module in the sense that they are free over CU(m-), co-free over CU(m+) where m= m+ffim- is a nilpotent algebra isomorphic to n+. These modules were introduced by Feigin and Frenkel, 50> where they were called "twisted Verma modules". In a lot of respects they resemble the Fock space modules of affine Kac-Moody algebras which we will introduce later, so we will discuss them separately in § 2.4. 2.2. Intertwiners The aim of this section is to discuss intertwiners between Verma modules. Suppose we have two Verma modules MA and MA', and we want to determine the set Homvcu>(MA, MA'). We will do this in several steps. Requiring the invariance under CU(n_) leads to the following lemma: Lemma 2.1. Given A, A'E t*, there exists a 1-1 correspondence between CU(n_) and Homvcn-l(MA, MA'). Proof Since Verma modules are free over CU(n-) on one generator VA (the highest weight vector), the maps ¢EHomvcn-lMA, MA') are completely determined by their action on VA. Explicitly, suppose ¢EHomvcn-lMA, MA') and that P. Bouwknegt, J. McCarthy and K. Pilch 76 (2·15) ¢(vA)=xvA' for some xECU(n-), then if>(YVA)=yif>(VA)=yXVA', \;:fyECU(n-). (2·16) Conversely, every xECU(n_) determines a map ¢EHomu<niMA, MA') by (2·16). Suppose we define a representation p of CU(n_) on MA by right multiplication i=1, ···, f' (2·17) and a "translation operator" TAA' : MA ~ MA' by TAA'(yvA)=yvA', (2·18) then the above reasoning shows that there exists a 1-1 correspondence between CU (n-) and Homv<niMA, MA') given by xH p(x)Tf'. (2·19) 0 Of course, not all xECU(n-) will give rise to an element ¢EHomv<ulMA, MA') through (2·19). The condition that the map ¢should intertwine with CU(t) is however easily incorporated-it just requires ¢ to preserve isospin. This precisely means that we should restrict ourselves to elements in CU(n_) of isospin ..l=A'- A, and thus every p(j;) should be accompanied by a change in highest weight A~ A'= A+ a;. Clearly, at this point it is convenient to introduce the operators (2·20) Then every polynomial in the s;'s will give rise to a map that intertwines with CU(t)EBCU(n-). Because of their role in the infinite dimensional case we will refer to the operators s; as the screening operators of g. It is obvious, but worth noting, that the screening operators generate an algebra isomorphic to n-. Requiring finally the invariance under CU(n+) leads to 2.2. There is a 1-1 correspondence between Homv<u>(MA, MA') and singular vectors in MA' of weight A. Moreover, every such intertwiner is injective, i.e. defines an embedding MA "-+ MA'· THEOREM Proof Requiring that ¢ of (2·16) intertwines with CU(n+) simply amounts to the requirement (2·21) i.e. XVA' should be a singular vector in MA' of weight A. Injectivity follows immedi0 ately from (2 ·16). Now let us make the above discussion of intertwiners somewhat more concrete by making explicit use of the Fock space realizations of Verma modules introduced in § 2.1. In the Fock space realization of MA we have of course an explicit representation of TAA' in terms of the coordinate-momentum operators (p, q) as 77 Free Field Approach to 2-Dimensional Conformal Field Theories (2·22) TAA'=ei<A'-AH, while the operators p(f;) are represented by polynomials in /:1 and abuse of notation we will denote these also by p(/;)). Example 2.5. MA for su(3). r (with a slight Let us determine the operators p(f;) in the Fock space realization of We use the same notation as in Example 2.1. We have e.g. p(f1)v~·q,r=- f/fsqf2r/1VA = -(f/+1fsqf2r + rf/fsq+l/2r- 1)VA (2·23) and in the same way (2·24) In the Fock space realization these are thus represented by p(/1)= -(r1_ rs /:12), p(/2)=- r2 , p(/s)=-r3, (2·25) which clearly satisfy the commutators of n-, e.g. [p(/1), p(/2)]=-p(/s). Similarly, for the other examples of § 2.1 su(n) (2·26) so(5) p(/1) = _ ( r1_ r12 fF + ~ r122 p2 p2) , p(/2)= -r2 • (2·27) The subset of polynomials in the screening operators that intertwine with the action of CU(g) could in principle be determined by explicit evaluation of commutators. Let us first evaluate the commutator [e,, sJ]. Lemma 2.3. We have - - uij s- (p , [ e;, Sj ] - Proof a; v) e ia,.q . (2·28) By acting on a generic vector YVAEMA we obtain ([e;, sJ)YVA = -e;(yfiVA+a;)-sie;YVA). (2·29) Because [e;, y]ECU(t)EBCU(n_) (we took a simple root generator), the two terms combine into 78 P. Bouwknegt, J. McCarthy and K. Pilch (2·30) which proves the lemma. 0 (Note that the commutator [e;, p(/;)] would have been considerably more difficult to calculate.) Let us now explicitly exhibit the prototype of intertwining operator. Suppose A EP is such that (A, a;v)EZ+, then we claim that s;<A+p,a,v>EHomucg>CMn*A, MA). The proof is a straightforward calculation. Let vEMnM then [e;, S;(A+p,a,v>]v=- ~ O:<:j:<:(A, a,') slA,a,V)-J(p, a;V)eia,.qs/v =0. (2. 31) Note that in the context of Theorem 2.2 the example above corresponds to the singular vector j/A+p,a,v>vA of weight r; *A in MA. The other intertwiners can, in principle, be found by making use of the basic commutator (2·28) to obtain (on MA) (2·32) where aj=a;;,+l +··· +a;;n and -denotes omission. An alternative way to find explicit formulas for the intertwiners that correspond to reflections in other than simple roots is to make repeated use of Lemma B.1 (for q =1) (see Example 2.7). The singular vector structure of MA was investigated in detail by BernsteinGel'fand-Gel'fand.73' Let us recall their results as far as relevant for the construction of the resolution of an irreducible highest weight module for dominant integral A. Recall thereto that the length l(w) of a Weyl group element wE W is defined as the minimal number of reflections r; in simple roots a; such that w = r., ... r,k (see also Appendix A). Then define the following partial ordering, the so-called Bruhat ordering, on W. For w', w" E W we write w'--> w" if w' =raw" for some aELh as well as l(w')=l(w")+l. Let w'sw" iff there exist W1, ... , whEW such that w'->w1_.. .. --> Wk-> w". THEOREM 2.4. 73 > For AEP+ we have (2·33) Moreover, for w s w', every such intertwiner is a multiple of the canonical embedding lw',w: Mw*A ~Mw'*A· Example 2.6. For g=su(3) and AEP+ the theorem above gives rise to the following directed graph of submodules of MA, and intertwiners between them (see also Ref. 74)) Free Field Approach to 2-Dimensional Conformal Field Theories /' Mnr•*A --+ Mrt*A '\. MA . ~ Mr1r2rz*A 79 '\. /' Mr•rt*A --+ Mr.*A (2·34) Due to the uniqueness of the intertwiners the diagram is commutative (up to proportionality) and every other intertwiner between modules of the form Mw*A is a composite of the ones depicted in this diagram (see Refs. 75) and 71) for an elucidation of this fact*>). The Verma modules Mw*A occurring along a vertical line in this diagram correspond to Weyl group elements w of fixed length, while between consecutive vertical lines the length differs by one. Example 2. 7. Consider the intertwiners of Example 2.6. Denote l;=(A + p, a;), z = 1, 2, 3, then we have already seen that ( ¢w',w : Mw*A--+ Mw'*A) (2·35) Lemma B.l applied to A=s1, B=s2 (for q=l) now gives (2·36) where s3=[s2, sd and b(m, n; j) m!n! j!(m- j) !(n-j)! (2·37) It follows that (2·38) Similarly we have (2·39) 2.3. The EGG-resolution In this section we will discuss how to obtain resolutions of the irreducible modules LA (for A dominant integral) in terms of Verma modules MA, the so-called BernsteinGel'fand-Gel'fand resolutions, by suitably combining the intertwiners of § 2.2. Define thereto (2·40) We have *l We thank V Dobrev for sending his papers74 >'75 >' 71 > and correspondence on this issue. 76 > 80 P. Bouwknegt, ]. McCarthy and K. Pilch 2.5. 49 ) For AEP+ we have a resolution of LA in terms of Verma modules, i.e. a complex (MA, d) THEOREM (2·41) with cohomology H~il(MA)~{Lo A for i=O otherwise, (2·42) where t=IL1+I=l(wo), M1il is given by (2·40). Proof (sketch) Consider the collection of modules MA ={Mw*AiwE W}. According to Theorem 2.4 there exist intertwiners cf>w.,w, : Mw,M ~ Mwz*A for WI~ Wz, and they are unique up to a multiplicative constant. Every other intertwiner within MA is a composite of the "elementary intertwiners" c/>w,,w. for WI~ wz. To get control over the "composite intertwiners" we need some results regarding the structure of the Weyl group. 49 l The first result is that given WI, wzE W such that l(wi)= /(wz)+2, the number of elements w'E W such that WI~ w' ~ Wz is either zero or two. Let us call the quadruple (wi, w', w", wz) a square if there are two elements w' =I= w" E W such that WI~ w' ~ Wz and WI~ w" ~ Wz. The idea is now to construct the differential d of a complex by combining the intertwiners {cf>w.,w., WI~ wz} and forcing d 2 =0 by cancellation of the maps within a square of the Weyl group. This requires the following result: To each arrow WI ~ W2 we can assign a number s(wi, w2)= ±1 such that for every square (wi, w2, Wg, w4) the product of numbers assigned to the four arrows occurring in it is equal to -1. Now, recalling that every intertwiner cf>w.,w, is a multiple of the canonical embedding tw.,w, we can define an intertwiner d<il: M1il~ M1i+Il by its components l(wi)=-i, l(w2)=-(i+1) (2·43) The above mentioned results ensure that d<i+IldUl=O. The proof of (2·42), which we will not reproduce here, can be given directly 49 l or, more easily, by establishing an isomorphism to the so-called weak resolution.77l D Let us illustrate the usefulness of this resolution by giving a derivation of the Weyl character formula for an irreducible highest weight module with dominant integral highest weight A. Using the algebraic Lefschetz theorem (1·4) we obtain = ~ ~ WEW (- 1)t<wl TrM... e2>riB·h= ~ ~ WEW 21ri8•(W*A) (- 1)t<wl II e (1 e 21rr8•a) . . aeA+ - Using the Kostant partition function K( · ), defined by (2·44) Free Field Approach to 2-Dimensional Conformal Field Theories 81 the Weyl character formula immediately leads to Kostant's formula for mA(A), the degeneracy of a weight A in the irreducible representation LA, (2·46) Twisted Verma modules, realizations and resolutions Twisted Verma modules were introduced by Feigin and Frenkel 50l as slightly better finite dimensional analogues of the Fock space modules (Wakimoto modules) of affine Kac-Moody algebras to be discussed later. In this section we will discuss their free field realizations, intertwiners and resolutions. The discussion closely parallels the one of the untwisted case as given in the previous sections. Instead of defining twisted Verma modules through their cohomological properties50l we will define a twisted Verma module MAw of highest weight A, for any given Weyl group element wE W, by the following properties: i) MAW is free overCU(n- wn n-) and co-free overCU(n- wn n+), where n- w= w· n-· w- 1• 2.4. ii) (2·47) The second property is essentially the statement that MAw is generated by one vector, namely the highest weight vector VA. Note that with these conventions MA 1 ~ MA and M:t 0 ~(MA)*, where Wo again denotes the longest Weyl group element. For convenience we will also introduce the notation FAw=M:two (FA 1 =FA), and refer to these modules as twisted Fock space modules. Realizations of these modules on polynomial spaces are obtained as follows. First, we construct a module which is free over CU(n- w) by Weyl rotation of the Verma module. Thus, choose the vector VwA satisfying n+ wVwA =0 and build the g-module CU(n- w)VwA. Identify the elements in a PEW-basis with monomials in rwa, aELh through the replacement Ia w= w · Ia • w- 1 ~ rwa. Then determine the induced action on these Fock space monomials, replacing wA....., pw in the result. (In practice, since all n- w are isomorphic we obtain the results by renaming the realization computed for one case, say w=l.) Of course, this is not yet a realization with highest weight. To achieve that we simply replace rwaH /3-wa, /3waH- r-wa for aE w- 1(LL) n Lh, and also shift pw~p=pw+(wp-p). Now consider the new result acting on the Fock space generated by the ra's on the vacuum lA> such that P;IA>=kiA>. As a consequence of the replacements above this is a realization on a g-module with highest weight lA> which is no longer free over CU(n- w), but instead free over CU(n- wn n-) and co-free over CU(n- wn n+). The shift pw ~ p is required so that the highest weight of this module will be equal to A. To see this explicitly, note that on CU(n_ w)VwA we have (2·48) After changing rwa ~ /3-wa, /3wa ~- r-wa, h;=a;V•PW- ~ aew-•(4-)n4+ vaE w- (LL) n Ll+ we have (wa, a/)/3-war-wa+ 1 ~ aew-•(4+)n4+ (wa, a;Y)rwa/3wa 82 P. Bouwknegt, ]. McCarthy and K. Pilch (2·49) where we used78 > L: aew(Ll+)nil- a=wp-p. (2·50) Thus, to make it into a realization with highest weight A, we simply have to shift as above-of course not only in the generators h,, but also in the generators of n- w. Note that after this procedure the realization of h becomes independent of the particular twist w. Thus the character of the constructed module is clearly given by (2·47), so we have indeed constructed a realization of MAw as required. Example 2.8. Let us illustrate the above for the twisted module MJi'r =su(3). We have nr 2 r'={e3, A e2}, and thus (see Example 2.1) 2 ~ FJ,t of g e1 = -(a1. p)(:J1- r1 (:Jl (:11 + r2 (:13, /1 = r1 + r3 (:12 , J2= _ (a2•P)r2+ (a1•p)( r2 + r3 (:11) + r2( _ r1 p1- r2 (:12- r3 (:13) + r1r3 p1 p1 , Ia= -(a3· P)r3_ r1r2_ r3(r1 (:11+ r2fJ2+ r3fJ3)' (2· 51) while the generators h; are the same as in Example 2.1. The construction of the intertwiners between twisted Verma modules MAw again proceeds in several steps. First, using exactly the arguments of the untwisted case, one establishes a 1-1 correspondence between Homv<n-•)(CU(n- w)VwA, CU(n- w)VwA') and CU(n- w). The intertwining properties with CU(t) then imply that every such intertwiner is a polynomial in the screening operators (2·52) To go to the free field realization of intertwiners between twisted modules we now make the replacements ywa H p-wa, (:Jwa H - y-wa for aE W- 1(L1-) n L/+ and pw----+p= pw +(wp-p). Then with the same notation after replacement, we have (2·53) which commute with n- wffit by construction, and further, [e;w, Sjw]=-8.:;(p-(wp-p), wa;V)e'wa,.q. (2·54) The proof of (2·54) parallels that given for the Verma module case, up to the replacements described above. The replacements can easily be effected as a similarity transformation between operators, thus it is clear from the above discussion that all intertwiners between twisted modules may be found as polynomials in the s,w. In fact, we have Free Field Approach to 2-Dimensional Conformal Field Theories 2.6. There exists an Homv(u)(Mw-I*A, Mw-I*A'). THEOREM isomorphism 83 between Homv<ulMAw, M:f,) and Proof Through the similarity transformation on the screening operators that compose the intertwiners Homv(u)(CU(n- w)VwA, CU(n- w)VwA') and similar reasoning as in the untwisted case (see Theorem 2.2), one establishes a 1-1 correspondence between Homv<ulMAw, M:f,) and n+ w-singular vectors in CU(n- w)VA'-<wP-P> of weight A -(wp- p) (recall the shift in A!). These, in turn, are in 1-1 correspondence with n+-singular vectors in the Verma module CU(n-)vw-l*A' of weight w- 1 *A by a Weyl reflection. D Again some intertwiners are easily constructed explicitly. We claim that for m =(A+ p, wa/)EZ+ we have (s,w)mEHom 'U(u)(M J!:,.,M, MAw). Clearly (s,w)m maps MJ!:,.,M to MAw since (2·55) so the statement follows from =0. (2·56) To formulate the analogue of Theorem 2.4 we introduce a "twisted length lw" on Wby (2·57) where the first term on the r.h.s. is suggested by Theorem 2.6 above-in particular, the elementary intertwiners (s,w)m decrease the twisted length by one unit. The second term is chosen to insure the "normalization" lw(1)=0, V wE W. Note in particular that Mw)=l(w) and lw(w)= -l(w). Moreover, it follows from woLI±=Lh: and Lemma A.2 that lwwo(w')= -lw(w'), thus relating twisted lengths relevant for twisted Verma modules and twisted dual Verma modules. We proceed with the definition of a "twisted Bruhat ordering" on W. First define w'~ww" iff there exists an aEL/+ such that w'=rwaw" as well as lw(w') =lw(w")+l. Then define w'sww" iff there exist W1, ···, WkE W such that w'->wwl~w ···~wwk~ww". Then noting that w'sw" iff w- 1 w's_w- 1 w" we find THEOREM 2.7. For AEP+ we have if w'sww" otherwise. (2·58) We should emphasize however that, contrary to the untwisted case, the inter- 84 P. Bouwknegt, ]. McCarthy and K. Pilch twiners that exist for w' s ww" are no longer injective as a consequence of interchanging the role of ra and /3a for various roots a. Example 2.9. As an example we take g=su(3), AEP+ and a twisting by w= rzr1 (see also Example 2.8). We find the following directed graph of intertwiners MJ.•n MT2T! rt*A ~ ? Mr2r1 r2*A \, X: \, ? Mi~~!*A ~ M~~~i*A' (2·59) M~~~!r1*A where again the diagram is commutative up to proportionality and every intertwiner between modules of the form M:;,•n is proportional to a composite of the ones depicted above. The twisted Verma modules M:;,•,n occurring along a vertical line all have the same twisted length lr.r/w) and the length differs by one unit for consecutive vertical lines. Again, the intertwiners corresponding to reflections in other than simple roots can be found by making repeated use of Lemma B.1, which simply amounts to replacing s, by s,w in the expressions for the intertwiners in the untwisted case. Finally, a resolution of LA for AEP+ in terms of twisted Verma modules MAw is obtained by combining the intertwiners of Theorem 2.7 exactly as in the untwisted case. Let (2·60) We have 2.8. For AEP+ we have a resolution of LA zn terms of twisted Verma modules, i.e. a complex (MAw, d) THEOREM (2·61) with if i=O otherwise, (2·62) Proof By using the fact that W1 ~ wWz iff w- 1W1-> w- 1wz we directly obtain from the previous section that for lw( w1) = lw( wz) + 2 the number of elements w' E W such that W1 .... ww' .... wWz is either zero or two. In case there are two such elements, w' and w" say, let us call the quadruple (w1, w', w", wz) a w-square. To every arrow W1 ~ wWz we can now assign a number sw( W1, wz) = ± 1 such that for every w-square the product assigned to the four arrows occurring in it is equal to -1, simply by choosing sw( w1, wz)=s(w- 1w1, w- 1wz), where the s(·, ·)are given in§ 2.3. Now for W1->wwz let Free Field Approach to 2-Dimensional Conformal Field Theories 85 Zw-•w,,w-•w, be the canonical embedding of Mw-•w•*A into Mw-•w•*A written in terms of screening operators s,. We have seen that by replacing s, by s,w we obtain the corresponding element r/Jw,,w, in Homv<u>(M~•*A, M:g•*A). Now choose lw(WI)=- i, lw(wz)= -(i + 1) (2·63) Then d<•+I>d<i>=o. Given the weak resolution in terms of twisted Verma modules 50> the cohomology (2 · 62) can be proved by establishing an isomorphism of the weak resolution to (2·61), exactly as in the untwisted case (see Ref. 77) for details). 0 Finite dimensional coset models In this section we will discuss finite dimensional G/H coset theories, which simply correspond to the decomposition of finite dimensional irreducible representations of a semisimple Lie group G with respect to its subgroup H. The goal is to illustrate how the free field resolutions introduced in § 2.4 can be used to calculate the branching rule multiplicities for finite dimensional algebras. Our approach is an extension of a rather well-established method for computing these multiplicities (as explained e.g. in Ref. 79), Ch. 18), and provides a cohomological interpretation for the corresponding "branching function" formula. 80 > Suppose G is a semisimple Lie group and LAG a finite dimensional irrep with highest weight A. We assume that the embedding of H in G is regular, and choose the triangular decomposition of algebra h=n_HffitHffin+H to agree with that of g, i.e. tH c t G and n± Hc n± G. Denote by Z the centralizer of H in G. As a Z X H module, 2.5. (2·64) where the sum runs over a finite set of dominant integral weights of H. The problem is to determine the multiplicities bA,A'-defined as dimension of the space L~~~,-with which a given H irrep VJ, occurs in the decomposition (2·64). Clearly, L~~~, is the subspace of H-singular states (annihilated by n+ H) in LAG, with the H-weight A'. The following theorem gives a characterization of L~~~, in terms of a (twisted-) resolution of LAG· [Recall that a twisted Fock space FAw corresponds to the twisted Verma module MX'wo.] 2.9. Let (FAw, d) be a resolution of the irrep LAG of G given in Theorem 2.8. For each FX'(i>, denote by S:f.Sil the subsPace of H-singular states with the H-weight A'. Then 1. d: S:f.Si.l-> S:f.Sit 1 >, i.e., (S:f,A', d) is a subcomplex of (FAw, d). 2. The cohomology of the subcomplex (S:f,A', d), with A' a dominant integral weight of H, is THEOREM )- s-z,OLGIH H d<il(Sw A,A' -u A,A'. (2·65) We will often refer to complexes (FAw, d) and (S:f,A', d) simply as the complex and the subcomplex, respectively. 86 P. Bouwknegt, J. McCarthy and K. Pilch In§ 2.4 we saw that an important property required from any resolution was that the cohomology of the corresponding complex was nontrivial only in one dimension. This allowed us for instance to calculate the dimension of LAc using the algebraic Lefschetz theorem. Part 2 of Theorem 2.9 asserts that the same holds for the subcomplex (S~.A', d), which thus provides a resolution of the "coset module" L~~;t•. Proof The first part of the theorem is quite obvious, since the generators of g-and so, in particular those of n+Hcn+c-commute with d. Thus for any H-singular vector¢, n+ H· ¢=0, we haven+ H · d¢=0, and d¢ has the same H-weight as¢. In fact this shows that for any H-weight A' (not necessarily dominant!) the subspace S~.~~ is mapped by d into S~.~t 1 l, or, equivalently, that (S~.A', d) is a subcomplex. Also, we have s~.~~= {W'E EB s:g.*A,A' , Wllw(W')=i} (2·66) where s:g'*A,A' is the subspace of H-singular states with the H-weight A' in the twisted Fock space F:g'*A· A priori, unlike in the case of the complex, the cohomology of the subcomplex need not be concentrated in the 0-th dimension. For, although it is true that any closed element tj;ES~.~~. d¢=0, of the subcomplex is also a closed element of the complex, its cohomology class in the subcomplex may be nontrivial-even if it is trivial in the complex. That is, even if tf;=dx, xEF:t<•-1), we may not be able to find fES~.~;-ll such that tf;=di. But, before we prove part 2 of the theorem in general, let us consider an example. Example 2.10. rank H=rank G, H=SU(2)X U(l)N, A'=(ja, a) We will show that for j-::::.0 the cohomology of the subcomplex can be nontrivial only in the 0-th dimension. Let (e, h, /) denotes the standard basis in su(2)ch. Thereto consider ¢ES~.~~ whose cohomology class in the complex is trivial, i.e. tf;=dx and e¢=0. The question is whether one can find a deformation ox such that dox=O and i=x+ox is H-singular. Assume thus ex=#=O and observe that, since eEn+H C n+ c has positive G-weight and the set of weights in F:t<il is bounded from above, there exists a smallest n"2::.1 such that enx*O and en+lx=O. For such x we can construct ox as follows : Consider a one parameter family of vectors of the form xr(t)= x+ t/ex, tER. Clearly, dxr(t)=¢ because d/ex= /e¢=0. Moreover, (2·67) where we used the identity [en,!]=nen- 1(h+2), and hx=2jx. We see that for j-::::.0 one can always choose t=tl=-l/2n(j+l), such that enxl(tr)=O. Repeating this process n times we obtain (2·68) This shows that for j-::::.0 the cohomology class of¢ in the subcomplex is nontrivial if and only if it is nontrivial in the complex, thus proving that H~il(S~.A·)CH1P(FAw). 87 Free Field Approach to 2-Dimensional Conformal Field Theories One may note that the r.h.s. of (2·68) coincides with the projection of x onto the eigenspace of the quadratic Casimir operator Cz=(l/2)h(h+2)+2/e of su(2). This suggests a general proof along these lines. To streamline the proof in the general case it is convenient to introduce a notion of projective modules. 47> We say that a module Pis projective in some category of modules if, given any two modules N and N and a surjective morphism 6: N--> N, we can "lift" any homomorphism rp: P--> N to a homomorphism ip : P--> N such that rp = 6° ip, i.e. the following diagram is commutative : (2·69) p -----. N We will extensively use the following standard fact about the category (') of modules. 49 > g 2.10. For any finite dimensional semi-simple Lie algebra g, Verma modules MA, where A is a dominant g-weight, are projective in the category (') of g-modules. THEOREM Proof 81 > Consider arbitrary modules N, NE ('), an epimorphism (i.e. a surjective morphism) 6EHomv(ul(N, N) and a homomorphism rpEHomv<ulMA, N). The image of the highest weight state VA EMA under rp is a singular vector v= rp(vA) in the module Nat weight A, and thus an eigenvector of the universal quadratic Casimir operator CzECU(g) with an eigenvalue cz(A)=(A, A +2p). By diagonalizing Cz on N, we can find a vector fJEN such that 6( v)=v and Cz v =cz(A) v. Consider the set {CU(n+)· v} eN. Note that because the set of weights in N is bounded from above (N is a module in the category (')) this set must have at least one singular vector, v, with the weight A+ ,8, where ,8= z;.;n;a; is a linear combination of simple roots with nonnegative integral coefficients. Clearly, we have Cz v = cz(A) v because v is in the submodule generated by v ; but also, Cz v = cz(A + ,8) v because v is singular. We obtain (A, A +2p)=(A + ,8, A+ ,8+2p) =(A, A +2p)+2z;.n;(p,+a,)+ z;.bi,jn;nJ, Z (2·70) Z,J where p;=(p, a;) >0, a;=(A, a;) are non-negative when A is dominant, while the matrix (bii)=((a;, aJ), related to the Cartan matrix (aii) by bii= -}(a;, a;)ai,j, is positive definite. Thus we see that all n,'s must vanish, i.e. ,8=0, which implies that in fact v = v. Having shown that vis singular, we define ip by setting ip(vA)= v and extending ip to the entire MA so that it is an CU(g)-homomorphism. It is obvious that ip is a sought after lift of rp. 0 We now return to the proof of the theorem. First we will show that H~0 (S'i,A·) We proceed as in the example. Let <P=dx, <f!ES'f(,<j,) and xEF:t<•- 1>. cH~0 (FAw). 88 P. Bouwknegt,]. McCarthy and K. Pilch Consider H-modules Nand N generated by x in Ff<•- 1>and rf; in Ff(i>, respectively. Then d restricted toN maps onto N. Also it is clear that Nand N belong to the category () of H-modules. Since rf; is H-singular, there exists a unique homomorphism cp from the Verma module MA' of H onto N, which, using Theorem 2.10, can be lifted to a homomorphism cp: MA'--+ NcF:<•- 1>. Then i= cp(vA') is H-singular and di=dcp(vA')=cp(vA')=rf;, which shows that cohomology class of rf; vanishes in the subcomplex. To prove the equality in (2·65) we must still show that any H-singular element of LAG has an H-singular representative in Ff<0 >. In general, one can only expect that if vELAG isH-singular then its representative rf; in Ff<0 >, drf;=O, v=[rf;], satisfies earp =dx, aEL1+ 8 , where xEF:f<•- 1>will in general depend on a and rf;. [Of course, this problem does not arise in the case of the untwisted resolution!] Let us now takeN to be an H-module which consists of elements in Ker(d< 0 >) that project onto the H-submodule N in LAG generated by v. One easily verifies that once more we are in a position to use Theorem 2.10 to deduce that there exist ¢ E S~f,<J.l such that [ ¢] = [rf;]. This concludes the proof of Theorem 2.9. D We have shown that the complex (SJ:,A', d) provides a resolution of the coset module L~~;[,. The reader will observe that if His abelian the difficulties discussed above do not arise, because in this case the subcomplex is simply a restriction of the complex to a given H-weight (or better, charge). The cohomology of the subcomplex is then nonvanishing only for i=O and coincides with a subspace of LAG which consists of states with the H-weight A'. Also, it follows from the above remarks that if we restrict the Ff(i>•s to the states with a given H-weight A', not necessarily dominant, we will obtain a subcomplex (F;f.A', d) with a nontrivial cohomology only at i=O. It is only the further restriction of this subcomplex to H-singular subspaces that the dominance of A' becomes important. When His abelian, the spaces SJ:,A' occurring at each step of the resolution are simply a restriction of the free field Fock spaces to a subspace of definite charge. However, when H is nonabelian, a further projection onto H-singular states yields subspaces for which a parametrization in terms of free fields is less obvious. In the present case of finite dimensional coset models there are two methods to accomplish the projection onto the subcomplex of H-singular states within the context of free fields. The first one is to use the well-known BRST techniques, and further extend the complex by introducing ghost oscillators. The second one, which seems to work only for the resolution in terms of non-twisted Fock spaces, is to explicitly solve for the H-singular states, and show that they correspond to free Fock spaces of certain new fields. As we discuss at the end of this section, in the finite dimensional models the two methods are equivalent. However, it seems that only the first method has an infinite dimensional generalization. Thus we will discuss it in more detail now. Introduce a set of IL1+ 8 ! conjugate pairs of free fermionic "ghost" oscillators (c-a, ba), {c-a, bP}=8a.o, a, j3EL1+ 8 • The corresponding Fock space Fgh is generated by the c-a•s acting on the vacuum !O> which is annihilated by all the ba's, ba!O>=O. Of course, for a finite number of ghost pairs, Fgh is finite dimensional. One can introduce a gradation in the ghost Fock space Fgh=E.BnF~~>, where F~~> is the eigenspace of the 89 Free Field Approach to 2-Dimensional Conformal Field Theories ghost number operator Ncb= ~aELI•• c-aba. For an arbitrary H-module M we consider a module M®Fgh which is graded according to the cb-ghost number, (M®F8 h)<n> =M®F~~>, in particular M®F~g>~M. Then we may construct a nilpotent operator, Q: M®F~~>~ M®F~~+l>, (2·71) where ea acts on M®Fgh as ea®l. It is clear that the cohomology H~0 >(M®Fgh) is isomorphic to the space of H-singular vectors in M, in fact Q is the BRST operator associated with the constraints ea=O, aELhH. To see how this works let us consider an example. Example 2.11. h=su(2), M=LJa, j>O. In the above example we have one set of ghosts (c, b) and Fgh is spanned by two states, IO> and ID=ciO>. The BRST operator is simply Q=ce. The closed states are of the form lj>®IO> and LJa®ID. From those only lj>®IO> and 1- j>®ID are not in the image of Q. Thus we find that both H~0 >(Lj®Fgh) and H~1 >(Lj®F8h) are !-dimensional. Note that if we replace LJ by the Fock space F., the second cohomology class "disappears" because the state 1- j) in F., is in the image of e and thus of Q. The idea now is to use Q to project onto the subcomplex of the resolution. Consider first the resolution (FA, d) of an irrep LAG, and (FA,A', d) the subcomplex with an H-weight A'. Given the H-ghost Fock space we may form a double complex (FA,A'®Fgh, d, Q) in which d acts as d®l and Q is given by (2·71). The spaces in this complex are labelled by two degrees, i and n, F~~~,tg;F~~>, where i=O, ···, l(wo) and n=l, ···, 2 LJ+H As discussed in Appendix C, to such a double complex we can associate a single complex 1 1• (KA,A', D), D=Q+(-l)Ncod. (2·72) The main result of this section is 2.11. Let (FA, d) be a non-twisted resolution of an irrep LAG, and A' a dominant H-weight. Then the cohomology of the complex (KA,A', D) associated with the double complex (FA,A'®Fgh, d, Q) is THEOREM (2·73) Proof The proof follows from Theorem C.l in Appendix C and the following technical result which is proved in Appendix D. 2.12. Let FA be a non-twisted Fock sPace G-module with the highest weight A (not necessarily dominant), and SA the sPace of all H-singular states in FA. Then the ERST cohomology of FA considered as an H-module is given by THEOREM P. Bouwknegt, J. McCarthy and K. Pilch 90 (2·74) Observe that in the double complex (FA,A'®Fgh, d, Q) all the columns complexes, labelled by n, are identical and their cohomology simply reproduces the H-weight space L~,A' of LAG· On the other hand the cohomology of the row complexes, labelled by i, using the above theorem reproduces the subspaces of H singular vectors. In the notation of Appendix C we can summarize this as follows : tO.F, )- N,OLGA,A'\CI tO.F,(n) H ·(z,n>(F d A,A'\01 gh - U gh , Vn, (2·75) (2·76) Let us restrict A' to be a dominant H-weight. On introducing the associated single complex (K, D) we can use Theorem C.1 in Appendix C to determine its cohomology. In fact we can compute it in two ways, first with respect to Q and then d, or other way round. The result is that the nontrivial cohomology can only occur in degree leSS than min(l(wo), 214 1), and iS given by •H (2·77) or, equivalently, (2·78) The first result (2·77) proves that the complex (K, d) is a resolution for the coset module L~~;[,. This completes the proof of Theorem 2.11. D For a twisted resolution (FAw, d), we introduce a double complex (FAW.A'®Fgh, d, Q), where the (i, n) space, i=O, ·--, l(wo) and n=1, ···, 2w 1, is equal to FJ,~;-t<w»rg;p~~>. [We shifted the labelling in i by -l(w) to bring the double complex to the first quadrant.] Let (Kf.A', D) be the associated single complex. We then have a generalization of Theorem 2.11. 2.13. For any twisted resolution of LAG and a dominant H-weight A', the cohomology of the single complex (Kf.A', D) associated with the double complex (FX:A' ®Fgh, d, Q) is THEOREM (2·79) Proof First note that the argument leading to (2·77) in the proof of Theorem 2.11 is not valid if we take instead a twisted resolution. The reason is that the cohomology of Q on a twisted Fock space need not be concentrated in a single dimension. However, for the double complex (FAW.A'®Fgh, d, Q) we still have an analogue of (2·75), (2·80) Moreover, (2·78) suggests that if we take another resolution of LAG then the cohomology of the corresponding double complex should not change. In fact using Theorem C.1 we obtain Free Field Approach to 2-Dimensional Conformal Field Theories if p< l(w), 91 (2. 81) and otherwise (2·82) But then comparing with (2·78) and (2·77) we see that (2·82) is nontrivial only when p=l(w) and is equal to the coset module as stated in Theorem 2.13. D In the non-twisted case there is an explicit parametrization of the resolution (SA,A', d) in terms of "perpendicular" Fock spaces. In § 2.1 we explained that there exists a natural parametrization of the non-twisted Fock spaces Fw*A in the resolution of LAG, such that the generators of n+ 8 are realized only in terms of r,,a, !3 11 a, aEL1+ 8 • [Theorem 2.14 below constitutes the general discussion of this parametrization for the case G x G/G.] With this choice all states independent of r,/, aELl+ 8 -i.e. the subspace Fil,M generated on lw *A> by r a, aELJrtH-are manifestly H-singular. In Appendix D we are then able to prove the following sharpening of Theorem 2.12. j_ THEOREM 2.12'. If the generators ea of n+ 8 are realized only in terms of parallel variables ra, !3a, a ELl+ H then (2·83) Thus SA,A' coincides with the subspace with the H-weight A' of the Fock space FA.l_ of perpendicular variables, and (Fl.A', d) is thus a resolution of the coset module L~~;f,. We conclude this discussion with a simple application of the resolution (FA,A', d, Q) and/or (Fl.A', d) of L~~~, to compute multiplicities (branching functions) bA,A'· Example 2.12. rank H=rank G Let us decompose the set of roots of G with respect to Has Ll+ G=Ll+ 8 UL1r 18 , Ll+ 8 nLJrtH=0. The calculation of multiplicities is most easily performed in two steps. First we use the Lefschetz formula in the complex (FA®Fgh, Q), where A is an arbitrary G-weight, to derive the character of SA, the space of all H-singular vectors in FA. Identifying weights of G with those of H, we obtain using (2·47) 21.oi+HI chsA= L: ( -1)n chFA( n=O L: npe-P) {n,=O,ll:2::,e4,.n,= n) (2·84) where KG 1u( ·) is the Kostant partition function on the lattice spanned by the coset roots LJrtH. We read off from (2·84) that 92 P. Bouwknegt, ]. McCarthy and K. Pilch dim SA,A'=KctH(A- A'). (2·85) Then from the resolution (SA,A', d) we obtain for the multiplicity bA,A'=dim L~~X,=dim H~0 >(sA,A') (2·86) which is the branching function formula derived in Ref. 80) in the case of equal ranks and a regular embedding. Observe that we could have also derived (2·85) directly using the realization of SA as a Fock space of perpendicular variables FAl.· We will now show how to choose the "perpendicular" variables for the coset G X G/G, G simple, and explore more carefully the consequences of doing so. As usual we will distinguish between the groups by subscripts or superscripts in round brackets. As discussed in § 2.1 for arbitrary G/H coset, the appropriate PBW -basis of CU(n- G) in the untwisted Verma module realization is that respecting the decomposition CU(n-H)®CU(n9.'H). Specifically for G X G/G we will take the basis (2·87) where Pa, Qa are non-negative integers and the order of the roots in the products is specified as in (2 · 7). Identify this with a Fock space basis via (2·88) One may now derive the realization of the generators in these coordinates via the action on this basis. There are several salient features of the result for arbitrary G which should be emphasized. One is just that the diagonal subgroup generators are realized in terms of parallel variables only-which is of course their raison d'etre -since the action of these generators from the left obviously does not disturb the perpendicular variables. Similarly, since the screening charges correspond to the action of n- from the right, those of c<o are purely expressed in terms of perpendicular variables. Both of these remarks apply immediately to the untwisted Fock space realization obtained by conjugation of the above as described below Example 2.4 in § 2.1. Our final observation is specific to the untwisted Fock space resolution. When acting on states in the subspace F}m,A'"'' the screening charges of G< 2 > can be identified with the negative root generators of c< 1>; i.e. s~2 >=- /J1> on this subspace. This follows from the discussion above since the action of /J2> from the right can be commuted past the V(n9.'H) factor in the PEW-basis until it is to the right of the U(n-H) factor. Here we may write it as /J2 >=(1J1>+ /J2 >)-/J1>. The first term goes to the left and increases the degree of the subgroup generators in the basis, giving terms with non zero powers of r11 in the realization. The second term can be taken to the right, and of course is just the action of - /J1> on purely perp states. On conjugation the first set of terms have non-zero powers of /3 11 and thus vanish on the subspace Ff"',A'"'• leaving the Free Field Approach to 2-Dimensional Conformal Field Theories 93 desired result The following criterium is a consequence of our discussion. THEOREM LA/~LA. 2.14. 79 > Let LA 1 and LA. be two irreps of G. The degeneracy of LA. in is equal to the number of independent solution to the equations (2·89) i=1, ···, f, where vELA1 has weight Aa- A2. Proof The proof is left as an exercise to the reader. 2.6. D Finite dimensional vertex operators This section contains a study of the finite dimensional analogue for the free field representation of chiral vertex operators, which provides an algebraic setting for computing the Clebsch-Gordon (C-G) coefficients of Lie algebras. Given three dominant integral weights A,, i = 1, 2, 3, of a semisimple Lie algebra g we introduce the formal "chiral vertex operator" as a set of operators {<l>v: LA1 ~LA., vELA.}, which transform according to the irrep LA. under g, i.e., [x, <Z>v]= <Z>xv, xEg. (2·90) It is clear that, given the standard inner product, matrix elements <vai<Z>v.lvl>, V;ELA., are proportional to the corresponding C-G coefficients. Further, {<Z>vlvELA.} is specified by any of its components <Z>v, and any non-zero matrix element of this component fixes the normalization of the vertex. One choice of the component used to represent the vertex might be more convenient depending on the method in which the irreps of g are constructed. For instance, in the context of twisted modules, we note that the component <Z>w<vA,h where wE Wand VA. is the highest weight vector of LA., will be particularly convenient since it commutes with all the generators of n+ w. Suppose irreps LA1 and LA. are given in terms of free field resolutions (FAW.., d) and (FAW., d'), respectively, in the class of twisted modules defined by the choice of wE W. In the following we will study the free field representation of <l>w<vA•h which for convenience we will simply denote by a>. Given a mapping between cohomology classes of two complexes, it is a natural problem to look for a representative as a mapping between the complexes. 48 > In the present case such a representative is given by V: (FJl,., d)~(F)f., d'), defined as a collection of maps v<n: F~;{~F~;;, i=-l(w), ···, l(wo)-l(w), with components v<il={ Vwa,Wi EHom 'll(n.•)(Fwi*Ai, Fw.*A.), lw(wl) =lw(wa)=i}, satisfying d'(i) vu>= v<•+l)d(i)' i= -l(w), ···, l(wo)-l(w)' (2 ·91) such that the map induced on the cohomology classes agrees with a>, i.e. v<o>iL•. =a>. It is an easy exercise to check that (2·91) guarantees that V induces a well-defined mapping on the cohomology classes. Clearly not all mappings between complexes which satisfy (2·91) induce a nontrivial mapping on the cohomologies. For that reason we say that a vertex V is trivial if there exits f={fu>EHom 'll(n.•)(F :flu>, F:f.<•-l>)} such that 94 P. Bouwknegt,]. McCarthy and K. Pilch (2·92) In particular one is interested in computing the dimension N:l[A. of the vector space of vertices V, modulo trivial vertices, for three given weights as above. Finally, the representation of other components of {(].)v} can be obtained by the action of the algebra. From §§ 2.2 and 2.4 we know that the components Vws,w of the vertex, Vw.,w, EHomvcn.wlF~.M., F~•*A.), are exactly the operators which can be built as polynomials in screening operators, s;w. To ensure the correct weight, wAz, under transformation (2·90) we must also have an appropriate translation factor. Thus we are led to consider operators of the form 1 e•wAz·q X [polynomial in screenings], (2·93) * * where the polynomial is homogeneous of degree W1 A1 + wAz- W3 A3 in the screenings. We will call them "screened vertex operators". Further, from the discussion in § 2.4 it is clear that the screened vertex operators for one class of twisted modules are operationally constructed from those of another by a set of invertible transformations. Thus without loss of generality it is sufficient to develop the theory for untwisted modules, which we will do unless otherwise stated. Example 2.13. su(2) vertices Consider untwisted modules, w=1, giving resolutions of su(2) irreps with highest weights ha and ha. The representation of a vertex (])is is V = {v<o>, v(l>} such that the following diagram is commutative 8 zh+I 0-----+ Fjz -----+ J. v<•J F-jz-1-----+ 0 (2·94) J. voJ s2is+l 0-----+ Fia -----+ F-js-1 -----+ 0 . A priori, (2·95) which requires that If h + jz- js-:?.2j1 + 1, we can factor s 2M 1 from v<o> which is then trivial. Thus we must have jz- j1~h. (2·97) However, in order to solve (2 · 91) for v<l) we need (2h + 1) + U1 + jz- j3) 2 2j1 + 1, i.e. (2·98) Clearly (2·96), (2·97) and (2·98) reproduce the tensor product rules for su(2). By their explicit form and the fact that the algebra of screening operators is Free Field Approach to 2-Dimensional Conformal Field Theories 95 isomorphic with n-,*> there is clearly a natural identification of components of screened vertex operators with elements of Verma modules, via (2·99) Using this identification, we are able to study the free field representations of the vertex <P using the BGG resolution for an irrep which we discussed in § 2.3. In particular we have the "extension property" as embodied in THEOREM 2.15. Given v<o> and V<1l such that (2·100) and (2·101) the remaining components v(i>, i=2, ···, l(wo), are uniquely determined by (2·91), up to addition of trivial pieces of the form (2·92). Proof The identification (2·99) restates (2·100) as a relation between states in MA,, namely that d'<o> v<o> is an element of the Verma submodule M1;:1l(see (2 ·40)). Similarly, if we act on (2·100) with d'< 1>from the left we obtain (2·102) which can be interpreted as the statement that d'< 1>v<I> is closed in M~~~> of the BGG resolution, and thus is exact. That is, there exists a v<z>EM~~z> such that (2·103) In the same way we may compute all of v<n, i=3, ···, l(wo). Clearly, solving (2·91) using the EGG-resolution is unique only up to "trivial elements", which add to a total ambiguity of the form (2·92). D The question of constructing a nontrivial vertex then reduces to the problem of finding V(O) and V(l) Obeying (2•100) and (2•101), and then COmputing V(i)' i=2, ···, l(wo) using (2·91). The following theorem characterizes the space of nontrivial vertices, v:I~A •. THEOREM 2.16. A nontrivial vertex operator VE v:I~A. exists precisely when the tensor product rules are satisfied, i.e., irrep LA, occurs zn the decomposition of the tensor product LA,0LA •. Proof Recall that the irrep LA, is the quotient of the Verma module MA, by its maximal submodule M1;: 1>, and thus we may restate (2·100) and (2·101) as requiring *> We will only be interested in the algebraic structure behind the screening operators, and thus the identification with n+ or n_ is a matter of convenience since both algebras are isomorphic. P. Bouwknegt, J. McCarthy and K. Pilch 96 that v<o> corresponds to a vector cjJEMA1 which is non-zero as a vector in LA1 at weight A3- Az, and satisfies the following equations i=1, ... , f. (2·104) Under the action of Wo on LA1 these equations are equivalent to (2·105) i=1, ···, f' where cjJ*=woc/J is an element of LA1 at weight Az*-Ag*, where Az*=-woAz and A3* = - woA3 are the highest weights of the conjugate representations. We recognize (2 ·105) as the condition stated in Theorem 2.14 79> under which L~. occurs in the decomposition of LA/i9L~ •• or equivalently that LA. occurs in the decomposition of LA/3SJLA.. 0 This shows that the dimension N:l:A. of the space of nontrivial vertices is equal to the multiplicity N:l:A. given by the nontrivial C-G coefficients. One might expect that it is possible to compute multiplicities N:l~"Az from some cohomological setup without solving for the vertices explicitly. Indeed, we observe that that vertices satisfying (2·91) can be considered as elements of a double complex that arises by the following standard construction :48> Given two resolutions (FA~> d) and (FA., d') we introduce the double complex (X, a, a') of maps of the form (2·93), X ={X<i,j>ixu,j>3x.:;: F5!{-+ F~ij, i, j=1, ···, t}, (2·106) The differentials a and a' have degrees (0, -1) and (1, 0), respectively, and they anticommute, aa'+a'a=O. As discussed in Appendix C we can define the associated single complex (K, D), where K ={K<P>iK<P>=EBi-j=PX(i·j), P=- t, ... , t} and D: K<P> -+K<P+l>, D=a+a'. In terms of the complex K one can interpret Eqs. (2 · 91) and (2 · 92) for the vertex as follows: Consider Vas an element of K< 0 >. Then (2·91) simply says that DV =0, whilst (2·92) implies that Vis a nontrivial element of the cohomology of K. The question is whether (K, D) provides a resolution of the space of nontrivial vertices. This is answered in affirmative by the following theorem. THEOREM 2.17. The cohomology of the complex (K, D) is Hlfl(K) = oP.Ovj~Az . (2·107) Moreover, (2·108) where 97 Free Field Approach to 2-Dimensional Conformal Field Theories Proof Once more by the algebra of screenings we may identify a component Xw,w' EX<l(w),l<w'))' w, w'E W, as an element of the Verma module Mw'*A• of the weight w*A3-A2. Then, for fixed w, the column complex in (X, a, a') coincides with the BGG resolution for the weight space w A3- A2 in the irrep LA, and we obtain * (2·110) The spaces in the complex (L1~.A., a') on the r.h.s. are (recall that we denote by LA(A) the subspace of LA with the weight A) (2·111) The differential a' is well defined on this quotient and is given explicitly in terms of polynomials in the generators of n- acting on the irrep LA,. Note that complex (L1~A., a') can also be obtained from the complex of the Fock space resolution of the irrep LA. by the following formal procedure: Take (FA,, d) and substitute (2·112) To compute the cohomology of this complex we will relate it to the double complex of the Fock space resolution of LA,®LAs* that arises in the G X G/G finite dimensional coset model, as discussed at the end of § 2.5. Acting with the Weyl element wo on LA, we obtain the following isomorphism (L A. At,A•, a')~( - Wo LA• A,,A., Wo a' Wo-1) . (2·113) The complex on the r.h.s. is obtained from the resolution (FA,*, d') by (2·114) We can now represent LA, at each point of this complex as the cohomology of the corresponding resolution (FA, d). This yields a new double complex in which the two differentials acting on Fock spaces are constructed as polynomials in the generators, e,, of n+ and polynomials in the screenings, s,, respectively. However, by Theorem 2.14 and the discussion above it such a double complex is then equivalent to the subcomplex of G-singular states in the double complex of the tensor product (FA,®F A,*, d, d') at the weight A2*. For a dominant weight A2, or equivalently A2*, we know by Theorem 2.9 that the cohomology of the associated single complex must be concentrated in 0-th dimension. Using Theorem C.1 we can go backwards to the original complex to obtain (2·107). The dimension formula (2·108) follows then the algebraic Lefschetz theorem (1·4) and (2·46). D Equation (2 ·109) is also called Steinberg formula. 72 ) Nate that the last line in (2·109) can also be obtained directly provided one knows that the cohomology of the complex (L1~A., a') is concentrated in a single dimension. In fact comparing the latter complex with (2 ·100) and (2 ·101) we see that its cohomology corresponds to the space of the v<o) components of nontrivial vertices. 98 P. Bouwknegt, J. McCarthy and K. Pilch § 3. 3.1. Free field approach to affine Kac- Moody algebras The Fock sj;ace realization After this rather lengthy exposition of free field realizations and their resolutions for finite dimensional Lie algebras we will now turn to the case of affine Kac-Moody algebras. There probably exist a huge set of potentially interesting realizations on free field Fock spaces but the specific realization that we are looking for, with the application to conformal field theory in mind, is a realization where the Kac-Moody currents themselves can be expressed in terms of a set of conformal fields. The most straightforward way to obtain such a realization is to attempt to "affinize" the realizations obtained in the previous chapter. In principle one can affinize all the twisted realizations. However, since the associated Fock space modules turn out to be of the same complexity we will try to avoid confusion by restricting ourselves to affinizing the dual Verma module only. Thus, the realization will be in terms of a set of conjugate first order bosonic fields (/3a(z), ra(z)) of conformal dimension (1, 0),82 > one such pair for every positive root a ELl+, and a set of scalar fields ¢/(z) as many as the rank .t of g. We will take the following convention for their (nonvanishing) operator product expansions aaa' ra(z)(3a'(w)= z- w ' (3·1) ¢i(z)¢/(w)=- oij ln(z- w). (3·2) Their mode expansions are given by ra(z)= ~ rnaZ-n' !3a(z)= ~ !3naZ-n-1' nEZ nEZ ¢i(z)=q'-ipi In z+i~ ani z-n. n*O n (3·3) Let us denote the Lie algebra of oscillators by a. The algebra a admits a Cartan decomposition a=a-EBaoEBa+, where a-={/3na, n<O, aEL1+}U{rna, n:::;;;o, aEL1+} U{ani, n<O, i=l, ···, .t}, ao={pi, i=l, ···, .t} and a+={/3na, n20, aEL1+}U{rna, n>O, a ELJ+}U{an;, n>O, i=l, ···, .t}. For a weight A let FA be the CZ.l(a)-module induced from a vector lA> satisfying a+IA>=O, pijA>=a+A;IA>. (The factor a+ is inserted for convenience, and will be determined later.) We will refer to this module as a Fock space module. The affinization of the finite dimensional realization consists of the following steps. Replace 13a~ /3a(z), ra~ ra(z), p;~(I/a+)ia¢;(z) and normal order the expression that one obtains. In the negative root currents e-a(z) add terms of the form : r··· ror: (z) (of the correct isospin) which have vanishing zero mode piece, such that the commutators work out, e.g. by requiring the correct central charge term in e;(z)/;(w). One can prove that this procedure is always possible and moreover unique. 14 > The affinization of the Cartan subalgebra generators (2·5) is easily accomplished Free Field Approach to 2-Dimensional Conformal Field Theories 99 These do satisfy the correct OPE, with central charge k, provided we set 1 (3·5) The expressions for the other generators of course depend on the basis we have chosen, so let us give the resulting realizations in some examples (compare Examples 2.2 and 2.3) Example 3.1. su (n) (3·6) Example 3.2. so (5) (3·7) To be able to make use of the above realization in solving a specific conformal field theory we should of course compute the stress energy tensor in the above realization. For the WZNW-models the stress energy tensor is given by the Sugawara construction, which can be shown40> to reduce to the free field form. THEOREM 3.1. Let xa, a=l, ···, dim G be an orthonormal basis of g, then P. Bouwknegt, J. McCarthy and K. Pilch 100 The central charge of the corresponding Virasoro algebra is given by kdim G k+hv ' (3·9) Finally, observe that by affinizing the finite dimensional realization we have constructed a module which is free over CU(iii niL) and co-free over CU(iii n ii+), where iii=(n+®C[t, t- 1 ])EB(t®tC[t]). This shows already that these Fock space modules resemble the twisted modules introduced in § 2.4 in a lot of respects. This, ultimately, was our main motivation for studying twisted Verma modules for finite dimensional Lie algebras. In fact, the analogy can be pushed further by observing that translation operators tr, rEM in the affine Weyl group act on the generators of ii as (3·10) So, if we choose some rEM such that (r, a)>O, \laELh then we formally have50 > iii= lim ii~·. N-oo (3·11) This suggests that we may, as in the finite dimensional case, analyze the Fock space modules by starting from known results for the Verma modules and "performing an infinite twist" limN-ootNr· Though this point of view seems to work to some extent (as confirmed by our results) it does not seem to be helpful in, for instance, the derivation of the realizations above. The reason is, as one easily convinces oneself, that the generators in the Verma module realization obtained similarly to § 2.1 will have terms of arbitrarily large order in the /3r-modes. 3.2. Intertwining operators The next step is the determination of all possible intertwiners between the Fock space modules. Intuitively, one might expect that the set of intertwiners is described by a condition similar to Theorem 2.7, by making use of the infinite twist limN-ootNr. This result would be in complete agreement with the analysis that we will present below, but for similar reasons to those alluded to in the previous section we have not been able to prove this rigorously. So, the approach we will take here is rather to study the affinization of the finite dimensional screening operators directly as natural candidates for the building blocks of the intertwiners. The intertwining property of certain combinations of screening operators will be established by an explicit evaluation of the commutators as we gave for illustrational purposes in the previous section. The affinization of the screening operators (2·53) is given by*> *> This result, as well as the other results in this section, are also valid for non-simply-laced Lie algebras. In particular one does not need to introduce additional fermions as is claimed in Ref. 38). Free Field Approach to 2-Dimensional Conformal Field Theories s;(z)=p(e;): e-ia.ad: (z), 101 (3·12) where p(e;)(z) is some polynomial in /3, r-fields, e.g. for su (n) (compare Example 2.5) (3·13) One readily checks that the screening operators (3·12) are primary fields of conformal dimension one with respect to the stress energy tensor (3·8). Contrary to the finite dimensional case these operators do not generate the algebra ii+, due to the nonlocality of s;(z) with respect to s;(w) for (a;, ai)=I=O_ To evaluate the OPE's of the screening operators with the Kac-Moody currents one first observes that the OPE of s;(z) with e;(w), h;(w) and /H;(w) at most contains a term of order 1/(z-w). This follows from the fact that all generators have conformal dimension one and the fact that there exist no spin-0 operators of the correct isospin to allow for an order 1/(z-w)Z-term. However, the C/(1/(z-w)) term only receives contributions from single contractions, i.e. ordinary commutator pieces, and hence vanishes due to the finite dimensional result. Similarly, the only nonvanishing OPE can be argued to equal (3·14) for some c-number a. The c-number piece can be calculated. Example 3.3. Consider su (n). Since the c-number piece a does not depend on the choice of basis it suffices to do the calculation for one particular value of i. For i =n-1(=1) we have (see (3·13)) (3·15) In this case the C/(1/(z-wY) term arises from the piece -(k+n-2)orn-ln(z) -a+ 1rn-ln(an-1' io¢)(z) in /;(z) (Example 3.1). One easily computes that this gives a =(k+n-2)+2=k+n. The result (in all known examples) is that the two terms nicely combine into - Vij i' 2(k+hv) f(i Z ) Sj ( W ) ( a;, a; ) aw(-1-e -ia.a,•(J(w)) . z-w (3·16) This suggests to try intertwiners of the form*> iiE{1, ... , f} *) (3·17) Conventionally,•> one likes to think of the "screening charges" Q,=fs,(z) as some operators that can be arbitrarily inserted into the conformal blocks without destroying the chiral algebra Ward identities. Though, for practical purposes, this may be a convenient way of thinking about the problem, it is, strictly speaking, incorrect. The "operators" Q, do not, in general, exist as operators with a welldefined action on the Fock spaces. It is only for certain combinations of screenings that one can find integration contours that give rise to operators acting on the Fock spaces. 102 P. Bouwknegt, J. McCarthy and K. Pilch for some suitably chosen contour r. This expression can be written somewhat more explicitly by the Campbell-Baker-Hausdorff formula (3·18) where "' ~ 'l'<<>l ( z ) = - z. "-' anz -n . n (3·19) n<(>)O Hence, by acting with (3·17) on a vector vEFA, we obtain a vector given by some integral whose integrand has monodromy properties determined by the factor IT IT (zk- Zt)a+•(a,.,a,,) l~k<l~n Zk -a+•(A,a,.) . (3·20) l~k~n Let us introduce contours Fzo as a set of nested contours taken counterclockwise from a basepoint Zo to Zo around 0 and nested according to lz1l > ··· > lzNI. The integrand is defined by the analytic continuation from the line passing through 0 and Zo where it is taken to be real (see Refs. 7) and 40)). If F=Fzo in (3·17) we will use the shorthand notation [s;,···s;Jzo- For the discussion of the intertwiners it suffices to restrict ourselves to say Zo=1 in which case we will omit the subscript Zo. Having fixed the contour and a set of simple roots a;1 , j=1, ···, N such that /3 = ~iai1 we may still obtain different operators [s;,···sd: FA~ FA-P by permuting the a;/s. There are three important lemmas that we will need in the determination of the subset of intertwiners. The first expresses the fact that not all of the permutations of screenings give rise to independent operators. It is the analogue of the statement that in the finite dimensional case screening operators s,, i = 1, · · ·, .t generate an algebra isomorphic to n+. Lemma 3.2. Within the contour integrals I: the screening operators s, satisfy the Serre relations of the quantum group CUq(n+) (see Appendix B) for the value q =exp(iTC/(k + hv)), z.e. [[ ··· ( a··] ~o ( -1)" [ 1-K I-a,J u qd, ) ]] =0 s,J-aii-"SiS/ ··· for av~O, (3·21) where the dots stand for arbitrary combinations of screenings, and d;= i-(a;, a;). The proof, as explained in detail in Ref. 40) for the simply-laced case, can be given by writing out the integrals in terms of integrals over the unit circle with a fixed ordering of the variables. The vanishing is then a consequence of the cancellations among the various phase factors one picks up by going around the other variables. The second lemma expresses the fact that not all polynomials in screenings give rise to non-trivial operators, but that there are some vanishing relations. Free Field Approach to 2-Dimensional Conformal Field Theories 103 Lemma 3.3. (3·22) Proof We can write (3·23) The phase factor vanishes for N=2(k+hv)/(a;, a;). D The third lemma, the analogue of the commutator (2 · 31), can be proved by using (3·16) and evaluating the boundary terms at arg z;=O and arg z;=27r, while being careful with the phase factors that one picks up by crossing the variables. Lemma 3.4. We have the following commutator while acting on a vector vEFA (3·24) ;;z;=i where (3·25) is independent of j, and as before (3·26) One observes that the combinatorics which makes the commutator vanish for specific polynomials in the screenings is similar to that in the finite dimensional case, the only difference being that one works with q-numbers instead of ordinary numbers. We therefore arrive at the following characterization of the intertwiners between Fock space modules 3.5. There exists a map from Homv,<ul(MJ, MAq) to Homv<ul(FA, FA') where the restricted quantum group Verma modules MA q are defined in Appendix B, and the quantum group parameter q and the central charge k of ii are related through q =exp(i7r/(k+ hv)). Explicitly, if we characterize an intertwiner in Homv,<ul(MJ, MAq) by the singular vector P[/]vA, of weight A', where P[!] is some polynomial in the generators f,, then the map is given by THEOREM P[!]vA H [P[k-~s;]]. (3·27) (Note that this map is well-defined because of Lemmas 3.2 and 3.3.) We would like to believe that this map is bijective. To prove surjectivity, we would have to show that all intertwiners between these Fock space modules are of the form (3 ·17), i.e. can be built from the screening operators s;(z) by taking appropriate contour integrals. To prove injectivity, we clearly need to understand the vanishing properties of integrals of the type (3·17). In the examples to be discussed below the nonvanishing of the Fock space intertwiners could be established by the explicit 104 P. Bouwknegt, ]. McCarthy and K. Pilch evaluation of the image of a certain vector. Let us explicitly exhibit some singular vectors in MAq, i.e. those that lie along the simple root directions. Let thereto wE Wand write w= taw for some aEM, wE W. Define if (w(A+p), a;v)>O, if (w(A +p), a/)<0. (3·28) Then by a straightforward calculation one checks that (/;) 1vw*A is a singular vector in M~*A of weight w'*A, i.e., [(s;) 1]EHomvculFw*A, Fw'*A), 40 l where ' {tardiJ = wr iiF'a; w= fa-a,YtW = t -ur'a,r ur'a, if (w(A+p), a/)>0 if (w(A + p), a;v)< o. (3·29) The successive application of this result would yield an intertwiner : F w*A--> Ft-a,•W*A, which however vanishes due to Lemma 3.3. Summarizing, we have reduced the problem of the determination of the Fock space intertwiners to finding the singular vectors in a certain (restricted) quantum group Verma module. The analysis of the singular vector structure for Vq(su(2)) was given in Ref. 58) (where also, for su(2), a direct relation to the Fock space intertwiners of Refs. 36) and 37) is suggested). For CfJ isu(3)) the singular vector structure was determined in Ref. 83). The structure that emerges (which is expected to hold for arbitrary groups g) is the following. For generic q the singular vectors are just q-deformations bf the singular vectors that exist for q=1, and are thus completely described by Theorem 3.5. For q a root of unity parametrized by q=exp (7ri/(k+hv)) (and integrable weight A) the singular vectors are in 1-1 correspondence with elements of the affine Weyl group of g. The directed graph given by the set of intertwiners is a tiling of .e'-dimensional space by the directed graph for q=1, i.e. translations of the q = 1 graph along the root directions. Let us formulate this more precisely. Inspired by (2·57) we introduce thereto a twisted length lr on the affine Weyl group W by [s~(k+hV)f(a,,a,)] (3·30) where rEM is chosen such that (r, a)>O, VaEL/+. Note that lr is well-defined because the limit is already obtained at a finite value of N since there are at most a finite number of cancellations (at most l(w)) that can occur between t-Nr and w. Also, one can show that lr does not depend on the choice of r (see Lemma A.3). su Example 3.4. For (2) we can take r=a, i.e., tr=ron. For w=(rori)m we have f-N,w=(riro)N-m provided N";?.m. Thus, l(t-N,w)-l(t-N,)=2(N -m)-2N= -2m for N";?.m, i.e. lr((rori)m)= -2m. Similarly lr((rori)mro)= -2m-1, (3·31) Free Field Approach to 2-Dimensional Conformal Field Theories 105 lr((rlro)m)=2m, lr((rlro)mr1)=2m+ 1. Example 3.5. (3·32) For su (3) we can take r= as, i.e. tr= ror1 rzr1. One easily verifies, e.g. lr(rlro)=O, (3·33) Using the twisted length we now define a twisted Bruhat ordering on W. Define w ~ r w' if there exists an aE .J+ such that w =raw' and lr( w) = lr( w') + 1. Then define WSr w' iff there exist W1, ···, WkE W such that w~r WI~r· .. ~r wk~r w'. Then we conjecture CONJECTURE 3.6. For AEl\ (integrable weight) we have - - {1 if w < r w' dim Hom'U,(ulMZ,*A, MZ,'*A)= 0 h -. ot erwzse. (3·34) As mentioned above, this conjecture has been proved for g=su(n), n=2, 3. Through the correspondence in Theorem 3.5, which was conjectured to be a bijection, this theorem gives a complete description of the Fock space intertwiners. Example 3.6. given by The directed graph of intertwiners for V q(su(2)) 58 > (or su (2) 36 >· 37>) is (3·35) and for V;(su(3)) 83 >· 40 > (or su (3) 40 >· 37 >) r1 ----+ /' '\. X ro r1 '\. r2 ro X X r1 r2 X /' rt '\. X r2 '\. X ro r1 r2 X X ro /' r1 '\. X r2 '\. X ro r1 /' r2 r1 ----+ /' r2 ----+ ro r2 ----+ /' ro ----+ '\. ro X '\. /' r1 ----+ /' T2 ----+ ro r2 ----+ /' '\. ro ro ro ----+* /' r1 ----+ r1 •----+ X '\. '\. /' r2 ----+ ro r2 ----+ /' ro ----+ X r1 ----+ X '\. X ro (3·36) where the arrows continue in all directions. For a similar hexagonal structure in the case of su (3) Verma modules we refer to Ref. 74). 106 P. Bouwknegt, }. McCarthy and K. Pilch The vertices of these diagrams, i.e. MZ,*A or Fw*A, are in 1-1 correspondence with elements of the affine Weyl group. If one associates simple reflections r; with the edges as in the picture, then the Weyl group element associated to a vertex can be obtained by combining the reflections that one encounters by taking a path along the edges from the middle vertex ( *) to the required vertex (e.g. the vertex ( •) is associated with w = r1 r2ro). The Weyl group elements associated to vertices along a vertical line have a fixed twisted length (compare Example 2.9) and the depicted intertwiners increase the twisted length by one unit. The intertwiners along simple root directions are explicitly given by (3·28). The other intertwiners can in principle be obtained by a successive application of Lemmas B.1 and B.2, exactly as in the finite dimensional case. Example 3. 7. d1,ro~ For su (3) we have e.g. (see Ref. 40) for more details) '£,_bq(T2, Ts;j)[(s1) 1d(ss)i(s2)l.-j], Q5;.j5;.[, (3·37) where l;=(A+p, a;), [;=(k+3)-l,, ss=-s1s2+q- 1s2s1, and i+Cm-i)(n-i) [m]q![n]q! bq ( m, n ., J")- q [J.] q.1[ m _ J.] q.1[ n _ .] J q.1 (3·38) A last remark concerns the affinization of the other twisted realizations of§ 2.4. One can go through the same steps as described above for the "affinized dual Verma module" with the result that the structure of the intertwiners now becomes the affinization of Theorem 2.7, e.g. for su (3), .M;•r, it becomes a tiling of the 2dimensional plane by the hexagon of Example 2.9. The resolution Collecting the material on intertwiners we come to a final conjecture on the existence of resolutions of irreducible highest weight modules LA, AEJ\ in terms of Fock space modules. 3.3. 3.7. For every AEP+ we have a resolution of LA in terms of Fock space modules, i.e. we have a complex (F, d) CONJECTURE (3·39) where (3·40) whose cohomology is given by (3·41) Free Field Approach to 2-Dimensional Conformal Field Theories 107 Let us make some remarks on the status of this conjecture. The fact that we can combine the intertwiners of Conjecture 3.6 into a complex has been proved for (2) in Refs. 36) and 37) and for (3) in Ref. 40). There is an important comment that we should make namely that unlike in the finite dimensional case, there exist elements w, w"E W such that the number of w'E W satisfying w-+r w'-+r w" equals one, namely w"= f-a,W. The nilpotency of the differential along these su(2) directions is ensured by Lemma 3.3. This feature does not have an analog in Verma modules but is purely a "quantum group" effect. This is one reason it cannot be completely trivial to relate resolutions through an "infinite twist". To get cancellations among the maps that form a (twisted) square in the affine Weyl group one has to make a clever choice of signs as in the finite dimensional case (see Ref. 40) for details on Sit (3)). Although the signs can be chosen periodically, this is not necessarily the same periodicity as present in the intertwiners themselves. We regard the mere fact that we have discovered enough intertwiners to build a complex as very suggestive that we may have found all, as conjectured in the previous section. As for the cohomology this has, to our knowledge, only been proved rigorously for (2). 36 >'37> Sample calculations for (3) have not revealed any contradiction. The most promising approach to prove Conjecture 3.7 seems to be to show the equivalence to the weak resolution of Feigin and Frenkel 50 > as can be done in the finite dimensional case, 77> and the infinite dimensional case for Verma modules.m This resolution, where the differential is the de Rham operator on the tangent bundle of a semi-infinite dimensional flag manifold, has a similar structure as the one described above, i.e. its terms are of the form (3·40). However, it has not been shown that this differential only maps between the Fock space modules as supposed in Conjecture 3.6, which would probably require a proof of the characterization of Homv<n>(FwM, Fw'*A). Also, the heart of the Rocha-Caridi's proof, which is the construction of a map between the two complexes is established by induction to i. This would have to be suitably modified due to the two-sidedness of the above complexes. One might wonder whether the analogous complex in terms of restricted quantum group Verma modules gives a resolution of the irreducible representation. This is not the case, however. In fact, it appears that the complex has trivial cohomology everywhere. This, we have been able to prove for CU q(su(n)), n=2, 3 (see Appendix E). In fact, triviality of the complex is crucial for the construction of the screened vertex operators. As far as we understand the construction of the BGG-type resolution for QG irreducible highest weight modules is still an important open problem. su su su 3.4. su Chiral vertex operators and fusion rules In this section we outline the construction of chiral vertex operators of the WZNW models in the framework of the free field resolution (3·39), in complete analogy with the finite dimensional case discussed in § 2.6. For three given irreducible representations LA., i = 1, 2, 3 of a chiral algebra Jl, we will introduce the chiral vertex operator84 H 6> as a set of (multivalued) conformal fields, <P"'(z): LA,-+ LA., ¢ ELA., such that if¢ is a primary state then <Prp(z) behaves as a primary field of Jl. In particular it is completely specified by its highest weight component, <P(z), and other components are obtained by the action of Jl. The "fusion rule", N1,"A., is simply the 108 P. Bouwknegt, ]. McCarthy and K. Pilch dimension of the vector space of such chiral vertex operator sets for the three given representations. It should be stressed that, unlike in the discussion in § 2.6, in the infinite dimensional case we cannot prove that homomorphisms between Fock spaces are always expressed in terms of screening operators. But since this is physically well motivated, we develop the theory below with this assumption. Thus, based on the finite dimensional case discussed in§ 2.6, the Virasoro minimal models, 7 > and the su(2) WZNW model, 36 > we now seek a representative of the highest weight component (J)(z) as a chain map between two resolutions (FA,, d) and (FA., d') defined by a collection of screened vertex operators { v< 0 (z), iEZ} whose components {Vw,w'(z)ll7 (w')=l 7 (w)=i} are of the form Vw,w'(z)= r dzr···dzn VA.(z) {ii"··in} ~ a;,. .. ;nS;,(zr)···s;n(Zn) Jr. = VA.(z) x [polynomial in screenings].., , (3·42) where (3·43) and the degree of each term in the polynomial is a;,+···+a;n=w'*Ar+Az-w*Aa. One verifies that Vw,w'(z) is a conformal primary field of the chiral algebra of the WZNW model with the weight Az, and the conformal dimension (Az, Az+2p) /2(k+hv). Because of the conformal invariance we may restrict our analysis to a specific value of the coordinate z. In the remainder of this section we set z=1, and do not write it explicitly. The requirement that the vertex V = {v< 0 } defines a chain map between two resolutions is given by a condition similar to (2·91) in the finite dimensional case, i.e. that the following diagram is commutative ···~ FA"";_ I) ~ ···~ v<-I> F1-;,IJ d(O) d(-1) ~ F1°[ ~ v<•> d'(-1) ~ ~ d'(O) F1°i ~ F1![ ~ v<'' F11i d(l) ~ F12[ ~ d'(l) ~ ~··· (3·44) v<•> F12j ~··· Similarly, we say that a vertex is trivial if there exists f={!Ci>, iEZ} with components Uw.w', lr(w')=lr(w)+1=i} of the form (3·42) such that (3·45) We will refer to the r.h.s. of (3·45) as a gauge transformation of the vertex. The space of screened vertex operators satisfying (3·44), modulo trivial vertices of the form (3·45), will be denoted by v1~,A., and N1,",A.=dimv1~,A •. The same arguments as in the construction of intertwiners in § 3.2 show that the screening currents inside [ ·] satisfy the algebra of the negative root generators of quantum group CU q(g ). Thus it is natural to identify components Vw,w' of a vertex with elements of the restricted QG Verma modules llfz,,M,, (3·46) Free Field Approach to 2-Dimensional Conformal Field Theories 109 or with elements of the transposed Verma modules MZll:A., (3·47) The transposed Verma module M:F is spanned by elements of the form VA/;,/;,_,···/;., and is isomorphic with the Verma module Mj. via VA/;J;,_,··-/;,~/;~···/;f_J[*vA•, where /;*= f-wo(«t)· These identifications, together with triviality of the complex (MAq, d), discussed in Appendix E, suggest that an analogue of Theorem 2.15 should hold. Indeed, we have 3.8. A nontrivial vertex component v<o> such that THEOREM VEv2~.A. is uniquely characterized by its 0-th (3·48) and (3·49) The remaining components vu>, i-=1=0, are completely determined by (3·44), up to addition of trivial pieces of the form (3·45). Proof First we will prove that given v<o> satisfying (3·48) we can solve (3·44) for the remaining components. Identifying {(d'<o> v<o>)w,w'llr(w')=O, w fixed} with an element of M:il 0 > we can interpret (3·48) as the condition that this element is closed (see, Appendix E). Since by Theorem E.1 the cohomology of (MAq, d) is trivial we conclude that there exists v< 1> such that (3·50) which satisfies (3·51) Repeating the same argument we determine all the vertices v<il, i?:.2. On the other hand if we identify {(V< 0 >dH>)w,w'llr(w)=O, w' fixed} with an element of Mi{, we can use the triviality of (Mi{, d), which obviously follows from Theorem E.1, to calculate v<-ll, and then the remaining vertices vu>, i-::;;. -2. It is also clear that the ambiguity one has at each step of this procedure adds up to the total ambiguity of the form (3·49). In particular, if we start with v<o> given by the r.h.s. in (3·49) we will end up D with a trivial vertex. Obviously, we could have chosen any other component v(i>, satisfying d'u> v<ildU-l>=o, and prove that it determines the vertex completely. In practice solving (3·48) for v<o> presents a formidable task because, in principle, one must still determine an infinite number of components Vw,w', lr(w)=lr(w')=O. This difficulty does not arise in the case of the su(2) WZNW model, which we will now discuss in some detail. Example 3.8. Screened vertex operators and fusion rules in the su(2) WZNW model. P. Bouwknegt, J. McCarthy and K. Pilch 110 su Recall from § 3.3 that a resolution of an (2) representation with a highest weight A= ja at level k (A is integrable for O~j ~ k/2) is given by a complex in which, for all i, Fjil is just a single space, p!il={ Fi-n(k+2), F-J-1-n(k+2l , J i=2n, i = 2n + 1 . (3·52) In terms of screening current s(z), we have d(i)= { [ s 2i+l] ' [sk+ 2-< 2i+1l], i=2n ' i=2n+ 1. (3·53) Given three integrable weights A;=j;a, i=1, 2, 3, the highest weight component of the vertex is specified by a set of screened vertex operators v<o : FJ\i>-> FX> of the form (3·42) which in this case simply gives (3·54) where a, are normalization constants determined by (3·44). By the general argument given above we only need to determine the screened vertex v<o>, which satisfies conditions (3·48) and (3·49). Clearly, to define v<o> we must have (3·55) Then (3·48) becomes (ao=I=O) (3·56) or, equivalently, [sk+2+Ua+i•-il>]=O, (3·57) which is satisfied provided (3·58) On the other hand (3·49) is given by (3·59) where j<o> and j<1l are of the form V,. times a power of the screening current, which must be positive if the corresponding I exists. Thus (3·59) simply requires that these powers of screenings are negative, which gives h+h+h~k. (3·60) h?:.h-h. (3·61) The solution to (3·55), (3·58), (3·60) and (3·61) is lh- hl~h~min(k-(jl +h), jl + j2), (3·62) Free Field Approach to 2-Dimensional Conformal Field Theories 111 which shows that a nontrivial vertex exists if and only if the su(2) WZNW model fusion rules87 > are satisfied. In the case of finite dimensional vertices it was convenient to relate the problem of constructing vertices to that of finding special states in an irrep of the algebra. We will now rederive Eqs. (3·55)'"'-'(3·61) from this point of view. Example 3.8. cont. Thereto, let us identify a nontrivial vertex, via its v<o> component, with the element rf; at weight h-h in the quotient, LJ,, of the restricted Verma module .zCt; by its submodule M!.j,-1, generated by the singular vector JZil+lvj,. For an integrable weight j1a, L'J, is an irreducible module of Vq(su(2)). We can now rewrite (3·48), which is equivalent to (3·50) and (3·49) as the following conditions for the state rf; (3·63) Since rf; = ajil+iz-i•vj,, this approach leads to the following interpretation of the inequalities above : h+j2-h~O, so that -h+h+h~O, so that ¢$.M!.J,-1, h-h+h~O, so that rf;EKer JZi•+l, h+h+hsk, so that rf;$.Im jk+2-(2i•+l>. rf;EMJ;, (3·64) Solving (3·50) we find a1=q- 2iz<2i•+l>ao, where the phase arises from moving s 2i•+l through Vj,. The remaining normalization constants are determined from (3·44) and are given by _ ( - 1)2iznao , a2n+1_ ( - 1)2jzna1 , n E a2n- z . (3·65) This shows that we can always choose a representative of a vertex satisfying (3·44) to be "periodic" in the sense of (3·65). Heuristically, we could have anticipated this result as follows: An element C =sk+ 2 lies in the center of CUq (su(2)), and C Vj, =( -1) 2jz Vj. c. Thus c v<O>=( -1)2i• v<O) c. On the other hand formally c v<O> = V( 2) c' which gives (3·65) for n=l. Once again we would like to derive (3·62) by a counting argument, not requiring an explicit construction of the vertex. We will now present two such calculations: the first one based on a double complex argument similar to that discussed in § 2.6, and the second one based on a cohomological interpretation of conditions (3·63) for the v<o> component of a nontrivial vertex. Example 3. 8. cont. First consider the double complex (X, a, o') build exactly as in the finite dimensional case (see (2·106)). It is easy to verify by counting powers of screenings that the cohomology of the associated complex (K, D) is trivial for P=l=-0. However, each 112 P. Bouwknegt, ]. McCarthy and K. Pilch K<P> is infinite dimensional, and the Lefschetz formula now simply gives an undefined alternating sum where each term is infinite. Thus, to make this procedure meaningful we must "cut the complex down to size". This is achieved by restricting to the space of "periodic" maps, xU,j>=( -1)2j•x<i+ 2 ,j+2>, Vi, jEZ, which clearly form a subcomplex since D preserves the symmetry. As we observed above the vertices we are interested in, i.e. elements of K' 0 >, always have such symmetric representatives. Moreover, it is clear that if f=Dg for a periodic collection of maps f, then we may choose g periodic (again each component of g lies in a one dimensional space). Thus the cohomology in this subcomplex is precisely equal to that in the complex (K, D). Now applying the Lefschetz formula to the periodic subcomplex we obtain ih•,i2= nEZ ~ [8(h + j2- h+n(k+2))- 8(h +h+ h+ 1 + n(k+2)) - 8(- j1-1 +h- h+ n(k+2))+ 8(- j1 + j2- ja+ n(k+2))], (3·66) which is known to reproduce fusion rules given in (3·62). The second method is to observe that the last two lines in (3·64) characterize ¢ as a nontrivial cohomology class in the following complex arising from the resolution with respect to the weight ha, (3·67) where all the spaces L'J,' 0 are isomorphic with LJ., and the action of differentials is simply that of the generators on the quotient. This complex has a weight space decomposition, and for each weight the corresponding subcomplex has non-zero spaces at most at two points. The latter follows from the integrability of weight ha and Jk+ 2 =0. Once more a straightforward power counting shows that a nontrivial cohomology of (3·67) at weight h-h can occur only at the 0-th degree. Thus, we can use the Lefschetz formula to compute the dimension of the space of nontrivial yco>•s, and the result agrees with (3·66). In the remaining part of this section we will discuss the general structure of vertices when f ~ 2. We formulate and illustrate on examples some conjectures which generalize properties established in the finite dimensional case in § 2.6 and in the (2) case above. One may try to calculate the dimension N:l,",A• using a double complex as above. In this case the "periodicity" condition should be su (3·68) We use obvious notation for the components of the vertex, e.g. Vlff~As,w'*A• = VJ,~~'- In the case of Sit (3) we have verified on several examples88 >that one can always choose a representative of a vertex in the class of gauge equivalent vertices, such that it is periodic in the sense of (3 · 68). However, a general proof of this property, that should follow from the basic equation (3·44) of the vertex, is not known. Assuming that we can restrict the double complex to periodic vertices, and that Free Field Approach to 2-Dimensional Conformal Field Theories 113 the cohomology of the restricted complex is concentrated in K<o> we obtain from the Lefschetz formula the dimension of the space of nontrivial vertices : (3·69) The last equality, in which we used (2·46), gives precisely the formula derived by the algorithm which was proven to reproduce the fusion rule coefficients_ 89 H 4 > In the above derivation we considered the vertex as a chain map between two complexes. On the other hand by Theorem 3.8 we know that the vertex is completely characterized by its v<o> component. In principle the characterization of the vertex given by Theorem 3.8 requires that we specify all of v<o>, i.e. an infinite number of components. One would certainly expect this to be unnecessary, since the cohomology of the Fock space resolution of LA can be concentrated at FA cF1°>, and only the Vi~~> component gives rise to a nontrivial mapping when we pass to the cohomology space. Thus one should be able to derive (3·69) by counting properly defined nontrivial Vi~~> components. We say that a vertex is diagonal if the off-diagonal components of v<o> vanish, i.e. V~~lo,=O for w=Fw'. For such diagonal vertex, equations in (3·50) which involve the Vi~~> component are (3·70) In order that the mapping induced by Vi~~> on cohomology classes was nontrivial we require that Vi~~)=F {wE _~ W[/,(w)~-1} df.--.J>j~~i + -~ {wE W[l,(w)~I} JlYod1J.i, (3·71) which is just the (1, 1) component of (3·49). 3.9. Any vertex is gauge equivalent to a diagonal vertex, i.e. given v<o> Which satisfies (3•48) there eXiStS a diagonal lf(O) SUCh that V(O)_ lf(O)=d'(-I)j<Ol +J<ll d<o> for some po> and J< 1>. CONJECTURE su This conjecture has been proven88 > in the case of (3) under a restriction that the weights Ar, Az and As satisfy the finite dimensional (!) tensor product rules, i.e., that the irrep of su(3) with the highest weight As occurs in the tensor product of irreps given by Ar and Az. Using a convenient basis in CU q(n~u<S>) (see Appendix E) we show by an explicit calculation that given v<o> which satisfies (3·48) there exists a gauge transformation such that the new (1, 1) component satisfies (3·70). Similarly, given w~) satisfying (3·70) one can use (3·44) to solve explicitly for the remaining components of a diagonal v<o>, and show that the solution is unique up to a gauge transformation. Combining these two facts we verify the conjecture in the restricted case. One can also check that in this case (3·71) simplifies and in fact is equivalent to 114 P. Bouwknegt, J. McCarthy and K. Pilch (3·72) because the remaining terms in the gauge transformation (3·71) can be recast into the form above. If we identify Vi~~> with an element ¢ in L1., the quotient of the restricted QG Verma module M:l, and the submodule generated by singular vectors j,<A•+p,a,v>vA., we may interpret (3·70) and (3·71) as the following conditions on ¢ f,(As+P,a;V)¢=0' ¢=I= _~ {wE Wll,(w)= -1) i=1, ···, f' dl,-;}>[/,]xw , (3·73) (3·74) By analogy with Theorem 2.14 we will call this a generalized Zhelobenko condition. One can try to count the diagonal vertices by constructing a complex similar to (3·68) in the su (2) case. For this we would need to prove the following: CONJECTURE 3.10. Let (L~~~A., d) be the complex obtained from the resolution (FA., d) by the substitution (3·75) If A1, A2 and A3 are dominant integrable weights of ii then the cohomology of (Lq:l~.A., d) is concentrated in the zeroth dimension, z.e. (3·76) It is easy to verify that in most cases complex (Lq:l~.A., d) has only one nontrivial space, and thus the conjecture is trivially satisfied. For instance this happens when the labels (A1, a; v) are small in comparison with (A3, a; v) and k + h v- (A3, a; v). In some sense, this situation corresponds to the classical limit, when we know that the analogous theorem is true (see remarks at the end of § 2.6). We can now calculate the dimension of the space of nontrivial Vi~~> vertices from the Lefschetz formula and, using (2·46), we obtain (3·77) This agrees with the previous calculation (3·69) and suggests that indeed all the information about the vertex is contained in its Vi~~> component. The explicit form of the Vi~~> component of the vertex, which satisfies (3·70) and (3·71), is needed for computing tree level correlation functions in the WZNW model. On the other hand, (3·70) and (3·71), or equivalently (3·73) and (3·74), provide a new algorithm for verifying whether for given three integrable weights A1, A2 and A3 of ii the fusion rules are satisfied. Let us illustrate some of this on examples for su (3). Example 3.9. is The relevant part of the resolution (FA, d) for su (3) (see Example 3.7) Free Field Approach to 2-Dimensional Conformal Field Theories Qf Fr,*A /' Qf' '\. Frz*A 115 /;=(A+p, a;v), i=l, 2, 3, l;=k+3-l;' i=l, 2, 3' Q/=[s/], i=l, 2, Qg 1 see (3·37) . (3·78) The operator Qg 1 is a homogenous polynomial in s1 and S2 of order la3 whose explicit form is given in (3·37). I. A1=8, A2=6, A3=3, k';;?:2 The most general screened vertex "VI~~> in this case is (3·79) where S12= -s1s2+q- 1S2SJ. Equations (3·70) and (3·71) are explicitly given by (3·80) (3·81) and (3·82) By the counting of screenings the most general form of the components of v<I> is (3·83) while, for k';;:::2 (as required by integrability of 6 and 8), the only non-vanishing component of the gauge transformation is (3·84) We can use this freedom to set c1=0, i.e. to diagonalize v<I> in (3·80). Substituting (3·83) in (3·80) we then obtain [si 2][a1S1S22+ a2S12S2] = [b1S1Sl+ b2S12S2][s1 2] , + a2S12S2]=[d1S1S2+ d2S12][s22], q- 2[s2][a1S1S22 (3·85) where the phase on the l.h.s. of the second equation arises from pulling S2 across VAz. Using the algebra of the generators of CfJ q(ntu< 3>), summarized in Appendix B, we find that there is a one parameter family of solutions a1=[j;qa, a2=a, b1=[~J:a, b2=-[ 3~qa, d1=[ 3ta, d2=q- 4 ~~~:a (3·86) with arbitrary a. In particular this proves that .N~.s=l. 116 II. P. Bouwknegt,]. McCarthy and K. Pilch Ar=8, A2=8, Aa=8, k=2, k~3 In this example the most general form of Vi~~>= Vi~~> is VA2[arsrs2+a2sd, (3·87) While the Only non-Vanishing COmponentS Of V(l) in (3·70) are vg!n and vg!r 2, Which are in form the same as the r.h.s. in (3·87), but with different coefficients which are uniquely expressed in terms of ar and az by solving (3·70). Moreover, the most general form of the r.h.s. in (3 · 71) for this case is ,c(O) + Qk+I ,C(O) + Qk-l ,C(O) + ,C(l) Q 2+ ,C(2) Q 2 Q1k+I Jr2rorz,l (3·88) 2 Jrtrort,l 3 Jro,l Jl,rt 1 Jl,rz 2 , and we see that for k~3 none of these terms contribute a gauge transformation to Vi~~>. This gives a 2-parameter family of vertices, in agreement with .N~.8 =2. On the other hand, for k=2 we have Qa 1 = q 3 [srs2] + q[3]q[sd , which allows one linear combination of the terms in shows that .N~.s=l for k=2. 3.5. (3 · 89) W~> to be gauged away. This Coset conformal field theories The coset construction51 >associates a RCFT to every pair (jj, ii) of Kac Moody algebras, where h is a subalgebra of g, with central charge kH related to kc by the Dynkin index j of the embedding hCg, kH= jkc. In this section we will only consider regular embeddings, i.e. j = 1. The energy momentum tensor (3·90) where Tc(z) and TH(z) denote the Sugawara tensors of G and H respectively, commutes with the generators of ii. It thus generates a "coset" Virasoro algebra with central charge c=cc-cH, and we may decompose a given irreducible highest weight module LAc of jj as (3·91) The sum runs over a finite set of integrable weights of ii. The modules L~:~, are of course highest weight modules of the coset chiral algebra-but not necessarily irreducible. The branching function is defined as (q = e 2 m) etH =Tr qa/H-c/24 bA,A'(3·92) • L~;f(. In line with our previous discussions, we might try to treat the coset model as a minimal model of some chiral algebra Jl. We would then construct a free field realization of Jl and a resolution of irreducible Jl-modules in terms of these free field Fock spaces. But in fact the chiral algebra of a generic coset model is not known. Thus we find it necessary to approach the problem more "extrinsically", starting from the known free field description of jj and projecting out the G/H model. As a first step we generalize the results of Theorem 2.9 and obtain a resolution of L~:~'· Clearly L~:~, can be identified with the set of vectors in L~ which are ii-singular Free Field Approach to 2-Dimensional Conformal Field Theories 117 (annihilated by ii+ H) with integrable ii-weight A'. This leads to the following resolution of L~~;[, in terms of a subcomplex of the resolution of L~. 3.11. Let (FA, d) be a resolution of the irrep L~, AEJ\, of {i given in Theorem 3.7. For each F~il, denote by S~?A' the subspace of ii-singular states with the ii-weight A'. Then 1. d: S~?A'-4 sy,-;:Y, i.e. (SA,A', d) is a subcomplex of (FA, d). 2. The cohomology of the subcomplex (SA,A', d), with A' an integrable weight of ii, is THEOREM H d(il(S A,A')-u.N,OLGIH A,A'. (3·93) The reader will notice the similarity to Theorem 2.9, and indeed the proof is almost identical. In particular the first part of the theorem follows from the fact that the generators of iii_f_c ii~ commute with d. Moreover, part two hinges on the following generalization of Theorem 2.10. 3.12. For any affine Kac-Moody algebra {i, Verma modules MA, where A is a dominant {i-weight, are projective in the category LJ of {i-modules. THEOREM Proof The proof is essentially identical to the one of Theorem 2.10 for Verma modules of finite dimensional Lie algebras. The only difference is that the Cartan matrix of a Kac-Moody algebra is positive semi-definite. Thus we can conclude that the last term in (2·70) is non-negative, but it may vanish even if some n, are non-zero. However, for a dominant weight A the coefficients p,+a, in the second term in (2·70) are strictly positive, and this enforces vanishing of all n;'s. A more complete discussion can be found, e.g. in Ref. 44). D We may now return to the proof of the theorem, which follows from the result above by the same reasoning as the one used in § 2.5 to obtain the twisted resolution of finite dimensional coset models. For the sake of completeness we recall the main steps. From Theorem 3.12 it follows that any ii-singular element of L~ has an ii-singular representative in F~0 >. It further follows that any state which is trivial in the complex is also trivial in the subcomplex, and thus the cohomology of the subcomplex vanishes, except at the restriction of F~0 >. There remains the problem of actually performing this projection onto the resolution of L~~;[,. The procedure which seems to generalize from our finite dimensional analysis in § 2.5 is the ERST projection. Introduce conjugate pairs of free fermionic ghost oscillators, one pair for each generator in ii+ H. Thus for the positive modes we have the set of conjugate pairs {(c~n, bnb), n >0} (a denotes adjoint indices of hand rab is the Killing metric of h) with {c~m, bnb}= Omn"Yab, and for the zero modes just {(coa, boa), aE.d+ H) with {coa, bl}= oaP. Denote by Fgh the Fock space generated by the c~n, n2:0, acting on the vacuum IO> annihilated by all the boa and bna. The ghost Fock space is graded by the cb-ghost number-i.e., Fgh=EBnEz.Fi~>. For an arbitrary ii-module M, we introduce the module M@Fgh, graded as (M@Fgh)<n> =M@Fi~>, with M@Fig>=.M. Then we may construct a nilpotent operator Q: M @Fi~>-4 M@F~~+l> via P. Bouwknegt, J. McCarthy and K. Pilch 118 where xE ii+ H acts on M@Fgh as x®l. That Q is nilpotent follows from associativity of ii+ H by a straightforward calculation. Further, it is clear that the cohomology H¢0 >(M@Fgh) is isomorphic to the space of ii-singular vectors in M. Given the resolution (FA, d) of a fi irrep LAG, denote the subcomplex with fixed ii-weight A' by (FA,A', d). Then we may form a double complex (FA,A'®F8 h, d, Q), where Q is given in (3·94) and d acts as d®l. The spaces in this complex are labelled by two degrees, i and n, Ft~,ti!)F~~>, where i EZ and nEZ+. As discussed in Appendix C, to such a double complex we can associate a single complex (K, D). The following is required if the BRST construction is to be applicable. Let (FA, d) be the resolution of an irrep L~, AEP+, and A' an integrable h-weight. Then the cohomology of the complex (KA,A', D) associated with the double complex (FA,A'®Fgh, d, Q) is CONJECTURE 3.13. (3·95) In § 2.4 we argued that a good finite dimensional analogue of the resolution (FA, d) is the one in terms of twisted Fock spaces. Conjecture 3.13 should then be viewed as an infinte dimensional generalization of Theorem 2.13. Recall that the proof of the latter required an indirect argument. More precisely, we showed that the cohomology of the complex (KX'.A', d) could be computed using a more convenient resolution of L~ via a complex of dual Verma modules, which are characterized by the co-free action of n+ H. Generalization of this proof to the infinite dimensional case is not straightforward. There exists a resolution of fi irrep L~ in terms of dual Verma modules 95 > for which, after restricting to a fixed fi-weight, one should be able to prove the analogue of Theorem 2.11. However, in relating the cohomology of this complex to that of (FA,A'®Fgh, d, Q) we are interested in, one encounters the same difficulties as in relating the Fock space resolution to the EGG-resolution. For this reason we do not pretend to have a rigorous understanding of the proof. This construction allows us to compute the branching functions. That can be seen from the following set of examples, which all use the Lefschetz sum to express bA,A'=q -h.·-c/24"' ~ "' ~(- zEZ n<oO 1)'( - 1)n T rFt>·®F,i::' ( Q fo) (3·96) where (3·97) projects onto the appropriate isospin, hA' is the eigenvalue of LoH on ii-singular states with h-weight A', and we have used explicitly that LgiH=LoG-LoH. Moreover, hoi and La are just the D-invariant extensions of hoi and LoG, respectively, io=LoG+ ~( ~ (nc:::bna+nc~nb;;-a)+ ~ nc!-nbni), n>O aELJ+H I:::;;:iS::l Free Field Approach to 2-Dimensional Conformal Field Theories 119 The calculations use the identity96 l-99 ' 1 II (1-qne2~ri6•a)(1-qn-le-2"i6•a) _ _____;1c____ ~ e-27rtn6•a¢n. II (1- qn)2 nEZ (3·99) n;;;,l n;;;,J Here we have introduced the quantity </Jn= ( -1)mq(l/2)m(m+I)+nm. ~ (3·100) m;;;,o It is simple algebra to check that this may also be written </Jn= _ ~ ( -1)mq(112)m(m+I)+nm, (3·101) m<O and thus, as a consequence, we obtain (3·102) </J-n=qn</Jn. Example 3.10. G simple, simply-laced, H = U(1)1 These are the "(generalized) parafermion" theories/ 00 >- 102 ' and hA'=(A', A')/2k. The ghost contribution in (3·96) exactly cancels that from the non-zero modes of ¢', i=1, ···, f. Thus _ q L1A,A·-c/24 w~c ~ (-1)1(W) [ bA,A,- 1 ) de ( ----.=II.,----.=II.--(,-::1---q-=-n e 2Jri6•(w*A-A') e'2c:;;",rr.6:-;;·"")(7::1---q-=-n'1-e-2""~r::oo;6""'·""') aEL1+c n;;;,J ' (3·103) where LJA,A' (A, A+2pc) 2(k+ hc v) (3·104) The integral over B-i.e. the projection onto the h-isospin A'-is easily computed by using (3 · 99), and we find Note that by construction this expression must agree with the more familiar expression for the parafermionic characters 101 > r/c,A in terms of the Kac-Peterson string functions. 103 ' There is an interesting subtlety arising in this example; namely, it would be incorrect to interchange the summation over naEZ, and the summation that defines <Pn•• due to an infinite degeneracy in the energy. The interchange may be made after splitting the sum and using (3 ·102). Example 3.11. G simple, H=®H,, rank(G)=rank(H)=f For this case hA'=~;(A'(il, A'(i'+2pH,)/(2(k+h'k)), where A'<il=A'IH,. The ghost contribution in (3·96) is now seen to exactly cancel that from the /3", r", aELJ+H, and the nonzero modes of¢\ i=1, ···, f. The projection onto the h-isospin A' gives 120 P. Bouwknegt, J. McCarthy and K. Pilch where LJA,A' (A, A+2pc) 2(k+ he v) (3·107) The example of G x G/G can be found in Ref. 44). Although we have not worked out the most general coset model in all detail, there are no conceptual difficulties in applying this procedure to obtain the character of any coset model. There is one subtlety, however, that it may be necessary to introduce a character valued partition function in the intermediate steps to make all the sums well-defined.*> This typically occurs when the centralizer of H in G is nonzero. It is well-known that the conformal dimension hA,A' of a coset primary field ([)A,A' is, in general, not just the difference between the conformal dimensions of G and H, but differs from this by the addition of a positive integer. One might wonder how this comes about in our derivation of the character since, e.g. Eqs. (3·106) and (3·107) seem to suggest otherwise. The answer is that if negative coefficients na, aELi+ are needed to satisfy A- A'= L: naa then, due to the identity cP-n = qnc/Jn, the ground state energy will be raised by lnal. Example 3.12. Concretely, in the example of G/H above (3·108) where LJA,A' is given in (3 ·107). However, to prove that the representation of conformal dimension hA,A' is present in the coset, one should show that the branching function bA,A' is nonzero. It is conjectured (and proved in several cases) in Ref. 104) that bA,A' is nonzero as soon as there exist na, aELJ;IH such that A-A'=L:naa. We have presented a procedure to obtain free field resolutions that works for arbitrary coset models. It can be a complicated problem to go further for a specific model in this way. However, there is certainly one case where we can give a complete treatment already, and that is the generalized parafermions. The reason is that in this case-and only this case-the projection may be achieved by a separation into parallel and perpendicular fields exactly as for the untwisted resolution in the finite dimensional problem. Then for a given irreducible coset module L~;If the required resolution is just the restriction of an appropriate "perpendicular subcomplex" of (FA, d) to H-isospin A. For this purpose it is convenient to bosonize the ,By-systems in the realization of {j. Following Ref. 82), we have *l As of course is also true in the WZNW-model itself. Free Field Approach to 2-Dimensional Conformal Field Theories /3a=aee•wa=a(e-;xa)e'wa, 121 (3·109) are conjugate spin (1, 0) fermionic first order fields, x bosonizes the and w is a symplectic boson. As is well-known, however, the Fock space of the xw-fields is not isomorphic to the /3r-Fock space. The bosonization process introduces additional degrees of freedom, namely ~oa. Equivalence of the Fock spaces is achieved by removing the zero modes ~oa and the identification of the xa and wa momentum operators. 82 >'96 >-99> The problem of the removal of ~oa, or equivalently the projection on nKerr;oa, has an elegant solution in the context of "resolutions". One can extend the intertwining operator r;oa=fe•xa to a 1-dimensionalline of Fock spaces. This 1-dimensionalline obviously defines a complex (due to the fermionic character of r;oa) which has trivial cohomology, and whose intertwiners commute with the intertwiners coming from {j (as they are independent of ~oa). Hence, the projection onto nKer r;oa is achieved by taking a half-infinite alternating sum. 21 > This will be made more explicit now. For G simple, the Cartan subalgebra currents of {j always are realized as h;(z) =Jk+hv ia¢;+~ae.:~.(a;, a)ra(z)/3a(z), and thus after bosonizing we obtain where r;~ r;~-system (3·110) Hence we may realize fi in terms of =!ka;ji(z), by the redefinition .t "parallel" bosonic fields ;p simply as h;(z) (3·111) A convenient set of "perpendicular" free fields, vanishing 2-point functions are w and i, such that the only non <¢i(z)¢i(w)>=-(a;, ai)ln(z-w), <ia(z)ia'(w)>= -saa' ln(z- w), <wa(z)wa'(w )>= saa' ln(z- w)' (3·112) (and thus w and x are trivially fi-invariant) is conveniently given by 18 > (3·113) In these fields the energy-momentum tensor for the G/H parafermion theory is indeed "free-field" form ~ T GtH = £..J ae<l+ (l( ·a ~a)2 _ _l_ z·a2 x~a_lt ·a2 ~a-( J k+ hv -If)( ) ·a2 ~a) 2'z·a w~a)2+_l_ v Jk v p, a z w . 2 zx 2 2z w h +h (3·114) Further, the {j-screening currents s; are independent of ¢-that is obvious since they commute with the generators of {j, in particular h,-and thus the intertwiners in the 122 P. Bouwknegt, ]. McCarthy and K. Pilch resolution (FA, d) of L~ simply reduce to "purely perpendicular" operators. This shows explicitly that the Fock spaces FA.L of the perpendicular fields indeed form a subcomplex. Denoting the value of the zero modes Pwa=- Pxa of wa, xa by na we have FA.L=EBn,ezF~·l.L. Then from (3·113) the projection onto a fixed H-isospin ,.\ is achieved by restricting the sum over naEZ to those satisfying A-t!=~ae.:~+naa. Thus we are led to the complex (3·115) where (3·116) This complex is not yet a resolution of the irreducible coset representation since by the bosonization we have introduced additional degrees of freedom l;oa. As explained above these can be removed by considering a (half-infinite) extension of the complex by the intertwining operators r;oa=feix', 0 0 ,1, ,1, d(-1) ···~FtP.L~ J, ···~ d(O) ~ d(l) F1~~.L ~ ,1, 7]oa ~ ~ ,1, ,1, F1o>.L ,A J, 7]oa 0 ,1, ··· 7Joa ~··· (3·117) ,1, By construction the cohomology of this resulting (multi-dimensional) complex is concentrated in F1~~.L, where it is isomorphic to the irreducible module L~~lf. We may, of course, apply this resolution to rederive the formula for the character given in (3·105). Using Lox=(1/2)PiPx+1), it is clear that the Fock space one reaches from F~:}.L by taking m steps with the intertwining operator r;oa has a highest weight vector whose conformal dimension differs from that of F~:J..L by the amount 1 1 1 -f<m+na)(m+na+ 1)---zna(na+ 1)=--zm(m+ 1)+nam. (3·118) Hence, taking the alternating sum of the zero modes contributing to the x trace over the 7Joa-directions of the complex, produces the factors of ¢n in the result. The alternating sum over the part of the complex coming from ii results, in the usual way, in the sum over the affine Weyl group. As an aside, note that the result (3·102) can now be understood as a consequence of the trivial 7Joa-cohomology! Also, the above derivation of the character constitutes another proof of the identity (3 · 99). The point of the reformulation was not the computation of characters. Rather, it allows the construction of representatives of chiral vertices between irreducible parafermion modules in the reduced free field space, as we now briefly indicate. Take first a representative of the highest weight state in LA•·••· This is easiest when A2 is in the finite dimensional irrep with highest weight A2, since the representative can Free Field Approach to 2-Dimensional Conformal Field Theories 123 then be taken to be an exponential in free fields-otherwise there will be derivative prefactors as discussed above. Now, since the screening operators only depend on the perpendicular fields, the construction continues precisely as discussed in § 3.4. For a generic coset model this problem seems difficult to attack using the BRST projection technique we have presented here, and there is a clear need for further investigation. It seems likely that progress might be made if the BRST projection was modified along the lines of Refs. 105)'"'"'107). Acknowledgments We would like to thank the following: V. Dobrev, V. Dotsenko, B. Feigin, G. Felder, M. Frau, E. Frenkel, K. Gaw~dzki, A. Lerda, G. Lusztig, N. Reshetikhin, S. Sciuto, ]. Sidenius for discussions; D. Nemeschansky for collaboration on Ref. 46). The Aspen Center for Physics where the writing of this work began. K. P. thank:; M. I. T. for hospitality during various stages of this work. Appendix A - - Some Lemmas on the Weyl Group - - In this appendix we establish some lemmas regarding the concept of twisted length of Weyl group elements introduced in §§ 2.4 and 3.2. to make contact with previous approaches. 40 J,soJ We recall that the usual length of a Weyl group element wE W (for both finite dimensional Lie algebras g and affine Kac-Moody algebras ii) is defined as the least number of simple reflections that are required to write w as a string of these simple reflections. An elementary result is the following : Lemma A.1.73l l(w)=i(J)(w)l, (A·1) where (A·2) We have the following generalization of Lemma A.1, which shows that definition (2·57) of the twisted length lw coincides with the one given in Ref. 50) Lemma A.2. (A·3) where (J)w±(w')=(w(Ll±) nLI+) n w'(LJ_) = w(Ll±) n (J)(w'). Proof We have l(w- 1 w') -l(w- 1)= ILl+ n w- 1 w'(LI-)I-ILI+ n w- 1(L1-)I = lw(Ll+) n w'(LJ_)I-Iw(Ll+) n LI-1 (A·4) 124 P. Bouwknegt, J. McCarthy and K. Pilch =I w(L1+) n w'(LJ_) n .a+l +I w(L1+) n w'(LJ_) n L1-l -lw(L1+) n LJ_ n w'(L1+)1-Iw(L1+) n LJ_ n w'(L1-)I =I (/)w +( w')l-1 (/)w -( w')l . D For affine Weyl groups W we have the additional possibility of taking (at least formally) infinite twists limN-ootNr characterized by some vector r, as explained in§ 3. For these a similar lemma holds. Define thereto .J~+>={a=n8+a, aEL1+, n~O}, .J~->={a=n8-a, aEL1+, n>O}. (A·5) Note that formally .J~±>=(limN-oofNr.J±) n .J+. Introduce furthermore (J)±(w)=.J~±>n w(.J_). (A·6) Then we have Lemma A.3. (A·7) where the twisted length lr is defined in (3·30). The proof is exactly the same as that of Lemma A.2 in the finite dimensional case. In particular it shows, as mentioned in § 3.2, that lr does not depend on the choice of r EM provided (r, a) >0, V aEL1+, and that the definition of lr coincides with that given in Refs. 40) (where it was called modified length) and 50). Appendix B - - Quantum Group Identities-In this appendix we collect some notations and lemmas which are used throughout the paper. Let qE C be such that q 2 =Fl. We use the following definitions from q-number analysis: qn-q-n q -ql, n [n]q!=IT[k]q, k=l [m] n q [n]q ! [m- n]q ! ' (B·1) known as the q-number, q-factorial and q-binomial, respectively. Let us review briefly the definition of the quantum group 'l.Jq(g). 108H 10> Suppose g is a finite dimensional Lie algebra with Cartan matrix aii=(aj, a;v) of rank .f. Fix a normalization such that (8, 8)=2 for the highest root 8 of g and defined;= {-(a;, a;) such that d;aij=djaj;. Fix a complex number q such that q 2d'=F1 (1~i~.f). Then, 'lJ q(g) is the associative C -algebra with generators e,, /,, MI, (1 ~ i ~f) (k, ~qa'), and relations (we use the conventions of Ref. 111)) Free Field Approach to 2-Dimensional Conformal Field Theories l~ij -1)"[1- az:;J j,l-ao-K /;/," =0 ( K=O K ' qdi 125 (B·2) This algebra is endowed with a co-multiplication, co-unit and antipode which makes it into a Hopf algebra. We refrain from giving their definitions as we will not need them here. We now describe the definition of the "quantum group Verma module" MAq. 112 > We define MAq=CfJq(g)vA where VA is a (highest weight) vector satisfying, i=1, ···,f. e;VA =0, (B·3) The space MAq has an (overcomplete) basis consisting of monomials j,···/,nVA, and is a CfJ q(g) module under the action j;(/,···f,nVA)= j;j,···f,nVA, V )=q(A,at)q-dt(a<t,+···+aunlj ···ftnA, V k z·(fl1 ···ftnA it1 (B·4) where a> is defined in (3·26). This module is integrable for AEP+ and reduces to the conventional Verma module for gin the limit q-+1, i.e., is a deformation of MA. 112 > For q a root of unity, say q=exp(7ri/(k+hv)), the algebra CfJq(n_) contains an ideal .5 q generated by the elements Ia 2 <k+hVJt<a,a>, a ELl+. Let us define q] q(n_) =CfJin-)/ Sq. Then we might also consider the "restricted quantum group Verma module" MAq=q] in-)vA, where VA satisfies the identities (B·3). This becomes a CfJq(g) module under the action (B·4). The following lemma proves to be useful for the explicit determination of the intertwiners : Lemma B.1. 113 > Consider the associative algebra with two generators A, B and defining relations A 2 B-(q+q- 1 )ABA+BA2 =0, (B·5) AB2 -(q+q- 1 )BAB+B2 A=O. (B·6) Define C=-AB+q- 1 BA, then (i) AC=qCA, qBC=CB; (B·7) At At-> C> Bk-J Bk 2: q>+(k-j)(l-j) • (ii) - - - - - - = [k]q ! [l]q ! o,;;,,;;mm(k,/) [!- J ]q ! [j]q ! [k- j]q ! (B·8) (iii) (B·9) (iv) (B·10) 126 P. Bouwknegt, ]. McCarthy and K. Pilch If in addition q is a root of unity, then there are additional relations. Lemma B.2. 40 l Let A, B and C be as in Lemma B.l. = M- k for 0 < k < M, then for 0 < l < k < M we have Let qM = -1, and define Ai;.+IBI=(-1)1 ~ qH<I-j)(k-j) [l]q ![k]q! Al-jCjBI-JAk. [j]q ![/-j]q ![k-j]q! ' (B·ll) (ii) AIB.k+t=( -1)1 ~ qH<I-j)(k-j) [l]q ! [k]q ! Bk Al-jCjBI-j [j]q![/-j]q![k-j]q! . (B·12) (i) O~j~/ O~j~/ k Appendix C - - Cohomology of Double Complexes - - In this appendix we summarize briefly elementary facts about the cohomology of double complexes of the type discussed in §§ 2.5 and 2.6. For a more detailed exposition and proofs the reader may consult standard textbooks on homological algebra, e.g., Refs. 48) and 114). C.l. A double complex (X, d', d") is a bigraded module X={X<Mllp, qEZ} with differentials d': X<Ml--> x<P+l,q) and d": X<Ml--> X<M+ll such that DEFINITION d'd'=O, d"d"=O and d'd"-d"d'=O. (C·1) If we represent a double complex as a family of modules in the p-q plane then the first two conditions d' d'=O and d" d"=O say that each row and each column is a complex. Denote the corresponding cohomology groups by H~f.ql(X) and H~f,.ql(X), respectively. [The reader should not be mislead by the notation. Depending on the case, p and q label the P-th cohomology group in the q-th complex in H~f.ql(X), or the other way round in H~f,.ql(X).] Using the las,t condition d' d"- d" d'=O one verifies easily that d": H~f.ql(X)--> H~f.q+ll(X) and similarly d': H~f,.ql(X)--> H~+l,ql(X) are well-defined and nilpotent. The cohomology groups of these two complexes are denoted by H~f,.qJ Hd'(X) and H~f.qJ Hd"(X), respectively. With any double complex one can also associate a single complex (K, D), where K={K<nl=EJjp+q=nX<Ml} and D=d'+( -1)Pd" on x<Ml. Properties (C·l) then guarantee that DD=O. A natural question that arises is to what extent can one determine the cohomology of the complex (K, D) given some information about the cohomologies of the row and column complexes in the double complex. The answer can usually be obtained by examining the so-called spectral sequence, and the results we review in the remainder of this appendix are in fact the most elementary consequences of such analysis. Let us restrict to a double complex in the first quadrant, i.e., such that x<Ml=O if P<O or q<O. In §§ 2.5 and 2.6 we make frequent use of the following theorem. THEOREM C.l. Let (X, d', d") be a double complex such that the cohomology groups H~f,.ql(X)=O for q=t=qo. Then n<qo, n?:::.qo. (C·2) Free Field Approach to 2-Dimensional Conformal Field Theories Similarly, if H~f.q>=o H}r>(K) 127 for P=I=Po, then ={H~t;,o.n-Po>Hd·(X~ : n<Po, n?:Po. (C·3) Proof We use the notation of Ref. 114), Ch. III). Since H~t;,.q>=t=o only for q=qo we have mM>(X)=H~f.q>Hd"(X)=O for q=l=qo. But then d2=0 on E£*·*>(X), which implies that in fact Ei*·*>(X)=E2(*-*>(x). Then (C·2) follows from the fact that the cohomology of the single complex is given by (Ref. 114), Theorem 14.14, p. 165) (C·4) D The proof of (C·3) is analogous. Remark The double complexes that are discussed in§§ 2.5 and 2.6 can be brought to the first quadrant by a suitable relabelling. An interesting consequence of Theorem C.1 is Lemma C.2. If H~f.q>(X)=O for P=I=Po and H~·q>=o for q=l=qo then (C·5) Proof Take n=p+qo in (C·2) and (C·3). Appendix D --Proof of Theorems 2.12 and 2.12' - - THEOREM 2.12. Let FA be a non-twisted Fock space G-module with the highest weight A (not necessarily dominant), and SA the space of all H-singular states in FA. Then the ERST cohomology of FA considered as an H-module is given by (D·1) Proof In this proof we make use of standard techniques as elaborated for a similar problem in Ref. 43). Since we identified the positive root generators of h with those of g, we see that the general form of the representation of ea, aELl+ H in FA, which we discussed in § 3.1, is ea =,a a+ higher degree terms . (D·2) On the space FA®Fgh we can introduce additional grading according to the "total ghost number", i.e. we define a degree (deg) such that deg(ya)=-deg(,Ba)=1, aELJ+G, deg(c-a)= -deg(ba)=1, aELl+ H. (D·3) (D·4) Then the "higher degree terms" in (D·2) have deg> -1. Also, on defining (D·5) we can split Q as 128 P. Bouwknegt, J. McCarthy and K. Pilch (D·6) Let NH="'2.aE.tJ.•(c-aba-ra/3a) be the total "H-ghost number" operator. that all eigenvalues of this operator are non-negative. Then we have Lemma D.l. Note The cohomology of Qo on FA@Fgh is H~~'(FA@Fgh)-::::<on,oFA.l., (D·7) where FA.l. is the subsPace of FA annihilated by /3a, aE.Ll+ H. Proof Using the identity {Qo,- "'2, bara}=NH' (D·8) aELJ+H we see that any closed state ¢, NH¢=N¢, N=i=O, is also exact. Indeed, (D·9) Thus the cohomology of Qo must be concentrated at n=O on the states which are annihilated by all /3a, aELJ+H· These states are in 1-1 correspondence with FA.l.. D To proceed further we must introduce yet another gradation in the space FA@Fgh. Since the embedding is regular we can consistently assign G-weights (!) +a and -a to the ghosts ba and c-a, respectively. The complex has then G-weight direct sum decomposition, FA@Fgh="'i},;.F;., and each weight space F;. is mapped by Q and Qo into itself. Thus we can consider each subcomplex (F;., Q) separately. Moreover, since Fgh, and similarly each weight space in FA, are finite dimensional, we see that the F/s must also be finite dimensional. We will now prove that for n > 0 the cohomology of Q is trivial. Observe that ¢ EF}n> is of the form¢= rf;t+ ··· + ¢L, rf;t=i=O, f>O, where vectors¢, have definite degree, deg(rf;i)=i, i=f, , L, and Lis the maximal degree occurring in F}n>. Call ord(¢) =deg(rf;t)=f. Since (D·10) Q¢=0 implies Qor/Jt=O. But then using Lemma D.1 for n>O we find r/Jt=QoXt, where deg(xt) =deg( rf;t). Consider now ¢- Qx~. which represents the same cohomology class. Since ord(¢-Qxt)>ord(¢), it is clear that repeating the same reasoning L-f more times we will find vectors Xm, ···,XL such that rf;=Q(xt+···+xL). In the last step one uses that Q>xL=O, because ord(Q>xJ>L. Thus we have shown that the cohomology of Q is concentrated at n=O, and, since (D·ll) it is given by (D·1). This concludes the proof of Theorem 2.12. D In the above proof we used only a part of the result in Lemma D.1, i.e. the triviality of the Qo cohomology for n >O (which implied N>O). If we parametrize the Fock space FA such that the generators of n+ H are realized using only the parallel variables, ra, /3a, aELJ+H then we can prove the following sharpening of Theorem 2.12. Free Field Approach to 2-Dimensional Conformal Field Theories THEOREM 2.12'. 129 If the generators ea, aELh H are realized only in terms of parallel variables then (D·12) Introduce an H-degree (deg') and the corresponding H-order (ord') by restricting a in (D·3) to lie in LJ+ H. [Obviously, this H-degree coincides with the H-ghost number.] We can repeat the entire proof of Theorem 2.12 with deg and ord replaced with deg' and ord'. We find that a nontrivial cohomology of Q must be concentrated on states with ord'=O. The cohomology states of Qo are in 1-1 correspondence with vectors ¢EFA such that deg'(¢)=0 (Lemma D.1). Clearly all such vectors define also cohomology states of Q. On the other hand if ¢1 and ¢z, Q¢1 = Q¢z=O, have the same component of deg' = 0, then ord'( ¢1- ¢z) > 0, which implies [ ¢1] = [ ¢z] in H~o). This shows that both cohomology of Q and Qo are realized by the same vectors which have vanishing total H-ghost number. D Proof Appendix E --Restricted QG Verma Modules-- In this appendix we argue that the cohomology of a naive generalization of the BGG complex to restricted QG Verma modules is trivial. We will consider only the simply-laced case. To define a BGG-like complex for restricted QG Verma modules we observe that the standard BGG complex, (MA, d), can be obtained from the Fock space resolution, (FA, d), of an irrep LA of a semisimple Lie algebra as follows: Given (FA, d) each component dw',w: Fw*A ~ Fw'M of the differential d corresponds to a singular vector, dw',w[s,Hj,]vw*A• in the Verma module Mw*A (Ref. 40) Theorem 3.6), and thus defines an embedding dw,w': Mw'*A ~ Mw*A· The BGG resolution is constructed by simply replacing Fock spaces with Verma modules (with the same highest weight) and reversing all the arrows in the complex. Similarly, given the Fock space resolution of an irrep LA of Kac-Moody algebra fj we have found (Theorem 3.5) that each component dw',w of the differential corresponds to a singular vector in the restricted Verma module MZ,*A of the quantum group, CU q(g ). By the same procedure as in the finite dimensional case we then obtain a complex (MA q, d) of CU q(g) restricted Verma modules. Our discussion of the CU isu(2)), CU isu(3)) and CU isu(N)) cases, which is presented below, suggests that the following is true : THEOREM E.l. The cohomology of (liiAq, d) is trivial. By definition the restricted Verma module MAq, AEP+, is isomorphic with the quotient CfJq(n_)=CUin-)/ .J Aq, where .J Aq is the (double-sided) ideal generated by the elements t:+hV, aELl+. For g=su(2), q] in-) is generated by a single element, /, satisfying jk+ 2=0, and thus, as a vector space, is spanned by 1, /, ···, jk+I. By inspection we obtain Proof Lemma E.2. For ¢EV q(n~u<z)), 130 P. Bouwknegt, ]. McCarthy and K. Pilch cJ;r=o iff cf;=xJk+Z-n' n:::;;:k+1 (E·l) for some xEq) q(n~u<z>). The proof of the theorem in the CfJ q(su(2)) case follows directly from this lemma. The algebra Cf)q(n'!!< 3 >) is generated by/,, i=1, 2, which satisfy (E·2) together with the Serre relations (B·2). The PBW theorem gives then two convenient bases in CfJ q(n~u< 3 >), 115>' 116 > which consist of elements of the form Jr!t.f/, i=l=j, O:::;;:m, n, p:::;;:k+2, (E·3) one with i=1, j=2 and the other with i=2, j=l. The analogue of Lemma E.2 in this case is cf;//=0 iff cf;=x/.r 3-n, j=1, 2, n:::;;:k+2 (E·4) for some xEq) q(n~u<s>). To prove it one expands cf; in one of the bases (E·3) and explicitly constructs x so that (E·4) is satisfied. In the context of the oomplex (MAq, d) the result of Lemma E.3 can be interpreted that the restriction to the line along a simple root direction yields a complex with trivial cohomology. More precisely, given MZ,*A, wE W, and a;, the restricted complex is of the form (E·5) where t;=(w *A+ p, a;)mod k+3 and T;=k+3- t;, with the normalization of the embeddings determined by that of d, as in (3·39) (see also Ref. 40)). Introduce a filtration · · · 2 Kz 2 K12 Ko 2 K-12 ·· · of the complex (MA q' d) such that factor complexes Kn+l/Kn coincide with complexes (E·5). Explicitly, we may construct spaces Kn, nEZ as follows. Let A, be a fundamental weight in P+. Then K21-1= lEZ. (E·6) The complex (MAq, d) has a weight space decomposition. As usual let (MA,A, d) denotes the subcomplex with weight A. Using vanishing conditions (E·2) one deduces that there can only be a finite number of spaces in this complex which are non-zero. Thus the filtration (E·6) induces a finite filtration {Kn(A), nEZ} of (MA,A, d). Moreover, we have just shown that for each n the quotient complex associated with this filtration has a trivial cohomology. A standard way to proceed (see, e.g. Ref. 114), pp. 157-158) is to consider a long exact sequence in cohomology that arises from the short Free Field Approach to 2-Dimensional Conformal Field Theories 131 exact sequence 0 ~ Kn(.-1) Since ~ Kn+l(.-1) ~ H~i>(Kn+l(.-1)/Kn(.-1))=0, Kn+l(.-1)/Kn(.-1) ~ 0. (E·7) Vi, we find that H~i)(Kn+l(..t))~H~i>(Kn(.-1)), Vi, nEZ. (E·8) However, for a fixed .-1, Kn(.-1)={0} provided n is large enough, which together with (E·8) implies that in fact H~;>(Kn(.-1))=0 for all i and n. Thus the cohomology of (MA,;., d), and consequently of (MA, d) is trivial. It is straightforward to extend the above arguments to CU isu(N)) using the induction on N. Choosing N -1 simple roots one may define a filtration such that the quotient complex corresponds to a complex of CU isu(N -1)). The reminder of the proof proceeds then as above. In fact it is clear that the same result should hold for any quantum group CU q(g). D In§ 3.4 we consider a quotient module LAq=MAq/MJH>, where as usual ifx<-I> is the submodule generated by the singular vectors flA+p,a,v>vA. In fact one can introduce such a quotient for an arbitrary value of q=I=O (when q is not a root of unity one takes the quotient of the full Verma module by its maximal submodule). For AEP+, LAq is an irreducible module of the quantum group/ 17> and it is clear that for a generic q the structure of LAq should be exactly the same as of the irreducible representation of the underlying finite dimensional Lie algebra (see e.g. Ref. 112)). However, it is less clear when q is a root of unity, because the overlaps between submodules generated by different singular vectors may in principle change due to some peculiar identities between q-numbers. On the other hand if we consider weights which are integrable, i.e. (A+ p, 8)< k+ hv, it is reasonable to expect that the structure of LAq will be similar to its finite dimensional counterpart, because in this case only "low powers" of the generators are present. In fact the latter observation is true. We will now demonstrate this in the case of CU q(su(2)) and CU q(su(3)). E.4. Let A be a dominant integrable weight of CU q(g). Then there is a 1-1 correspondence between the states in LAq and in the corresponding representation LA of the finite dimensional Lie algebra g. THEOREM Proof We will present a somewhat pedestrian approach in the case of CU q(su(2)) and CUq(su(3)). In the case of CUq(su(2)) the states of L]a, 2j~k, are spanned by (E·9) so the theorem is obvious. In the case of CU q(su(3)) we will construct explicitly the basis in LAq using the bases (E·3) in the restricted QG Verma module. We will see that in comparison with the finite dimensional case the only complication is to keep track of vanishing conditions (E·2). Let t1=n1a1 +n2a2 be a weight, ;\~A, with the corresponding weight spaces MAq(.-1) and MA(.-1) in the Verma modules MAq and MA, respectively. Let l;=(A+p, a;). We 132 P. Bouwknegt, ]. McCarthy and K. Pilch need to consider four cases : 1. n1 < /1 and n2 < l2 ; 2. n1 < /1 and n2;;::.: !2 ; 3. n1;;::.: /1 and n2 < !2 ; 4. n1zl1 and n2 z/2. In the first case all the states -rn•-J112 -{J f:nz-J · 0, ···,m1n · ( n1, n2 ) , 11 2 VA, ;= (E·10) that span MAq(ll) and/or MA(Il) are linearly independent in LAq and/or LA. In the second case we find the same correspondence, except that n2 -!2 < j s. n1, where the lower range restriction removes the states in (E·10) which are descendants of the singular vector j.f•vA. Clearly if n1S.n2-l2 there will be no states in the quotient. Similarly, using the other basis in (E·3), we show the 1-1 correspondence between the states in LAq(ll) and LA(Il) in Case 3. In Case 4 let us assume that n1S.n2. Then in the Verma modules we have bases: (E·ll) (E·12) Vectors Vj in (E·ll) with jS.n2-l2 span the submodule MMA(Il) of MA, while Vj in (E·12) with j S.minCn2-l2, k+2) span the submodule M~•*A(Il) of MAq. The submodules MnM(I!) and M~•*A(Il) are spanned by a set (possibly overcomplete in the case of M~.*A(Il)) of vectors (E·13) max(O, n2- k-2)S.j sminCn1-l1, k+2). Let us first consider MMA(Il). (E·13) into the basis (E·10) (E·14) Using identity (B·8) with q=1 we can expand (E·15) Note that all coefficients in the sum on the r.h.s. are nonvanishing, and the highest term in the expansion of iJJ is vJ+li· If /1 S. n2-l2+ 1, we can use (E·15) to express all basis vectors (E·ll) as a linear combination of iJ/s and vectors in Mrz*A(Il). Thus in this case LA (I!) is trivial. If /1 > n2 -12 + 1, we first observe, by the highest term in the expansion (E·15), that none of iJJ lies in Mrz*A(Il). Thus MAHl(l!)= MMA(Il) ffiMrz*A(Il). Then it is clear that the vectors Vnz-lz+l, ···, Vz,-~ form a basis in LA(Il). In M~•*A(Il) we obtain using (B·8) (E·16) Once more all the coefficients on the r.h.s. are non-zero if the j's are within the indicated range (E·14). Thus, if /1S. n2-l2+ 1 the same argument as above shows that LAq(ll) is trivial. If l1>n2-l2+1, the integrability of A implies that n1sk+2. Indeed, if we had n1 > k + 2 then also n2 > k + 2 and therefore I!:::; woA. Since woA is the lowest weight in LA, we deduce that LA(Il) is trivial and thus we must have /1 sn2-l2+1, which is a contradiction. Since n1sk+2, all iJ/s in (E·14) are linearly Free Field Approach to 2-Dimensional Conformal Field Theories 133 independent, and, exactly as in the finite dimensional case, we show that Vnz-tz+l, ···, V1,-1 form a basis in LAq(t\). 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