Free fermion resolution of supergroup WZNW models
aa r X i v : . [ h e p - t h ] J un Free fermion resolution of supergroup WZNW models
Thomas Quella and Volker Schomerus Korteweg-de Vries Institute for Mathematics, University of Amsterdam,Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands DESY Theory Group, DESY Hamburg,Notkestrasse 85, D-22603 Hamburg, Germany
Abstract
Extending our earlier work on
P SL (2 | GL ( M | N ) along with several close relatives such as P SL ( N | N ), certain Poincar´e supergroups and the series OSP (2 | N ). This remarkable progressrelies on the use of a special Feigin-Fuchs type representation. In preparation for the field theoryanalysis, we shall exploit a minisuperspace analogue of a free fermion construction to deduce thespectrum of the Laplacian on type I supergroups. The latter is shown to be non-diagonalizable.After lifting these results to the full WZNW model, we address various issues of the field theory,including its modular invariance and the computation of correlation functions. In agreementwith previous findings, supergroup WZNW models allow to study chiral and non-chiral aspectsof logarithmic conformal field theory within a geometric framework. We shall briefly indicatehow insights from WZNW models carry over to non-geometric examples, such as e.g. the W ( p )triplet models. Keywords:
Conformal Field Theory, Logarithmic Conformal Field Theory, FreeField Constructions, Supergroups, Lie Superalgebras, Representation TheoryDESY 07-074 NSF-KITP-07-128KCL-MTH-07-06 arXive/0706.07441 ontents
Two-dimensional non-linear σ -models on supermanifolds have been a topic of considerable interestfor the past few decades. Their realm of applications is vast, ranging from string theory to statisticalphysics and condensed matter theory. In the Green-Schwarz or pure spinor type formulation ofsuperstring theory, for example, supersymmetries act geometrically as isometries of an underlyingspace-time (target space) supermanifold. Important examples arise in the context of AdS/CFTdualities between supersymmetric gauge theories and closed strings. Apart from string theory,supersymmetry has also played a major role in the context of quantum disordered systems [1, 2, 3, 4]and in models with non-local degrees of freedom such as polymers [5]. In particular, it seems to bea crucial ingredient in the description of the plateaux transitions in the spin [6, 7] and the integerquantum Hall effect [8, 9, 10]. 2n addition to such concrete applications there exist a number of structural reasons to beinterested in conformal σ -models with target space (internal) supersymmetry. On the one hand,being non-unitary, the relevant conformal field theory models exhibit rather unusual features suchas the occurrence of reducible but indecomposable representations and the existence of logarithmicsingularities on the world-sheet. In this context, many conceptual issues remain to be solved, bothon the physical and on the mathematical side. These include, in particular, the constructionof consistent local correlation functions [11], the modular transformation properties of characters[12, 13], their relation to fusion rules [14, 15, 16], the treatment of conformal boundary conditions[17, 18] etc. On the other hand, the special properties of Lie supergroups allow for constructionswhich are not possible for ordinary groups. For instance, there exist several families of cosetconformal field theories that are obtained by gauging a one-sided action of some subgroup ratherthan the usual adjoint [19, 20, 21, 22]. The same class of supergroup σ -models is also known toadmit a new kind of marginal deformations that are not of current-current type [23, 24]. Finally,there seems to be a striking correspondence between the integrability of these models and theirconformal invariance [25, 26, 21, 22].In this note we will focus on the simplest class of two-dimensional conformal σ -models, namelyWZNW theories, in order to address some of the features mentioned above. The two essential prop-erties which facilitate an exact solution are (i) the presence of an extended chiral symmetry based onan infinite dimensional current superalgebra and (ii) the inherent geometric interpretation. While(ii) is common to all σ -models, the symmetries of WZNW models are necessary to lift geometricinsights to the full field theory. Both aspects single out supergroup WZNW theories among mostof the logarithmic conformal field theories that have been considered in the past [27, 11, 28] (seealso [29, 30] for reviews and further references). While investigations of algebraic and mostly chiralaspects of supergroup WZNW models reach back more than ten years [31, 32, 33, 34, 35] it wasnot until recently that the use of geometric methods has substantially furthered our understandingof non-chiral issues [36, 24, 37]. In the last three references the full non-chiral spectrum for the GL (1 | P SU (1 , |
2) and the SU (2 |
1) WZNW models has been derived based on methodsof harmonic analysis. The most important discovery in these articles was the relevance of so-called projective covers and the resulting non-diagonalizability of the Laplacian which ultimatelymanifests itself in the logarithmic behaviour of correlation functions.This paper will put these results on a more general and firm conceptual basis by consideringrather arbitrary supergroup WZNW models based on basic
Lie superalgebras of type I . The defin-ing properties of these Lie superalgebras are (i) the existence of a non-degenerate invariant form(not necessarily the Killing form) and (ii) the possibility to split the fermionic generators into two multiplets which transform in dual representations of the even subalgebra. The first feature isnecessary to even spell out a Lagrangian for our models. Our second requirement can be exploitedto introduce a distinguished set of coordinates in which the Lagrangian takes a particularly simpleform. These arise from some Gauss-like decomposition in which a bosonic group element is sand-wiched between the two sets of fermions. The construction resembles the free field construction ofbosonic models [38, 39, 40, 41, 42], but its fermionic version turns out to be easier to deal with In contrast to some appearances in the physics literature we will use the word “indecomposable” strictly in themathematical sense. According to that definition also irreducible representations are always indecomposable sincethey cannot be written as a direct sum of two other (non-zero) representations. Instead of referring to the names “Kac-Moody superalgebra” or even “affine Lie superalgebra” which are fre-quently used in the physics community, we will stick to the notion current superalgebra by which we mean a centralextension of the loop algebra over a finite dimensional Lie superalgebra. The generators of the underlying current superalgebraof our WZNW model are thus constructed from currents of the bosonic subalgebra along with anumber of free chiral fermionic ghost systems which equals the number of fermionic generators.As was observed in [24] already, at least for the example of
P SU (1 , | simple Lie superalgebras for a moment our analysis covers threetypes of infinite series, namely A ( m, n ) = sl ( m | n ) (for m = n ), A ( n, n ) = psl ( n | n ) and C ( n +1) = osp (2 | n ) [45]. But, widening Kac’s original usage of the qualifiers “basic” and “type I”,most of our results also apply to non-(semi)simple Lie superalgebras such as various extendedPoincar´e superalgebras, the general linear Lie superalgebras gl ( m | n ) or supersymmetric extensionsof Heisenberg algebras. We wish to stress that our general results below contain a solution of the
P SL ( n | n ) WZNW models. What makes these particularly interesting is the fact that their volumeis an exact modulus, in contrast to bosonic non-abelian WZNW models [23, 22]. It is also worthemphasizing that the isometries of flat superspace, AdS-spaces and many projective superspacesfall into the classes mentioned above. We thus expect our work to be relevant for these models aswell. A few comments in that direction can be found in the conclusions.The plan of this paper is as follows. In the next section we shall provide a detailed account ofLie superalgebras of type I and the associated representation theory. Particular emphasis is puton the structure of projective modules, i.e. typical Kac modules and projective covers of atypicalirreducible representations. Afterwards we present the supergroup WZNW Lagrangian in section 3and use a Gauss-like decomposition in order to rewrite it in terms of a bosonic WZNW model,two sets of free fermions and an interaction term which couples bosons and fermions. This freefermion resolution is shown to have an algebraic analogue on the level of the current superalgebrawhich constitutes the symmetry of the supergroup WZNW model. The analysis of the zero-modespectrum in the large volume sector is performed in section 4 using methods of harmonic analysis.Most importantly, we shall determine the representation content for the combined left right regularaction on the algebra of functions over the supergroup. To achieve our goal, we use a reciprocitybetween atypical irreducible representations and their projective covers. On the way we also prove See [43, 44] for a related approach. WZNW models based on Heisenberg algebras may be used to describe strings on maximally symmetric planewaves [46, 47]. L and ¯ L . At the end of section 5, we propose a universal partition functionresembling a charge conjugate invariant and gather some thoughts about the possibility of havingnon-trivial modular invariant partition functions. In the concluding section 6 we argue that thesolution of the logarithmic triplet model [11] formally fits into the framework outlined before. Thisobservation is used to speculate about the structure of general logarithmic conformal field theories.Most of the statements which appear in the main text can be turned into mathematicallyrigorous propositions. This applies in particular to all algebraic manipulations. In our discussionof spectra, however, we focus on models based on finite dimensional representations. The mostinteresting supergroups, on the other hand, are based on non-compact and occasionally on non-reducive groups. While we believe that our discussion may be extended to such cases, a fullycomprehensive presentation would have required to carefully distinguish between different realforms. In the present note, we rather preferred to put the emphasis on the algebraic structuresthat – in our opinion – are equally relevant for all type I supergroup WZNW models. The main actress of this paper, the Lie supergroup G , is best introduced in terms of its underlyingLie superalgebra g . We will assume the latter to be finite dimensional, basic and of type I. Theattribute “basic” guarantees the existence of a non-degenerate invariant metric and is needed inorder to exclude certain pathological cases which would even rule out the existence of a WZNWLagrangian. The predicate “type I”, on the other hand, implies the existence of two multipletsof fermionic generators and will simplify the interpretation of the chiral splitting in the conformalfield theory we are considering.In the remainder of this section we shall first present the commutation relations of a general(possibly non-simple) basic Lie superalgebra of type I. Afterwards we summarize their represen-tation theory following the beautiful exposition of Zou [48] (see also [49]). The reader who is notinterested in the mathematical details might wish to skip over parts of this section in the firstreading. A Lie superalgebra g = g ⊕ g is a graded generalization of an ordinary Lie algebra [45]. Thereare even (or bosonic ) generators K i which form an ordinary Lie algebra g , i.e. they obey thecommutation relations [ K i , K j ] = if ijl K l , (2.1)with structure constants that are antisymmetric in the upper indices and that satisfy the Jacobiidentity. In addition, type I Lie superalgebras possess two sets of odd (or fermionic ) generators S a and S a , a = 1 , . . . , r (generating g ) which transform in an r -dimensional representation R of g and its dual R ∗ , respectively [50]. Rephrased in terms of commutation relations, this statementmay be expressed as[ K i , S a ] = − ( R i ) ab S b [ K i , S a ] = S b ( R i ) ba . (2.2)5he symbol R i is an abbreviation for the representation matrix R ( K i ). In a type I superalgebra,the anti-commutators [ S a , S b ] and [ S a , S b ] vanish identically [50]. On the other hand, generators S a do not anti-commute with S b . Before we are able to spell out their commutation relations, weneed to introduce the supersymmetric bilinear form h K i , K j i = κ ij h S a , S b i = δ ab . (2.3)We assume κ ij (not necessarily the Killing form) to be invariant with respect to g and, moreover,to be non-degenerate such that its inverse κ ij exists. The latter is a crucial ingredient in thedefinition [ S a , S b ] = − ( R i ) ab κ ij K j . (2.4)The structure constants which appear in this relation are uniquely determined by the requirementthat the metric (2.3) is invariant, i.e. h [ X, Y ] , Z i = h X, [ Y, Z ] i . The supersymmetry and non-degeneracy of the metric on the full Lie superalgebra g follow immediately from the definition.The commutation relations above preserve the fermion number S a ) − S a ). Hence g andalso its universal enveloping superalgebra U ( g ) have a natural Z -grading (localized in three degrees)which is consistent with the intrinsic Z -grading [50]. It is this property which distinguishes type ILie superalgebras among all Lie superalgebras. Let us also emphasize that the Cartan subalgebraof g will always be identified with that of g in what follows. This will be important below whenwe introduce highest weight representations.Before we end this subsection on the definition of type I superalgebras, let us reflect a bit onhow restrictive their structure is. In fact, in building a Lie superalgebra one cannot just come upwith any bosonic subalgebra g and hope to extend it by adding fermions transforming in somerepresentation R of g . There is an additional constraint, namely the graded Jacobi identity. Whilethe latter is by assumption identically satisfied for g and the mixed bosonic/fermionic commutatorsdo not impose any new conditions, there is a non-trivial restriction arising from the commutator[ S a , [ S b , S c ]] and its cyclic permutations. This leads to the requirement( R i ) bc κ ij ( R j ) ad + ( R i ) ac κ ij ( R j ) bd = 0 . (2.5)An equivalent formulation is to demand that the quadratic Casimir vanishes on the symmetric partof the tensor product R ⊗ R . Alternatively, the constraint on the choice of R may be rephrased byrequiring that the tensor A abcd = ( R i ) ac κ ij ( R j ) bd (2.6)is antisymmetric in the upper two as well as the lower two indices. Although the property (2.5)(or (2.6)) looks rather innocent it will be a crucial ingredient in many of the equalities we shallencounter. In the analysis of supergroup WZNW models there are a variety of representations of the underlyingLie superalgebra which play a role. The aim of this section is to provide a brief summary of therelevant modules of finite dimensional type I Lie superalgebras following Zou’s exposition [48].For definiteness, all the definitions and statements that follow below will be formulated for finitedimensional representations. It is understood, though, that our definitions can be extended toinfinite dimensional representations (discrete and continuous) as well. Whether this is also true fortheir properties, however, remains to be investigated.6 .2.1 Kac modules and their duals
Let us denote by Rep( g ) the set of isomorphism classes of irreducible representations of the bosonicsubalgebra g . The basic building blocks in the representation theory of Lie superalgebras of type Iare Kac modules K µ , µ ∈ Rep( g ) [45, 50]. They are induced from irreducible representations V µ ofthe bosonic subalgebra g . More precisely, the representation is extended by letting one multipletof fermionic generators S a act trivially on the vectors v ∈ V µ . The remaining states in the Kacmodule are then created by acting with generators from the second multiplet of fermions, S a . Fromour verbal description we immediately infer the decomposition of Kac modules with respect to thebosonic subalgebra, K µ (cid:12)(cid:12) g = V µ ⊗ F = M ν (cid:2) K µ : V ν (cid:3) V ν . (2.7)Here and in what follows we assume all g -modules to be fully reducible and denote the resultingmultiplicities in terms of the square bracket [ M : N ] where M is an arbitrary (fully reducible) g -module and N an irreducible g -module. The g -module F = V ( S a ) appearing in the previousequation is the exterior (or Grassman) algebra generated by the fermions S a . Its structure as a g -module is determined by projecting tensor powers of the module R ∗ onto their fully anti-symmetricsubmodules, F = V ∗ ⊕ R ∗ ⊕ (cid:2) R ∗ ⊗ R ∗ (cid:3) antisym ⊕ (cid:2) R ∗ ⊗ R ∗ ⊗ R ∗ (cid:3) antisym ⊕ · · · ⊕ (cid:2) ( R ∗ ) ⊗ r (cid:3) antisym . (2.8)The n -fold tensor product here corresponds to a state involving n fermionic generators S a i , i =1 , . . . , n . The case of no fermionic generators leads to the one-dimensional trivial representation V = V ∗ . It is obvious that the series will truncate after the r -th tensor product since the fermionicgenerators S a anti-commute among themselves. Consequently, the dimension of Kac modules isalways given by dim( K µ ) = 2 r dim( V µ ).In close analogy to the previous definition we may also introduce dual Kac modules K ∗ µ bystarting with the dual bosonic representation V ∗ µ = V µ + . Deviating from the above construction wenow let the first set of fermionic generators S a act trivially on the corresponding vectors and use S a to create new states. Since the two sets of fermionic generators transform in dual representationsthe bosonic content is then obviously given by K ∗ µ (cid:12)(cid:12) g = V ∗ µ ⊗ F ∗ = (cid:0) V µ ⊗ F (cid:1) ∗ = M ν (cid:2) K ∗ µ : V ν (cid:3) V ν . (2.9)The dimensions of the modules K µ and K ∗ µ coincide and it may easily be seen that the representa-tions are indeed dual to each other.Let us conclude the discussion of Kac modules with a short comment about the last term inthe fermionic representation F , eq. (2.8). Innocent as it seems, it is important to stress that thehighest component [ R ⊗ r ] antisym need not be the trivial g -module V again, even though it certainlyis one-dimensional. The action of the bosonic subalgebra on this space can be calculated explicitly, K i · (cid:0) S · · · S r (cid:1) = − tr( R i ) S · · · S r . (2.10)In case g is semisimple, it admits a unique one-dimensional representation, namely the trivial g -module V . Hence, we conclude that tr( R i ) = 0 for Lie superalgebras with a semisimple bosonicsubalgebra. In the following we shall refer to the right hand side of this equation and other restrictions of g -modules tothe bosonic subalgebra g as the “bosonic content”. Hence, this phrase is not related to the Z -grading of therepresentation space. .2.2 Simple modules and their blocks Kac modules provide an important intermediate step to constructing irreducible representations .Finding their exact relation with irreducibles, however, requires good control over the structure ofKac modules. For generic labels µ , the (dual) Kac modules turn out to be irreducible. Thereby, theygive rise to what is known as typical irreducible representations L µ = K µ . But there exist specialvalues of µ for which the associated Kac module contains a proper invariant subspace. The so-called atypical irreducible representations L µ are obtained from such K µ by factoring out the uniquemaximal invariant submodule [50]. In contrast to the typical case, it is not straightforward to givea general formula for the dimension or the bosonic content of atypical irreducible representations,see however [48, 51]. As will be explained below the representations L as well as L R and L ∗ R = L R ∗ are always atypical.We shall assume that all irreducible representations of our type I superalgebra g emerge as(possibly trivial) quotients of Kac modules (cf. [50]). In other words, the set Rep( g ) of iso-morphism classes (or labels) of irreducible g -modules agrees with the one of its bosonic subal-gebra, i.e. Rep( g ) = Rep( g ). According to our previous remarks, it splits into two disjoint sets,Rep( g ) = Typ( g ) ∪ Atyp( g ), containing typical and atypical labels, respectively.Simple modules of a Lie superalgebra can be grouped into so-called blocks . By definition, blocksare the parts of the finest partition of Rep( g ) such that two simple modules belong to the same partas soon as they have a non-split extension (see, e.g., [52]). An intuitive way of understanding thisdefinition is to view the simple modules as vertices in a graph. There exists an edge between twovertices if and only if the corresponding simple modules admit a non-split extension. In this picture,the blocks correspond to connected components of the full graph. The property “being connected”defines an equivalence relation ∼ on Rep( g ). We will use the notation Γ( g ) = Rep( g ) / ∼ for the setof all blocks and [ σ ] ∈ Γ( g ) for individual blocks. Notice that each typical module forms a block byitself. Atypical irreducible representations, on the other hand, form constituents of larger blocks.This implies the decomposition Γ( g ) = Γ typ ( g ) ∪ Γ atyp ( g ) where Γ typ ( g ) = Typ( g ).It is easy to argue that each Lie superalgebra of type I possesses a (probably infinite) block[0] containing the trivial representation. Atypicality of the one-dimensional trivial representationalready follows on dimensional grounds since the dimension of Kac modules is always a multipleof 2 r . Let us continue to show that the representations L R and L ∗ R ∼ = L R ∗ which are based on the g -modules R and R ∗ lie in the same block [0]. It is straightforward to see that L is obtained asa quotient from the Kac module K where the subscript 0 refers to the trivial g -module. In orderto prove the atypicality of L R we consider the states in K which are obtained from the groundstate by applying precisely one fermionic generator. These states transform in the representation R of g . Since the Kac module K is atypical and its irreducible quotient is of dimension one, thisrepresentation has to decouple, i.e. the fermionic generators S a have to annihilate these states.We observe that the representation R can be part of at least two different supermultiplets: it maybe used to define a Kac module K R and it generates a submodule Q R of K . In both cases, thehighest weight conditions are exactly identical. But obviously the dimensions of Q R and K R donot coincide since dim Q R < dim K < dim K R . Hence, Q R has to be a non-zero quotient of K R ,proving the atypicality of the latter. The same reasoning could be repeated with at least one of the g -modules which appear in the (dual) Kac modules K R and K R ∗ and so on. Thereby we constructa presumably infinite chain of atypical representations L µ in the block [0]. The labels that are This statement only holds in this form if we restrict ourselves to finite dimensional representations. R ⊗ m ⊗ ( R ∗ ) ⊗ n forarbitrary powers m and n (the converse is not true, of course). Lie superalgebras possess a whole zoo of representations which cannot be decomposed into a directsum of irreducibles. We shall see some important examples momentarily. Let us recall before thatany g -module M possesses a composition series . The latter is determined by a special kind offiltration, in the present case an ascending set of submodules M i , i = 0 , . . . , n where M = 0 and M n = M , such that the quotients M i /M i − are simple modules. We will denote by [ M : L µ ] thenumber of irreducible g -modules L µ in this composition series of M .The most interesting class of indecomposables consists of the so-called projective covers P µ ofirreducibles L µ . The module P µ is defined to be the unique indecomposable projective module thatcontains the irreducible representation L µ as its head. By definition, the head of a representationis the quotient by its maximal proper submodule. For typical labels one has the equivalences L µ ∼ = K µ ∼ = P µ . For atypical labels, however, irreducible modules, Kac modules and projectivecovers are all inequivalent. In particular, they possess different dimensions.All projective modules P of a type I superalgebra are known to possess a Kac compositionseries [48], i.e. a filtration in terms of submodules whose quotients are Kac modules. We denoteby ( P : K λ ) the number of Kac modules K λ in the Kac composition series of P . In order to describethe precise structure of indecomposable projective modules we will rely on the following reciprocitytheorem [48, Theorem 2.7] (see also [49]) (cid:0) P µ : K λ (cid:1) = (cid:2) K λ : L µ (cid:3) . (2.11)This important equation relates the multiplicities of Kac modules in the Kac composition seriesof a projective cover to the multiplicity of irreducible representations arising in the compositionseries of Kac modules. Hence, the structure of projective covers is completely determined by thatof Kac modules. The statement is trivial for typical labels but it contains valuable informationin the atypical case. Note that a small technical assumption underlying Zou’s proof of eq. (2.11)seems to be overcome if one uses the approach of [49].There is one simple construction that is guaranteed to furnish projective modules and it isexactly this construction through which the latter will enter in our harmonic analysis later on. Theidea is to induce representations from irreducible representations V µ of g by letting both sets offermionic generators S a and S a act non-trivially, i.e. B µ = Ind gg ( V µ ) . (2.12)These modules are projective and reducible [48]. Indeed, under reasonable assumptions on g allfinite dimensional g -modules are projective, and this property is preserved by induction. Forlater use, let us write down the decomposition of the representations B µ into their indecomposablebuilding blocks. We start with the observation that their bosonic content is given by B µ (cid:12)(cid:12) g = V µ ⊗ F ⊗ F ∗ . (2.13) The attribute “projective” is used here in the sense of category theory and should not be confused with thenotion of projective representations that is used when algebraic relations are only respected up to some multipliers(cocycles). It should be stressed that this property is not true for type II Lie superalgebras. A counter-example is providedby D (2 , α ) whose representation category is discussed in [53]. B µ are given by ( B µ : K ν ) = [ K ∗ µ + : V ν ] . For the actual decompositioninto indecomposables we use our knowledge that B µ is projective. This implies that it may bewritten as a direct sum of (typical) irreducible Kac modules and (atypical) projective covers. Whilenothing remains to be done for typical representations, the correct description of the atypical sectorrequires combining the corresponding (non-projective) Kac modules into projective covers. In orderto achieve this goal we note the equality [ K ∗ µ + : V ν ] = [ K ∗ ν : V µ + ] which holds because both sidescorrespond to the number of g -invariants in the tensor product V µ ⊗ V ∗ ν ⊗ F ∗ . Now we can usethe following simple consequence of the duality relation (2.11),[ K ∗ µ + : V ν ] = (cid:2) K ∗ ν : V µ + (cid:3) = X σ (cid:2) K ∗ ν : L ∗ σ (cid:3) (cid:2) L ∗ σ : V µ + (cid:3) = X σ (cid:0) P σ : K ν (cid:1) (cid:2) L ∗ σ : V µ + (cid:3) , (2.14)to arrive at the final result B µ = M ν ∈ Typ( g ) (cid:2) K ∗ µ + : V ν (cid:3) K ν ⊕ M σ ∈ Atyp( g ) (cid:2) L ∗ σ : V µ + (cid:3) P σ . (2.15)This formula will be one of the main ingredients in the harmonic analysis to be performed belowin section 4.2. It is interesting to note that every indecomposable projective module arises asa subspaces of some B µ [48]. This means that the category of representations considered here“contains enough projectives”.Let us elaborate a bit more on the distinguished role that projective modules – direct sums oftypical irreducibles and projective covers of atypical irreducibles – play for the representation theoryof Lie superalgebras. In fact, in many ways they take over the role of irreducible representations inthe theory of ordinary Lie algebras. Most importantly, it can be shown that the tensor product ofany module with a projective one is projective again. In other words, projective modules form anideal in the representation ring. Moreover, the Clebsch-Gordon decomposition for tensor products ofprojective modules can be determined through a variant of the Racah-Speiser algorithm. Considerfor instance two projective g -modules A and A . Being projective, they have a Kac compositionseries and hence their bosonic content is given by A i (cid:12)(cid:12) g = X µ m iµ V µ ⊗ F . (2.16)For the bosonic content of the tensor product A ⊗ A this implies (cid:0) A ⊗ A (cid:1)(cid:12)(cid:12) g = X µ m µ m ν h V µ ⊗ V ν ⊗ F i ⊗ F . (2.17)The last F should be interpreted as the fermionic factor that is guaranteed to be present in everyprojective module, due to the fact that they possess a Kac composition series. All we need to do isto decompose the factor V µ ⊗ V ν ⊗ F into irreducibles of g . This provides us with a list of all Kacmodules in A ⊗ A along with their multiplicities. Typical Kac modules correspond to irreduciblerepresentations appearing in the tensor product while atypical Kac modules must be re-combinedinto projective covers. This final step is performed based on formula (2.11) and it leads to anunambiguous result. Our discussion shows how the Clebsch-Gordon decomposition of the tensorproduct A ⊗ A may be played back to the bosonic subalgebra. The decomposition of V µ ⊗ V ν ⊗ F can be tackled with the usual algorithmic tools from the representation theory of Lie algebras.10 .2.4 The quadratic Casimir element One of the most important objects in representation theory are the
Casimir elements , i.e. elementsof the center of the universal enveloping algebra U ( g ). For our concrete choice of generators andinvariant form we have a natural quadratic Casimir C = K i κ ij K j − S a S a + S a S a . (2.18)It may easily be checked that this operator acts as a scalar on Kac modules K µ . For a vector v ∈ V µ in the defining irreducible bosonic multiplet one finds Cv = (cid:0) C B − tr( R i ) κ ij K j (cid:1) v , (2.19)where C B = K i κ ij K j is the quadratic Casimir of g associated to its non-degenerate metric. Sincethe second term inside the bracket commutes with g as well, the irreducibility of V µ implies that C acts as a scalar on the whole multiplet V µ . Using the commutativity of C with g , this action maybe extended to the complete Kac module K µ . We will denote the corresponding eigenvalue of theCasimir by C µ = C ( K µ ). Because irreducible g -modules are defined as a quotient of Kac modulesthis immediately implies C ( L µ ) = C ( K µ ).The observation that several representations may have the same Casimir eigenvalues can beseen to generalize. In fact, it just takes a moment of thought to convince oneself that one has C µ = C ν (and the same for other Casimirs) whenever the simple modules belong to the same block, µ ∼ ν . It seems plausible that also the converse holds, i.e. that the set of Casimir operators may beused to separate different blocks. If this assertion was true, then choosing µ and ν from differentblocks, one would be able to find a Casimir (not necessarily quadratic) whose eigenvalues on L µ and L ν disagree.The previous comment that Casimir eigenvalues are constant on blocks has interesting impli-cations for indecomposables. By definition, the composition series of an indecomposable containsirreducibles belonging to one and the same block. Therefore, within any indecomposable, no matterhow complicated it is, all generalized eigenvalues of the Casimir elements are the same. The ad-ditional qualifier “generalized” is necessary because a Casimir element need not be diagonalizablewhen evaluated in an indecomposable representation. This phenomenon is particularly commonfor the projective covers of atypicals. We shall see later that – at least for a type I Lie superalgebra– the quadratic Casimir (2.18) cannot be diagonalized in any of the projective covers P µ . Further-more, there exists at least one series of projective covers, the ones associated to the block [0] ∈ Γ( g )of the trivial representation, for which the generalized eigenvalues, i.e. the diagonal entries in theJordan block, vanish identically. In this section we will introduce the WZNW model using its Lagrangian formulation. We willemploy a Gauss-like decomposition in order to rewrite the Lagrangian in terms of a bosonic WZNW Diagonalizability might be true for other Casimir operators though. For gl (1 | E is diagonalizable in all weight modules. Note however that E is not related to a non-degenerate invariant formas in eq. (2.18). Certain type II superalgebras such as e.g. D (2 , α ) are known to also possess projective covers with non-vanishinggeneralized eigenvalues [53]. Given the Lie superalgebra g as defined in (2.1)-(2.4), we can combine its generators with elementsof a Grassmann algebra in order to obtain a Lie algebra which can be exponentiated. In physicist’smanner we shall define the supergroup G to be given by elements g = e θ g B e ¯ θ (3.1)with θ = θ a S a and ¯ θ = ¯ θ b S b (this parametrization has been termed “chiral superspace” in [54]).The coefficients θ a and ¯ θ b are independent Grassmann variables while g B denotes an element ofthe bosonic subgroup G B ⊂ G obtained by exponentiating the Lie algebra generators K i . Theattentive reader may have noticed that the product of two such supergroup elements (3.1) will notagain give a supergroup element of the same form. We shall close an eye on such issues. For us,passing through the supergroup is merely an auxiliary step that serves the purpose of constructinga WZNW-like conformal field theory with Lie superalgebra symmetry. Since Lie superalgebras donot suffer from problems with Grassmann variables, the resulting conformal field theory will bewell-defined.The WZNW Lagrangian for maps g : Σ → G from a two-dimensional Riemann surface Σ tothe supergroup G is fully specified in terms of the invariant metric on g and it reads S WZNW [ g ] = − i π Z Σ h g − ∂g, g − ¯ ∂g i dz ∧ d ¯ z − i π Z B h g − dg, [ g − dg, g − dg ] i . (3.2)The second term is integrated over an auxiliary three-manifold B which satisfies ∂B = Σ . Notethat the measure idz ∧ d ¯ z is real. The topological ambiguity of the second term possibly imposesa quantization condition on the metric h· , ·i or, more precisely, on its bosonic restriction, in orderto render the path integral well-defined. Given the parametrization (3.1), the Lagrangian can besimplified considerably by making iterative use of the Polyakov-Wiegmann identity S WZNW [ gh ] = S WZNW [ g ] + S WZNW [ h ] − i π Z h g − ¯ ∂g, ∂hh − i dz ∧ d ¯ z . (3.3)The WZNW action evaluated on the individual fermionic bits vanishes because the invariant form(2.3) is only supported on grade 0 of the Z -grading. The final result is then S WZNW [ g ] = S WZNW [ g B , θ ] = S WZNW [ g B ] − i π Z h ¯ ∂θ, g B ∂ ¯ θ g − B i dz ∧ d ¯ z . (3.4) Note that for WZNW models based on bosonic groups one usually explicitly introduces an integer valued constant,the level, which appears as a prefactor of the Killing form. For supergroups the Killing form might vanish. Hence thereis no canonical normalization of the metric. Moreover, we would like to include models whose metric renormalizesnon-multiplicatively (see below). Under these circumstances it is not particularly convenient to display the levelexplicitly and we assume instead that all possible parameters are contained in the metric h· , ·i . g . The latter implies for instance that the scalar product vanishes if bosonicgenerators are paired with fermionic ones.It is now crucial to realize (see also [36, 24]) that we may pass to an equivalent description ofthe WZNW model above by introducing an additional set of auxiliary fields p a and ¯ p a , S [ g B , p, θ ] = S WZNWren [ g B ] + S free [ θ, ¯ θ, p, ¯ p ] + S int [ g B , p, ¯ p ]= S WZNWren [ g B ] + i π Z n h p, ¯ ∂θ i − h ¯ p, ∂ ¯ θ i − h p, g B ¯ p g − B i o dz ∧ d ¯ z . (3.5)Here, θ, ¯ θ and our new fermionic fields p = p a S a and ¯ p = ¯ p a S a all take values in the Lie superalgebra g . Our conventions may look slightly asymmetric but as we will see later this just resembles theasymmetry in the parametrization (3.1). Up to certain subtleties that are encoded in the subscript“ren” of the first term, it is straightforward to see that we recover the original Lagrangian (3.4)upon integrating out the auxiliary fields p and ¯ p .Let us comment a bit more on each term in the action (3.5). Most importantly, we need tospecify the renormalization of the bosonic WZNW model which results from the change in the pathintegral measure (cf. [55]). The computation of the relevant Jacobian has two important effects.First of all, it turns out that the construction of the purely bosonic WZNW model entering theaction (3.5) employs the following renormalized metric h K i , K j i ren = κ ij − γ ij with γ ij = tr( R i R j ) . (3.6)Note that this renormalization is not necessarily multiplicative. For simple Lie superalgebras therenormalized metric is always identical to the original one up to a factor. For non-simple Liesuperalgebras, however, this is generically not the case as can be inferred from the example of gl (1 | R (2) , S WZNWFT [ g b ] = Z Σ d σ √ hR (2) φ ( g B ) where φ ( g B ) = −
12 ln det R ( g B ) . (3.7)The same kind of expression has already been encountered in the investigation of the GL (1 | φ vanishes whenever g is a semisimple Lie algebra. Therefore, a non-trivial dilaton is a feature of theseries osp (2 | n ), sl ( m | n ) and gl ( m | n ) or, in other words, of most basic Lie superalgebras of type I.The precise reason for the claimed form of renormalization, i.e. the modification of the metric andthe appearance of the dilaton, will become clear in the following sections when we discuss the fullquantum symmetry of the supergroup WZNW model. At the moment let us just restrict ourselvesto the comment that the dilaton is required in order to ensure the supergroup invariance of thepath integral measure for the free fermion resolution, i.e. the description of the WZNW model interms of the Lagrangian (3.5). We assume this metric to be non-degenerate. Otherwise we would deal with what is known as the critical levelor, in string terminology, the tensionless limit. g B ¯ p g − B = g B S b ¯ p b g − B = S a R ab ( g B ) ¯ p b . (3.8)The result for the interaction term is S int [ g B , p, ¯ p ] = − i π Z p a R ab ( g B ) ¯ p b dz ∧ d ¯ z . (3.9)In an operator formulation, the object R ab ( g B ) should be interpreted as a vertex operator of thebosonic WZNW model, transforming in the representation R ⊗ R ∗ . We may consider the interactionterm p a R ab ( g B ) ¯ p b as a screening current. Note that the latter is non-chiral by definition, a featurethat is not really specific to supergroups but applies equally to bosonic models. Nevertheless, theexisting literature on free field constructions did not pay much attention to this point. Actually,the distinction is not really relevant for purely bosonic WZNW models because of their simplefactorization into left and right movers. In the present context, however, a complete non-chiraltreatment must be enforced in order to capture and understand the special properties of supergroupWZNW models. It is well known that the full WZNW model exhibits a loop group symmetry. More precisely, theLagrangian (3.2) (and hence also the functional (3.5)) is invariant under multiplication of the field g ( z, ¯ z ) with holomorphic elements from the left and with antiholomorphic elements from the right.Infinitesimally, each of these transformations generates an infinite dimensional current superalgebra,a central extension ˆ g of the loop superalgebra belonging to g . For the holomorphic sector the latteris equivalent to the following operator product expansions (OPEs). In the bosonic subsector wefind K i ( z ) K j ( w ) = κ ij ( z − w ) + if ij l K l ( w ) z − w . (3.10)The transformation properties of the fermionic currents are K i ( z ) S a ( w ) = − ( R i ) ab S b ( w ) z − w and K i ( z ) S a ( w ) = S b ( w ) ( R i ) ba z − w . (3.11)Finally we need to specify the OPE of the fermionic currents, S a ( z ) S b ( w ) = δ ab ( z − w ) − ( R i ) ab κ ij K j ( w ) z − w . (3.12)The previous operator product expansions are straightforward extensions of the commutation re-lations (2.1), (2.2) and (2.4). The central extension is determined by the invariant metric (2.3).The current superalgebra above defines a chiral vertex algebra via the Sugawara construction[56]. As usual, the corresponding energy momentum tensor is obtained by contracting the currentswith the inverse of a distinguished invariant and non-degenerate metric. The appropriate fullyrenormalized (hence the subscript “full-ren”) metric is defined by h K i , K j i full-ren = (Ω − ) ij = κ ij − γ ij − f imn f jnm h S a , S b i full-ren = (Ω − ) ab = δ ab + ( R i κ ij R j ) ab (3.13)14nd it is the result of adding half the Killing form of the Lie superalgebra g to the original classicalmetric (2.3). Note that some of the terms in the fully renormalized metric (3.13) can be iden-tified with the (partially) renormalized metric (3.6) which we introduced while deriving the freefermion Lagrangian. The energy momentum tensor of our theory involves the inverse of the fullyrenormalized metric, T = 12 h K i Ω ij K j − S b Ω ab S a + S a Ω ab S b (cid:3) . (3.14)Both, currents and energy momentum tensor, may similarly be defined for the antiholomorphicsector. The appearance of a renormalized metric in the Sugawara construction is a rather commonfeature. Supergroup WZNW models are certainly not exceptional in this respect.In order to complete the discussion of the operator content, we have to introduce vertex oper-ators Φ ( M ) ( z, ¯ z ). The latter carry a representation M of g ⊕ g , the underlying horizontal part ofthe current superalgebra of our model. If we assume for a moment that M = ( µν ) where µ and ν refer to Kac modules of the individual factors in g ⊕ g then primary fields are characterized by theoperator products K i ( z ) Φ ( µν ) ( w, ¯ w ) = − D ( µ ) ( K i )Φ ( µν ) ( w, ¯ w ) z − w S a ( z ) Φ ( µν ) ( w, ¯ w ) = 0 (3.15)¯ K i (¯ z ) Φ ( µν ) ( w, ¯ w ) = Φ ( µν ) ( w, ¯ w ) D ( ν ) ( K i )¯ z − ¯ w ¯ S a (¯ z ) Φ ( µν ) ( w, ¯ w ) = 0 . (3.16)In addition, there are fields ( S a · · · S a s ¯ S b · · · ¯ S b t Φ ( µν ) )( z, ¯ z ) which belong to the same represen-tation of the horizontal subsuperalgebra. The matrices D ( µ ) are representation matrices of g .As usual we may infer the conformal dimension of the primary fields from their operator productexpansion with the energy momentum tensor, T ( z ) Φ ( µν ) ( w, ¯ w ) = h ( µν ) Φ ( µν ) ( w, ¯ w )( z − w ) + ∂ Φ ( µν ) ( w, ¯ w ) z − w ¯ T (¯ z ) Φ ( µν ) ( w, ¯ w ) = ¯ h ( µν ) Φ ( µν ) ( w, ¯ w )( z − w ) + ¯ ∂ Φ ( µν ) ( w, ¯ w )¯ z − ¯ w . (3.17)Using the standard techniques one easily finds that the conformal dimensions are given by (renor-malized) Casimir eigenvalues, h ( µν ) = 12 C full-ren µ ¯ h ( µν ) = 12 C full-ren ν . (3.18)The corresponding Casimir is given by C full-ren = K i Ω ij K j + tr(Ω R i ) κ ij K j and should be thoughtof as a renormalization of eq. (2.19). It is important to stress once more that in our conventionsthe level is contained implicitly in the metric κ ij . Thus the conformal dimensions depend on thelevel. They vanish if the metric of the supergroup is scaled to infinity. In that limit the groundstate sector decouples, and it can be analyzed using methods of harmonic analysis. This will becarried out in section 4. Again, this renormalization does not need to be multiplicative, see for instance GL (1 | .3 Free fermion resolution Our next aim is to describe the current superalgebra defined above and the associated primaryfields in terms of the decoupled system of bosons and fermions that appear in the Lagrangian (3.5).As one of our ingredients we shall employ the bosonic current algebra K iB ( z ) K jB ( w ) = ( κ − γ ) ij ( z − w ) + if ijl K lB ( w ) z − w , (3.19)which is defined using the (partially) renormalized metric which has been introduced in (3.6). Inaddition, we need r free fermionic ghost systems with fields p a ( z ) and θ a ( z ) of spins h = 1 and h = 0, respectively. They possess the usual operator products p a ( z ) θ b ( w ) = δ ba z − w . (3.20)Fermionic fields are assumed to have trivial operator product expansions with the bosonic genera-tors. By construction, the currents K iB and the fields p a , θ b generate the chiral symmetry of thefield theory whose action is S [ g B , p, θ ] = S WZNWren [ g B ] + S free [ θ, ¯ θ, p, ¯ p ] . (3.21)Our full WZNW theory may be considered as a deformation of this theory, once we take intoaccount the interaction term between bosons and fermions, see eq. (3.9). The further developmentof this approach and its consequences will be the subject of section 5.But returning first to the decoupled action (3.21), it is easy to see that it defines a conformalfield theory with energy momentum tensor T = 12 h K iB Ω ij K jB + tr(Ω R i ) κ ij ∂K jB i − p a ∂θ a . (3.22)Note the existence of the dilaton contributions, i.e. terms linear in derivatives of the currents. Inaddition to the conformal symmetries, the action (3.21) is also invariant under a ˆ g ⊕ ˆ g current su-peralgebra. The corresponding holomorphic currents are defined by the relations (normal orderingis implied) K i ( z ) = K iB ( z ) + p a ( R i ) ab θ b ( z ) S a ( z ) = ∂θ a ( z ) + ( R i ) ab κ ij θ b K jB ( z ) −
12 ( R i ) ac κ ij ( R j ) bd p b θ c θ d ( z ) S a ( z ) = − p a ( z ) . (3.23)It is a straightforward exercise, even though slightly cumbersome and lengthy, to check that this setof generators reproduces the operator product expansions (3.10), (3.11) and (3.12). The only inputwe need is the Jacobi identity (2.5). The same identity shows that the quantity in (3.23) which isused to contract p b θ c θ d is in fact antisymmetric in the lower two indices. Obviously, a similar setof currents may be obtained for the antiholomorphic sector. Given the representation (3.23) forthe current superalgebra one may also check the equivalence of the expressions (3.14) and (3.22)for the energy momentum tensors. Algebraically, the calculation rests on the Jacobi identity (2.5)as well as on the equations(Ω − ) ij κ ij ( R l ) ab = ( R i ) ac (Ω − ) cb = (Ω − ) ac ( R i ) cb . (3.24)16he latter arise as invariance constraints for the metric h· , ·i full-ren as defined in eq. (3.13).The current superalgebra defined in (3.23) has a natural action on the vertex operators of theconformal field theory defined by the decoupled Lagrangian S . Once we include the interactionterm, the theory becomes equivalent to the full WZNW model. Hence, we must be able to mapthe vertex operators of the decoupled theory to the vertex operators of the WZNW theory. Theprecise relation turns out to be rather involved. Therefore, we postpone a more detailed expositionof this relation to section 5. Instead, we will continue with a semi-classical analysis of the spaceof vertex operators. This procedure allows us to clearly exhibit the subtleties of the full quantumfield theory in a simple and geometric setup. The WZNW model we introduced in the last section admits a semi-classical limit when the invariantmetric defined in (2.3) is scaled to infinity. This corresponds to choosing the levels of the underlyingbosonic WZNW model large. In this weak curvature regime we expect the conformal dimensions ofall primary fields to tend to zero and the higher modes to decouple. We will start with a discussionof the global symmetry of the WZNW model and how it is realized in terms of differential operatorson the space of quantum mechanical wave functions. Then we discuss the Laplacian, i.e. the waveoperator, on G and determine its (generalized) eigenfunctions and eigenvalues which approximatethe vertex operators and their conformal dimensions in the full conformal field theory. It is shownthat the spectrum contains non-chiral indecomposable modules on which the Laplacian is notdiagonalizable. One of the inherent properties of supergroup manifolds G is that they admit two actions of G onitself. These so-called left and right regular actions are defined by the maps L h : g hg and R h : g gh − . (4.1)Since the definition of the WZNW Lagrangian (3.2) only involves the invariant metric, both actionsare automatically symmetries of our model. In fact, in the present situation they are even promotedto current superalgebra symmetries as we have already seen in the previous section. In this sectionwe will just discuss the point-particle limit (or minisuperspace approximation) where only the zero-modes are taken into account and every dependence on the world-sheet coordinates is ignored. Thiscorresponds to quantum mechanics on the supergroup [57]. Our aim is to find all the eigenfunctionsof the Laplace (or wave) operator.Given the symmetry above we know that the state space of the physical system may be de-composed into representations of g ⊕ g . The corresponding symmetry can be realized in terms ofdifferential operators acting on the wave functions which are elements of some algebra of functions F ( G ) on the supergroup. These functions will naturally depend on a bosonic group element g B and on the fermionic coordinates θ a and ¯ θ a . By using a Taylor expansion with respect to the The naive definition of the algebra of function as elements of the Grassmann algebra in the fermions θ a and ¯ θ a with square integrable coefficient functions on G B leads to inconsistencies. A more detailed discussion of these subtlepoints and the explicit introduction of the correct algebra of function shall be postponed until section 4.3. F ( G ) may be represented as a complex valued functiondepending solely on g B multiplied by a product of Grassmann variables.The left and right regular action of the supergroup on itself, as given in (4.1), then induces theaction ( h L × h R ) · f : g f ( h − L gh R ) (4.2)on arbitrary elements f ∈ F ( G ). This in turn translates into the following differential operators, K i = K iB − ( R i ) ab θ b ∂ a S a = − ∂ a S a = R ab ( g B ) ¯ ∂ b + ( R i ) ab θ b κ ij K jB −
12 ( R i ) ac κ ij ( R j ) bd θ c θ d ∂ b , (4.3)for the infinitesimal left regular action. In addition to the various structure constants of the Liesuperalgebra, these expressions contain derivatives ∂ a = ∂/∂θ a and ¯ ∂ a = ∂/∂ ¯ θ a with respect tothe Grassman variables θ a and ¯ θ a . We have also introduced the differential operators K iB whichimplement the regular action of the bosonic subgroup G B . They involve derivatives with respectto bosonic coordinates only, but the precise form depends on the particular choice of coordinateson G B . Similar expressions can be found for the infinitesimal generators of the right action,¯ K i = ¯ K iB + ¯ θ a ( R i ) ab ¯ ∂ b ¯ S a = ¯ ∂ a ¯ S a = − R ba ( g B ) ∂ b − ¯ θ b ( R i ) ba κ ij ¯ K jB −
12 ( R i ) ca κ ij ( R j ) db ¯ θ c ¯ θ d ¯ ∂ b . (4.4)One can check explicitly that these two sets of differential operators form two (anti)commutingcopies of the Lie superalgebra g . Again, these calculations rely heavily on the Jacobi identity (2.5).The expressions for the differential operators exhibit some peculiar properties that we wouldlike to expand on. Note that, apart from purely bosonic pieces, the generators (4.3) of the leftregular action would only involve the Grassmann coordinates θ a and the corresponding derivatives– but no bared fermions – if it were not for the very first term in the definition of S a . Indeed,this term does contain derivatives with respect to the fermionic coordinates ¯ θ a . Obviously, thesituation is reversed for the right regular action. It is also worth stressing that the coefficients inthe first terms of both S a and ¯ S a are non-trivial functions on the bosonic group. Again this is insharp contrast to all the other terms whose coefficients are independent of the bosonic coordinates(though functions of the Grassmann variables, of course). It has been emphasized in [24] thatthe occurrence of the matrix R ( g B ) can spoil the normalizability properties of the functions thesymmetry transformations are acting on. This always happens if the target space is non-compactsince R is a finite dimensional representation and hence non-unitary in that case. Consequently,the product of an L -function from F ( G B ) with R ( g B ) will not be an L -function anymore.In view of these issues with S a and ¯ S a it is tempting to simply drop the troublesome terms.Even though that might seem a rather arbitrary modification at first, it turns out that the corre-sponding truncated differential operators K i = K i , S a = S a , S a = ( R i ) ab θ b κ ij K jB −
12 ( R i ) ac κ ij ( R j ) bd θ c θ d ∂ b (4.5)and their bared analogues also satisfy the commutation relations of g ⊕ g ! For the special caseof P SU (1 , | K i , S a and S a model the action of zero-modes of our currents(3.23) on ground states in the decoupled free fermion theory, i.e. before the coupling of bosonic andfermionic fields is taken into account. Note that the zero-mode of p ( z ) is a field theoretic incarnationof the derivative ∂ since p ( z ) is the canonically conjugate momentum belonging to θ ( z ). We shallnow proceed to argue that the original differential operators (4.3) and (4.4) encode a much moreintricate structure, namely the action of the zero-modes on primaries in the full interacting WZNWmodel. The algebra of functions F ( G ) furnishes a representation of g ⊕ g via the differential operators (4.3)and (4.4). Our aim is to write F ( G ) as a direct sum of indecomposable building blocks of the typediscussed in section 2.2. The final result can be found in eq. (4.7) below. But since the outcomeis rather complicated and somewhat hard to digest we would like to start the harmonic analysisby discussing the left and the right action of g separately. We claim that the space of functionsdecomposes under these actions according to F ( G ) (cid:12)(cid:12) g (left) = F ( G ) (cid:12)(cid:12) g (right) = M µ ∈ Typ( G ) dim( K µ ) K µ ⊕ M µ ∈ Atyp( G ) dim( L µ ) P µ . (4.6)The symbols Typ( G ) and Atyp( G ) denote the sets of typical and atypical irreducible representationsof the supergroup. The distinction between modules of G and modules of g is necessary since theremight exist representations of the Lie superalgebra which cannot be lifted to G . Under rathergeneral conditions (to be recalled below eq. (4.13)) the set Rep( G ) of supergroup representationscoincides with Rep( G B ) ⊂ Rep( g ), the set of all unitary irreducible representations of the bosonicsubgroup G B .As we see, the decomposition (4.6) clearly distinguishes between the typical and the atypicalsector of our space. In the typical sector we sum over irreducible Kac modules K µ = L µ with amultiplicity space M ( K µ ) of dimension dim K µ , a prescription which is familiar from the Peter-Weyl theory for bosonic groups. In contrast, the atypical sector consists of a sum over all theprojective covers P µ belonging to atypical irreducibles L µ and coming with a multiplicity space M ( P µ ) of the smaller dimension dim L µ < dim K µ . Note that the algebra of functions forms aprojective module and hence possesses a Kac composition series, i.e. a filtration in terms of Kacmodules. This immediately permits us to spell out the character of the g ⊕ g -module F ( G ) and itwill lead to a concrete proposal for the modular invariant partition function of the WZNW modelin section 5.Naturally, our formula (4.6) is the same for the left and the right action. This symmetry betweenleft and right regular transformations must certainly be maintained when we extend our analysisto the combined left and right action of g ⊕ g on F ( G ). In the typical sector the multiplicityspaces of the Kac modules have precisely the dimension that is needed to promote them to Kacmodules themselves, a prescription that is perfectly consistent with the symmetry between left andright action. On the other hand, the same symmetry requirement excludes that the individualmultiplicity spaces in the atypical sector are simply promoted to irreducible representations of g .Consequently, the left action must induce maps between various multiplicity spaces for the rightaction and vice versa. In this way, the atypical sector then consists of non-chiral indecomposables A similar expression already appeared in [58] in a more general context. [ σ ] which entangle a (possibly infinite) number of left and right projective covers whose labelsbelong to the same block [ σ ]. The final expression for the representation content of the algebra offunctions on G is thus of the form F ( G ) (cid:12)(cid:12) g ⊕ g = M µ ∈ Typ( G ) L µ ⊗ L ∗ µ ⊕ M [ σ ] ∈ Γ atyp ( G ) I [ σ ] . (4.7)The systematic study of the non-chiral representations I [ σ ] will be left for future work. Notethat similar and, in the specific cases of GL (1 |
1) and SU (2 | G = GL ( m | n ) is treated in the framework of Hopf superalgebras.Having stated the main results of this subsection we would like to sketch their derivation. Forthe proof of eq. (4.6), it is advantageous to enlarge the symmetry from g to an action g ⊕ g , i.e.to retain the bosonic generators of the right regular transformations if we analyze the left action.With respect to the combined action one finds F ( G ) (cid:12)(cid:12) g ⊕ g = M µ ∈ Rep( G B ) B µ ⊗ V ∗ µ F ( G ) (cid:12)(cid:12) g ⊕ g = M µ ∈ Rep( G B ) V µ ⊗ B ∗ µ . (4.8)In fact, from the Peter-Weyl theorem for compact semisimple Lie groups (or suitable generalizationsthereof) we deduce that the functionsdet R ( g − B ) (cid:2) D ( µ ) ( g B ) (cid:3) αβ θ · · · θ r ¯ θ · · · ¯ θ r (4.9)involving matrix elements of the representation D ( µ ) are part of the spectrum for all unitary ir-reducible representations µ of G B . The matrix elements of D ( µ ) transform in the representation V µ ⊗ V ∗ µ with respect to g ⊕ g . Since the product of the remaining factors multiplying D ( µ ) is in-variant under purely bosonic transformations, we conclude that the set of functions (4.9) transformsin V µ ⊗ V ∗ µ as well.All that remains to be done is to augment the action on the left from the bosonic subalgebra g tothe entire Lie superalgebra g . The supersymmetric multiplets we generate from the functions (4.9)by repeated action with all the fermionic generators S a and S a are isomorphic to the representation B µ of g . Similar remarks apply if we consider the action of g ⊕ g . Thereby we have establishedthe decompositions (4.8). In order to proceed from eqs. (4.8) to the decomposition formulas (4.6)the representations B µ must be decomposed into their indecomposable building blocks. This isachieved with the help of eq. (2.15) and results in eq. (4.6) after a simple re-summation. Ourderivation has actually furnished a slightly stronger result since it determines how the multiplicityspaces decompose with respect to the action of the bosonic subalgebra g . Given the decomposition of the algebra of functions into representations of g ⊕ g we can now addressour original problem of finding the semi-classical expressions of both the conformal dimensionsand the primary fields. In the semi-classical limit, conformal dimensions are given by (half) theeigenvalues of the Casimir operator acting on F ( G ). Since we are dealing with a space of functions20e will refer to the latter as “Laplacian” on the supergroup. The eigenvalues can be read off directlyfrom the decomposition (4.7). In the typical sector the Laplacian is diagonalizable and leads to theeigenvalues C ( K µ ). On the other hand, the Laplacian ceases to be diagonalizable on the non-chiralrepresentations I [ σ ] . Here, the Casimir may just be brought into Jordan normal form.The previous paragraph provides a complete solution of the eigenvalue problem but it doesnot yield explicit formulas for the (generalized) eigenfunctions. Since the latter are semi-classicalversions of the primary fields in the full CFT (see section 5 below), it seems worthwhile recallingthe elegant construction of eigenfunctions that was presented recently in [24]. The Laplace operatoron our supergroup G is given by∆ = 12 C = ∆ B −
12 tr( R i ) κ ij K jB − ∂ a R ab ( g B ) ¯ ∂ b . (4.10)Observe that only the last term contains fermionic derivatives, with coefficents which depend onbosonic coordinates. Let us also emphasize that the purely bosonic piece of ∆ differs from theLaplacian on the bosonic subgroup by the second term. This deviation is related to the presenceof the non-trivial dilaton contribution (3.7). Since the complete Laplacian is non-diagonalizable itwas proposed in [24] to perform the harmonic analysis in two steps. First an auxiliary problem issolved which is based on the purely bosonic Laplacian∆ = ∆ B −
12 tr( R i ) κ ij K jB . (4.11)This auxiliary Laplacian agrees with the Casimir operator obtained from the reduced differentialoperators K and S and, as we shall see, it is completely diagonalizable on the following auxiliaryspace F ( G ) = F ( G B ) ⊗ ^ ( θ a , ¯ θ b ) . (4.12)Here, the factor F ( G B ) denotes the algebra of square (or δ -function) normalizable functions on thebosonic subgroup and V ( θ a , ¯ θ b ) is the Grassmann (or exterior) algebra generated by the fermioniccoordinates. In the second step, the eigenfunctions of ∆ are mapped to generalized eigenfunctionsof ∆ using a linear map Ξ : F ( G ) → F ( G ). The latter adds “subleading” fermionic contributionsin a formal but well-defined way and thereby turns an eigenfunction of ∆ into a generalizedeigenfunction of ∆. Our prescription involves explicit multiplications with the matrix elements of R ( g B ) which, e.g. for non-compact groups G B , are not necessarily part of the unitary spectrum.Hence, the eigenfunctions of ∆ need not be normalizable in the original sense, i.e. when regardedas Grassmann valued functions on the bosonic subgroup. This is the main reason why we need todistinguish between the spaces F ( G ) and F ( G ) = Im(Ξ). Ultimately, the problem may be tracedback to the presence of the terms involving R ( g B ) in S a and ¯ S a . In fact, as we pointed out before,because of those terms the unreduced differential operators may cease to act within F ( G ).In order to gain some intuition into the structure of the function space (4.12) as a representationof the symmetry algebra g ⊕ g , it is helpful to restrict the action to the bosonic subalgebra g ⊕ g first. Since the differential operators K i and ¯ K i factorize in an action on the function algebra F ( G B )and on the Grassmann algebra V ( θ a , ¯ θ b ), we can decompose both factors separately. If the bosonic The auxiliary space F ( G ) should be thought of as the semi-classical truncation of the state space for the decoupledtheory S , see eq. (3.21). On the other hand F ( G ) corresponds to the semi-classical truncation of the full state spaceof the WZNW model. F ( G B ) (cid:12)(cid:12) g ⊕ g = M µ ∈ Rep( G B ) V µ ⊗ V ∗ µ , (4.13)where Rep( G B ) ⊂ Rep( g ) is the set of all unitary irreducible representations of G B . In moregeneral situations this formula will need a slight refinement concerning the content of Rep( G B ),although the structure will still be very similar. With regard to the fermions, the left action justaffects the set θ a , while the right action operates on the set ¯ θ a . Given the known transformationbehavior of a single fermion we thus find ^ ( θ a , ¯ θ b ) (cid:12)(cid:12) g ⊕ g = F ⊗ F ∗ . (4.14)Combining these simple facts and defining Rep( G ) = Rep( G B ) we conclude F ( G ) (cid:12)(cid:12) g ⊕ g = M µ ∈ Rep( G ) h V µ ⊗ F i ⊗ h V µ ⊗ F i ∗ . (4.15)Before we proceed to the supersymmetric extension, we would like to discuss the general form ofelements in the individual subspaces of (4.15). The space of functions is spanned by f ( µ ) a ··· a s ,αb ··· b t ,β ( g ) = (cid:2) D ( µ ) ( g B ) (cid:3) αβ θ a · · · θ a s ¯ θ b · · · ¯ θ b t , (4.16)where D ( µ ) denotes the representation of the bosonic subgroup G B on the module V µ .Our most important task is to determine how the bosonic representations that occur in thedecomposition (4.15) combine into multiplets of the full symmetry g ⊕ g . As a first hint on whatthe answer will be, we observe that the representation content in eq. (4.15) agrees with the bosoniccontent of Kac modules. And indeed, under the action of fermionic generators, the various bosonicmodules are easily seen to combine into our modules K µ . To see this we note that the purelybosonic functions (cid:2) D ( µ ) ( g B ) (cid:3) αβ are annihilated by S a and ¯ S a simultaneously and therefore theyspan the subspace V µ ⊗ V ∗ µ from which we induce the Kac module K µ ⊗ K ∗ µ . Consequently, weobtain the decomposition F ( G ) (cid:12)(cid:12) g ⊕ g = M µ ∈ Rep( G ) K µ ⊗ K ∗ µ . (4.17)Note that the sum runs over both typical and atypical representations, i.e. the space of functions isnot fully reducible. The Laplacian ∆ is completely diagonalizable on this space and its eigenvaluesare given by eq. (2.19).Let us now return to the analysis of the space F ( G ). We recall that a function Φ λ ∈ F ( G ) is ageneralized eigenfunction to the eigenvalue λ if there exists a number n ∈ N such that(∆ − λ ) n Φ λ = 0 . (4.18)Following [24], let us introduce operators A n ( λ ) which are defined through the relation A ( n ) λ = (∆ − λ ) n − (∆ − λ ) n . (4.19)In the sequel it will become crucial that each single term of A ( n ) λ contains at least one fermionicderivative. After these preparations we consider a function f λ ∈ F ( G ) which is an eigenfunction22f ∆ , i.e. which satisfies ∆ f λ = λf λ . We then associate a family of new functions Ξ ( n ) λ f λ to f λ through Ξ ( n ) λ f λ = ∞ X s =0 h − (∆ − λ ) − n A ( n ) λ i s f λ ≡ r X s =0 (cid:16) Q ( n ) λ (cid:17) s f λ . (4.20)Obviously, the sum truncates after a finite number of terms due to the fermionic derivatives whichoccur in all the operators A ( n ) λ . A formal calculation shows furthermore that the function Ξ ( n ) λ f λ is a solution of eq. (4.18). Using the definition (4.20) on each of the eigenspaces Ker(∆ − λ ) weobtain a family of maps Ξ ( n ) which formally exist on the complete function space F ( G ).The only problem with the maps Ξ ( n ) is that they might be singular on a certain subspace of F ( G ). In fact, a close inspection of our expression (4.20) shows that it requires to invert (∆ − λ )which may not be possible. If this happens, it signals the existence of functions in F ( G ) whichare not annihilated by (∆ − λ ) n for any λ , and therefore implies that some Jordan blocks of theLaplacian must have a rank higher than n . It may be shown by explicit calculation that the familyof maps Ξ ( n ) stabilizes for n > r and that the resulting limit map Ξ is well-defined on the completespace F ( G ) [24]. We then define the space F ( G ) = Im(Ξ) as the image of the auxiliary space F ( G ) under Ξ. This procedure provides an explicit construction of the eigenspaces and Jordanblocks appearing in the decomposition (4.7). It should also be recalled that the map Ξ acts as anintertwiner between the typical subspace of F ( G ) with the reduced action of g ⊕ g and the typicalsubspace of F ( G ) with the full action of g ⊕ g [24]. As before, reduced and full action refer to theuse of the differential operators ( K i , S a , S a , ¯ K i , ¯ S a , ¯ S a ) and ( K i , S a , S a , ¯ K i , ¯ S a , ¯ S a ), respectively.Within the present context we can actually convince ourselves that the quadratic Casimir is notdiagonalizable on any of the projective covers P µ . From the above it is clear that every projectivecover appears in the decomposition of the right regular action on the function space F ( G ) andthat the corresponding subspace M ( P µ ) ⊗ P µ contains functions of the form (4.9). We claim thatsome of the latter must necessarily be proper generalized eigenfunctions. In fact, all of them areeigenfunctions of ∆ with eigenvalue λ = C µ /
2. But in order for them to be eigenfunctions of ∆,the action of Ξ (1) must be well defined. This would require in particular that we can invert ∆ − λ on ∂ a R ab ( g B ) ¯ ∂ b det R ( g − B ) (cid:2) D ( µ ) ( g B ) (cid:3) αβ θ · · · θ r ¯ θ · · · ¯ θ r . (4.21)But this is clearly not the case if the Kac module K µ contains singular vectors that are reached fromthe ground states through application of a single fermionic generator. Hence, we have establishedour claim for all such labels µ . In case the singular vectors of K µ appear only at higher levels, onehas to refine the analysis and consider also higher order (in the summation index s ) terms in thedefinition of Ξ (1) . By now we have complete control over representation content and eigenfunctions of the Laplacianin the weak curvature limit of the WZNW model. In addition, we can also compute correlationfunctions in this limit. They are given as integrals over a product of functions on the supergroup.Integration is performed with an appropriate invariant measure, namely the so-called Haar measure dµ ( g ) of the supergroup. The easiest way to obtain dµ is to extract it from the invariant metric, ds = ds B − d ¯ θ a R ab ( g − B ) dθ b . (4.22)23ere, ds B denotes the standard invariant metric on the bosonic subgroup. The total metric hasa “warped” form since the fermionic bit has an explicit functional dependence on the bosoniccoordinates g B . We can now obtain the desired measure as the superdeterminant of the metric, dµ ( g ) = dµ B ( g B ) det (cid:0) R ( g B ) (cid:1) dθ · · · dθ r d ¯ θ · · · d ¯ θ r (4.23)where dµ B denotes an invariant measure on the bosonic subgroup. Once this expression has beenwritten down, we can forget our heuristic derivation and check the invariance explicitly. Note thatthe existence of the dilaton (3.7) in the WZNW Lagrangian (3.5) is directly related to the presenceof the factor det (cid:0) R ( g B ) (cid:1) in the measure.Suppose now we are given N generalized eigenfunctions of the Laplacian ∆ on the supergroup.According to the previous discussion, the space of eigenfunctions possesses a basis of the form φ a µ ; b = Ξ f a µ ; b = r X s =0 Q sλ f a µ ; b where f a µ ; b = f a ,...,a s µ ; b ,...,b t = f µ ( g B ) θ a · · · θ a s ¯ θ b · · · θ b t . (4.24)Here, f µ ( g B ) are eigenfunctions of the bosonic Laplacian ∆ with eigenvalue λ and Ξ = Ξ ( r ) , Q λ = Q ( r ) λ have been defined in eq. (4.20). The N -point functions of such semi-classical vertex operatorsare given by the integrals (cid:10) φ a µ ; b · · · φ a N µ N ; b N (cid:11) = Z dµ ( g ) φ a µ ; b · · · φ a N µ N ; b N = r X s =0 · · · r X s N =0 Z dµ ( g ) Q s λ f a µ ; b · · · Q s N λ N f a N µ N ; b N . (4.25)Most of the ( r + 1) N terms in this expression vanish due to the properties of Grassmann variablesand their integration. In fact the largest number of non-zero terms that can possibly appearis N · r + 1. This is realized if all eigenfunctions contain terms with the maximal number offermionic coordinates (along with the lower order terms that are determined by the action of Q sλ ).A particularly simple case appears when e.g. the first eigenfunction φ = φ , ,...,rµ ;1 , ,...,r contains leadingterms with r fermions θ and ¯ θ while all others are purely bosonic. In that case, the correlator issimply given by (cid:10) φ , ,...,rµ ;1 , ,...,r φ µ · · · φ µ N (cid:11) = Z dµ B ( g B ) det (cid:0) R ( g B ) (cid:1) f µ ( g B ) f µ ( g B ) · · · f µ N ( g B ) . (4.26)We shall see that very similar results can be established for correlators in the full WZNW on type Isupergroups. This is one of the subjects we shall address in the next section. After the thorough discussion of its symmetries and its semi-classical limit it is now only a smallstep to come up with a complete solution of the full quantum WZNW model. We first show thatthe free fermion resolution gives rise to a natural class of chiral representations. Subsequently,24e comment on the representation content of the full non-chiral theory, sketch the calculationof correlation functions and argue that the natural modular invariant partition function can beexpressed as a diagonal sum over characters of Kac modules. We conclude with some speculationsabout non-trivial modular invariants.
In section 3.2 and 3.3 we decribed in some detail the chiral symmetry of WZNW models on su-pergroups along with their construction in terms of free fermions. Our next aim is to introducerepresentations H µ of ˆ g . It is clear that free fermion resolutions provide a natural construction forrepresentations of current superalgebras. What is remarkable, however, is that these representationsturn out to be irreducible for generic (typical) choices of µ .According to the results of section 3.3 every representation of the decoupled system of thebosonic currents K iB and the fermions pθ defines a module of the current superalgebra via eqs.(3.23). In the bosonic part we shall work with irreducible representations V µ of ˆ g ren0 . If thegroup G B is compact there will be a finite number of physical representations (the “integrable”ones), otherwise one may encounter infinitely many of them, including continuous series. Weidentify the physically relevant representations with a subset Rep(ˆ g ren0 ) ⊂ Rep( g ) within therepresentation labels for the horizontal subalgebra g . This is possible since the ground states of V µ form the g -module V µ upon restriction of the ˆ g ren0 -action to its horizontal subalgebra g . Notethat the curvature of the background geometry leads to truncations which imply that Rep(ˆ g ren0 ) isgenerally a true subset of Rep( g ). The fermions, on the other hand, admit a unique irreduciblerepresentation V F . The latter is generated from the SL (2 , C )-invariant vacuum | i by imposingthe highest weight conditions ( p a ) n | i = 0 for n ≥ θ an | i = 0 for n > The irreduciblerepresentations of the product theory therefore take the form H µ = V µ ⊗ V F . (5.1)Given the free fermion realization (3.23), these spaces admit an action of the infinite dimensionalcurrent superalgebra ˆ g as defined in (3.10)-(3.12).The generalized Fock modules H µ provide the proper realization of chiral vertex operators asdefined around eq. (3.15). It is indeed evident from our construction that the ground states of H µ transform in the g -module K µ (recall that the ground states of V µ form the g -module V µ ) andthat they are annihilated by all positive modes of the currents and by the zero modes of S a ( z ).But there is another and much deeper reason for the relevance of the modules H µ . Observe thatthe current superalgebra ˆ g is a true subalgebra of the algebra that is generated from ˆ g ren0 and thefermions. Therefore, one might suspect that the spaces H µ are no longer irreducible with respectto the action of ˆ g . But for generic choices of µ this is not the case: The action of ˆ g on H µ istypically irreducible! This property is in sharp contrast to what happens for standard bosonicfree field constructions [38, 39, 40, 41, 42] and it characterizes the modules H µ as the naturalinfinite dimensional lift of Kac modules for the finite dimensional Lie superalgebra g . We take thisobservation as a motivation to refer to the generalized Fock modules H µ as Kac modules from now For c su (2) k , for instance, the integrable representations are λ = 0 , , . . . , k while there is no upper bound forunitary su (2)-modules. One could include twisted sectors where the moding of the fermions is not integer. But then the global super-symmetry would not be realized in the WZNW model since there were no zero-modes. g in section 2.2.1.Since it is a rather crucial issue for the following, we would like to spend some time to establishirreducibility of the representations H µ for generic labels µ . We shall assume for simplicity that theunderlying bosonic representation V µ is a highest weight module. The highest weight µ determinestwo seemingly different (but in fact equivalent) Verma-like modules of ˆ g . The first of them will bedenoted by Y ′ µ . It is obtained as a product Y ′ µ = Y (0 , ren) µ ⊗ V F of the Verma module Y (0 , ren) µ of ˆ g ren0 with the free fermion state space V F . We shall consider Y ′ µ as a ˆ g -module. The ˆ g -module H µ may be recovered from Y ′ µ by dividing out all the bosonicsingular vectors from the ˆ g ren0 -module Y (0 , ren) µ . But there is a second natural Verma-like module Y µ for ˆ g which is constructed directly by requiring that all the positive modes as well as the zero-modes ( S a ) annihilate the highest weight, i.e. Y µ is defined without any reference to the freefermion construction of ˆ g . Since the generators K in , S a ,n , S b,n and K iB,n , θ an , p a,n are in one-to-onecorrespondence with each other, the Verma modules Y µ and Y ′ µ are naturally isomorphic as vectorspaces. The natural isomorphism preserves the grading by conformal dimensions. Hence, thecharacters of Y µ and Y ′ µ agree. It is tempting to conjecture that Y µ and Y ′ µ are in fact equivalentas ˆ g -modules.In order to understand the equality of conformal dimensions we could simply refer to theequivalence of energy momentum tensors which has been proven in section 3.3. But there is also amore pedestrian way of seeing it. In the case of ˆ g ren0 , the current algebra involves the renormalizedmetric κ − γ while the bosonic subalgebra ˆ g of ˆ g is defined in terms of the metric κ . But accordingto the Sugawara constructions for ˆ g ren0 and ˆ g , the respective energy momentum tensor requiresan additional quantum renormalization of the metric in both cases. This extra renormalization isdifferent as well and the final result (the “fully renormalized metric”) coincides again. The previousstatement corresponds to the two different ways of introducing brackets in the following equation, (cid:16) κ ij − γ ij (cid:17) − f imn f jnm = κ ij − (cid:16) γ ij + 12 f imn f jnm (cid:17) . (5.2)The first term on both sides refers to the “classical” metric and the second term describes thequantum renormalization. In addition, the effect of the fermions in ˆ g has to be traded for thepresence of the dilaton in the ˆ g ren0 description.Let us now focus on the Verma-like modules Y µ . In general, these modules contain singularvectors, certainly of bosonic type but possibly also fermionic ones. Our goal here is two-fold: First,we would like to argue for a one-to-one correspondence of the bosonic singular vectors with thosein Y (0 , ren) µ . Moreover, we would like to show that the existence of fermionic singular vectors is anatypical event, occurring only for a small subset of weights µ .In principle, the structure of singular vectors in the module Y µ can be discussed using a suitablevariant of the Kac-Kazhdan determinant [50]. For simplicity we shall follow a more down-to-earthapproach here. The existence of a proper submodule Y ν in the representation Y µ requires that theweight ν can be reached from µ by (multiple) application of the root generators of ˆ g . We mayqualify this further with the help of two gradings, one with respect to the generator L and the The metric or the level(s), respectively, are assumed to be fixed once and for all. g (which is identical to that of g ). The latter impliesthat the weights µ and ν have to be related by ν = µ − mα where α is a positive root of g and m ∈ Z ≥ . If the energy direction is considered separately, one obtains a necessary condition of theform h µ − mα = h µ + nm , (5.3)where h denotes the conformal dimension and the root generator belonging to α is assumed toincrease the energy by n units.We will investigate condition (5.3) for bosonic root generators of ˆ g first. The latter are in one-to-one correspondence with those of ˆ g ren0 . Since, in addition, the conformal dimensions of highestweight modules Y µ and Y (0 , ren) µ coincide, we conclude that the associated decoupling equations (5.3)possess the same set of bosonic solutions. We consider this a strong hint that singular vectors inthe ˆ g -modules Y (0 , ren) µ ⊗ V F agree with those singular vectors of the ˆ g -modules Y µ which can bereached by application of bosonic root generators. If we assume this to be true, all bosonic singularvectors are removed when be pass from Y µ to H µ . Therefore, the singular vectors that remain in H µ are necessarily fermionic.Let us now look for the existence of potential fermionic singular vectors. We do not intend toformulate any precise rules for when they appear, but would like to argue that they must be rarecompared to their bosonic counterparts. To this end, we recall that the conformal dimension h is aquadratic expression of the form h µ = h µ, µ + 2 ρ i (the bracket denoting the non-degenerate scalarproduct that comes with the metric (3.13)). Hence, we can always solve eq. (5.3) for m , no matterwhich bosonic root vector α we insert. This ceases to be true for fermionic root generators. Sincethey are nilpotent, eq. (5.3) needs to be solved with m = 0 ,
1, something that rarely ever works out.Therefore, modules with fermionic singular vectors are called atypical. A more systematic studyof atypical representations is beyond the scope of this article. But the experience with severalexamples suggests that the composition series of the representations H µ is finite and that theypossess the same structure as the modules of the horizontal subsuperalgebra. In fact, we believethat the only possible fermionic singular vectors are those that appear on the level of ground statesand images thereof under the action of certain spectral flow automorphisms (see section 5.2).Given the structure of the Kac modules (5.1) it is straightforward to derive character formulasand their modular properties. Indeed, the characters simply factorize into χ H µ ( q ) = χ V µ ( q ) χ V F ( q ) . (5.4)The supercharacter of H µ has the same product form but with the fermionic factor χ V F beingreplaced by its corresponding supercharacter. Relation (5.4) may also be extended to a statementabout non-specialized characters since the fermions p a and θ a are charged under the bosonic gener-ators K i . If g is a simple Lie algebra the characters of the unitary ˆ g ren0 -modules V µ can be lookedup in [60, 61]. They form a finite dimensional unitary representation of the modular group. Thecharacter of the fermionic representation V F , on the other hand, is given by χ V F ( q ) = " q ∞ Y n =1 (1 + q n ) r = " ϑ ( q ) η ( q ) r . (5.5)Under the modular transformation τ
7→ − /τ the quotient ϑ /η is simply replaced by ϑ /η . Hence,all the non-trivial information about modular transformations resides in the behaviour of the char-acters for the bosonic algebra ˆ g ren0 . Consequently, the modular properties of Kac modules H µ H µ of ˆ g is very closely related to that of Kacmodules for the horizontal subsuperalgebra g . In specific examples it is usually straightforward toinvert the linear relations between characters resulting from such a composition series, i.e. to ex-press the characters of atypical irreducible representations through those of Kac modules. A moregeneral approach to this problem using Kazhdan-Lusztig polynomials has been presented in [48,Proposition 5.4] (see also [51, 62]). Recently it has been shown that the solution for the inversionproblem could be used to (re)derive the characters of irreducible representations for the affine Liesuperalgebras b sl (2 |
1) and c psl (2 |
2) [24, 37]. We expect that this observation extends to more generalcurrent superalgebras and that it will be helpful in the study of modular transformations. Repre-sentations of affine Lie superalgebras and their behaviour under modular transformations have alsobeen studied in [63, 64, 13].
In the previous subsection we have skipped over one rather important element in the representationtheory of current (super)algebras: The spectral flow automorphisms. As we shall recall momentar-ily, spectral flow automorphisms describe symmetry transformations in the representation theoryof current algebras. Furthermore, they seem to be realized as exact symmetries of the WZNWmodels on supergroups, a property that makes them highly relevant for our discussion of partitionfunctions below.Throughout the following discussion, we shall denote (spectral flow) automorphisms of thecurrent superalgebra ˆ g by ω . We shall mostly assume that the action of ω is consistent with theboundary conditions for currents, i.e. that it preserves the integer moding of the currents. In thecontext of representation theory, any such spectral flow automorphism ω defines a map on the setof (isomorphism classes of) representations ρ : ˆ g → End( V ) via concatenation, ω ( ρ ) = ρ ◦ ω : ˆ g → End( V ).In line with our general strategy, we would like to establish that spectral flow automorphisms ω of the current superalgebra are uniquely determined by their action on the bosonic generators.A spectral flow automorphism ω : ˆ g → ˆ g of the bosonic subalgebra ˆ g is, by definition, a linearmap ω (cid:0) K i ( z ) (cid:1) = ( W ) ij ( z ) K j ( z ) + w i z − (5.6)satisfying certain consistency conditions to be recalled below. The map W ( z ) = z ζ is definedin terms of an endomorphism ζ : g → g of the horizontal subalgebra. While the eigenvaluesof ζ determine how the spectral flow shifts the modes of the currents, the vector w i affects onlythe zero-modes. In order to preserve the trivial monodromy under rotations around the origin we We refrain from introducing a different symbol here such as ω . W ( z ) is a meromorphic function, i.e. that all the eigenvalues of ζ are integer.Inserting the transformation (5.6) into the operator product expansions (3.10) leaves one with theconstraints ( ζ ) ij = f ikl κ kj w l (5.7)and ( W ) ik ( z ) ( W ) j l ( z ) κ kl = κ ij , f ijk ( W ) kl ( z ) = ( W ) im ( z ) ( W ) j n ( z ) f mnl . (5.8)The first equation (5.7) in fact implies that the only free parameter is the shift vector w i . In thecase of a semisimple Lie algebra g (which leads to a non-degenerate Killing form) this argumentcan also be reversed and hence it allows to express w i in terms of ζ .We would now like to argue that equation (5.7) already implies the consistency of the spectralflow (up to the question whether ζ has integer eigenvalues), i.e. the validity of the equations (5.8).Given the concrete form of W ( z ), it can indeed be shown that the two relations (5.8) follow fromthe equations( ζ ) ik κ kj + ( ζ ) j l κ il = 0 f ijk ( ζ ) kl = ( ζ ) ik f kjl + ( ζ ) j k f ikl . (5.9)These relations are in turn just a consequence of (5.7) using the invariance of κ ij and the Jacobiidentity for the structure constants. Since the same idea will be used again below let us sketch theproof of our assertion that the eqs. (5.9) imply the eqs. (5.8). First of all, it is easy to see that onecan generalize the relations (5.9) to powers of ζ using induction. In the first case, this just yieldsan alternating relative sign, while in the second case it establishes some kind of binomial formula.Writing W ( z ) = exp( ζ ln z ) and expanding in powers of ln z one can then explicitly verify theequations for W ( z ). Any vector w i which leads to a matrix ζ with integer eigenvalues under theidentification (5.7) will accordingly be referred to as a spectral flow automorphism of ˆ g from nowon. Given the insights of the previous paragraphs it is now fairly straightforward to extend thespectral flow automorphism ω : ˆ g → ˆ g to the full current superalgebra. To this end, we introducethe element ζ = − R i κ ij w j . (5.10)It is crucial to observe that this matrix satisfies the relation( ζ ) ij ( R j ) ac + ( ζ ) ab ( R i ) bc = ( R i ) ab ( ζ ) bc , (5.11)an analogue of eq. (5.9). Following the discussion in the bosonic sector, we now introduce a function W ( z ) = z ζ . Using the same reasoning as in the previous paragraph, the equation (5.11) implies( R i ) ab ( W ) bc ( z ) = ( W ) ij ( z ) ( W ) ab ( z ) ( R j ) bc . (5.12)Now we can define the action of the spectral flow automorphism ω on the fermionic currents by ω (cid:0) S a ( z ) (cid:1) = ( W ) ab ( z ) S b ( z ) , ω (cid:0) S a ( z ) (cid:1) = S b ( z ) ( W ) ba ( z ) , (5.13)where W denotes the inverse of W . Once more, consistency with the operator product expansionsof the supercurrents is straightforward to verify. The only input is the definition (5.10) and theproperty (5.12). 29e would also like to argue that the spectral flow symmetry is consistent with the free fermionrepresentation (3.23). To be more specific, we shall construct an automorphism on the chiral algebraof the decoupled system generated by the currents K iB ( z ) and the free fermions p a ( z ) and θ a ( z )that reduces to the expressions above if we plug the transformed fields into the defining equations(3.23). In this context the most important issue is to understand how the renormalization of themetric κ → κ − γ affects the action of the spectral flow. As a consequence of eq. (5.11) we notethat ( ζ ) ik γ kj + ( ζ ) j k γ ik = tr (cid:0) [ R i R j , ζ ] (cid:1) = 0 , (5.14)where γ ij = tr( R i R j ), as before. Consequently, the data ζ which gave rise to a spectral flowautomorphism of ˆ g above, can also be used to define a spectral flow automorphism of the renor-malized current algebra, i.e. of the algebra that is generated by K jB with operator products givenin subsection (3.3). Only the shift vector w i of the zero modes needs a small adjustment such thatthe new spectral flow action reads ω (cid:0) K iB ( z ) (cid:1) = ( W ) ij ( z ) K jB ( z ) + w iB z − where w iB = w i + tr (cid:0) ζ R i (cid:1) . (5.15)In order to validate that this indeed defines an automorphism we need to check the analogue of thecondition (5.7) for the new metric κ − γ . But this constraint is trivially met, using w iB = ( κ − γ ) ij κ jk w k . (5.16)along with the invariance of both metrics κ and κ − γ . Note that ζ is not changed and hence ithas the same (integer) eigenvalues as before.In order to obtain an automorphism which is compatible with the free field construction we alsoneed to introduce the transformations ω (cid:0) p a ( z ) (cid:1) = p b ( z ) ( W ) ba ( z ) , ω (cid:0) θ a ( z ) (cid:1) = ( W ) ab ( z ) θ b ( z ) . (5.17)It is then straightforward but lengthy to check that the previous transformations define an auto-morphism of the algebra generated by p a , θ a and K jB that descends to the original spectral flowautomorphism ω of our current superalgebra ˆ g . During the calculation one has to be aware ofnormal ordering issues.In conclusion we have shown that any spectral flow automorphism of the bosonic subalgebraof a current superalgebra (related to a Lie superalgebra of type I) can be extended to the fullcurrent superalgebra. Furthermore, this extension was seen to be consistent with our free fermionresolution. Let us remark that even if we start with a spectral flow automorphism ω preservingperiodic boundary conditions for bosonic currents, the lifted spectral flow ω does not necessarilyhave the same property on fermionic generators. Only those spectral flow automorphisms ω : ˆ g → ˆ g for which W is meromorphic as well seem to arise as symmetries of WZNW models on supergroups.Nevertheless, also non-meromorphic spectral flows turn out to be of physical relevance. They canbe used to describe the twisted sectors of orbifold theories, see section 5.4 for details. Obviously, it is of central importance to determine the partition function and higher correlators ofWZNW models on supergroups. Here we shall explain how the calculation of these quantities maybe reduced to computations in the corresponding bosonic WZNW models. For the torus partition30unction we will provide a full expression in terms of characters of the (renormalized) bosoniccurrent algebra.All computations in the WZNW model on type I supergroups depart from the decoupled the-ory (3.21). The interaction between bosons and fermions is treated perturbatively. What makesthis approach particularly powerful is the fact that the perturbative expansion turns out to truncateafter a finite number of terms. The order at which the truncation occurs, however, depends on thesupergroup and the correlator to be computed. As a general rule, the number of terms to considerin the perturbative expansion increases with the number of vertex operators that are inserted.To begin with, let us describe the unperturbed theory (3.21) with a few concrete formulas.As we proceed it is useful to keep in mind that solving the unperturbed theory is a field theoreticanalogue of solving the truncated Laplace operator ∆ . Fields in the decoupled theory form a space H which is a field theoretic version of the semi-classical space F ( G ). The state space H naturallyfactorizes into bosonic and fermionic contributions, H = M µ ∈ Rep(ˆ g ren0 ) (cid:16) V µ ⊗ V F (cid:17) ⊗ (cid:16) ¯ V ∗ µ ⊗ ¯ V F (cid:17) . (5.18)For simplicity we assumed that the bosonic part has a charge conjugate modular invariant partitionfunction. The fermionic representation is unique if we restrict ourselves to the Ramond-Ramondsector. In case applications require to include fermionic fields with anti-periodic boundary con-ditions as well, they can be incorporated easily. According to eq. (5.18), vertex operators of thedecoupled theory possess a basis of the form V a µ ; b ( z, ¯ z ) ≡ V a ,...,a s µ ; b ,...,b t ( z, ¯ z ) = V µ ( z, ¯ z ) θ a ( z ) · · · θ a s ( z ) ¯ θ b (¯ z ) · · · ¯ θ b t (¯ z ) (5.19)where V µ are vertex operators in the bosonic WZNW model. We have noted before that the freefermion theory admits a current superalgebra symmetry ˆ g ⊕ ˆ g . The latter is given explicitly by theformulas in section 3.3. When analyzed with respect to this current superalgebra, the state space H assumes the form H = M µ ∈ Rep(ˆ g ) H µ ⊗ ¯ H ∗ µ (5.20)where H µ an H ∗ µ are the Kac modules and their duals, as defined in equation (5.1). It shouldbe kept in mind though that H contains an atypical sector (including, e.g., H ⊗ ¯ H ∗ ) which is notfully reducible. Nevertheless, the zero-modes L and ¯ L of the Virasoro-Sugawara fields are fullydiagonalizable.The true state space H of the interacting theory, on the other hand, is a field theoretic versionof the space F ( G ) in our minisuperspace theory. In particular, H agrees with H as a gradedvector space (with the grading provided by the generalized eigenvalues of L and ¯ L ) and even asˆ g ⊕ ˆ g -module. But when considered as a module of the left and/or right current superalgebra, H and H are fundamentally different. While, under the action of e.g. the right moving currents, H decomposes into a sum of typical and atypical Kac modules, H may be expanded into projectives.The corresponding multiplicity spaces, however, do not carry a representation of the left moving In case the consistency of the bosonic theory requires to consider spectral flow automorphisms, e.g. for non-compact groups, they should also be included in the definition of the labels µ . It is the dual which is relevant here since we assume the antiholomorphic current superalgebra to mimic thedifferential operators (4.4), not those in (4.3). Notice that the roles of S a and S a are exchanged in these expressions. I [ σ ] which entangle projective coversin an intricate way. Now recall that the Virasoro element L contains the (renormalized) Casimiroperator of g as a summand and it agrees with the latter on ground states. But since our harmonicanalysis revealed that the Casimir operator may not be diagonalized in the atypical subspace of F ( G ), the same must be true for the action of L (and ¯ L ) on H . This shows that supergroupWZNW theories are always logarithmic conformal field theories. After these remarks, let us address the partition function of the theory and its modular invari-ance. We have stressed above that H and H are isomorphic as ˆ g ⊕ ˆ g -modules. Hence, the partitionfunction of the interacting theory agrees with the partition function of the decoupled model andboth may be written as a sum over bilinears of characters of Kac modules. Thereby, the partitionfunction of WZNW models on type I supergroups takes the form Z G ( q, ¯ q ) = Z G B ren ( q, ¯ q ) · Z F ( q, ¯ q ) , (5.21)i.e. it is obtained as a product of the corresponding partition functions of the (renormalized) bosonicmodel with that of the free fermionic system. Each of the two factors corresponds to a well-definedand consistent conformal field theory. This shows that our proposal for the state space of thesupergroup WZNW model yields a suitable partition function.In theories with fermions one has to distinguish between the purely combinatorial partitionfunction which merely counts states and the torus vacuum amplitude which is the relevant physical quantity. Since the fermions anti-commute, the latter requires an insertion of the fermion numberoperator ( − F + ¯ F into the trace, thus turning characters into supercharacters. In our state spaces,bosonic and fermionic states always come in pairs, causing Z F ( q, ¯ q ) to vanish. Actually, this is theusual way in which modular invariance manifests itself in fermionic theories. To avoid dealing withtrivial quantities, one may switch to unspecialized characters. The latter lead to a non-vanishingphysical partition function.We claim that the expression (5.21) is the universal partition function for supergroup WZNWmodels similar to the charge conjugate one in ordinary bosonic models. We will indeed argue in thefollowing section that this modular invariant can be used as the basic building block to derive new,non-trivial partition functions using methods that are well-established in purely bosonic conformalfield theories.We wish to conclude this subsection with a few comments on the calculation of correlationfunctions. We have argued above that fields in the decoupled and the interacting theory are inone-to-one correspondence with each other. In fact, the transition from the auxiliary space H to theproper state space H of the supergroup WZNW model is implemented by a linear map ˆΞ : H → H .The latter generalizes and extends the map Ξ that we used in the semi-classical analysis to identifystates in F ( G ) and F ( G ). Let us denote the image of the field (5.19) under ˆΞ by Φ a µ ; b . According Note that the structure and number of ˆ g -blocks and hence of the indecomposables ˆ I [ σ ] in the field theory maydiffer from that in the minisuperspace theory, see eq. (4.7). The relation between the two may be established withthe help of spectral flow automorphisms. There might exist consistent truncations to diagonalizable subsectors for low levels, see the discussion in [37].Such phenomena appear to be very rare, though. Since the Cartan subalgebra of g was assumed to be identical to the Cartan subalgebra of g this statement evenholds for unspecialized characters and partition functions.
32o our general strategy, correlation functions in the interacting theory may be computed through (cid:10) Φ a µ ; b ( z , ¯ z ) · · · Φ a N µ N ; b N ( z N , ¯ z N ) (cid:11) = s max X s =0 s ! (cid:10) V a µ ; b ( z , ¯ z ) · · · V a N µ N ; b N ( z N , ¯ z N ) S s int (cid:11) , (5.22)where the correlators on the right hand side are to be evaluated in the decoupled theory. We shallshow below that correlators with s ≥ s max = N r insertions vanish so that the summation over s isfinite. Let us also recall that the interaction term is given by S int = − i π Z p a R ab ( g B ) ¯ p b dw ∧ d ¯ w . (5.23)Here, the expression R ab ( g B ) should be interpreted as a vertex operator of the bosonic WZNWmodel, transforming in the representation R ⊗ R ∗ .There are now two computations to be performed in the decoupled theory. First of all, wehave to determine correlation functions for the bosonic fields V µ i with additional insertions of s vertex operators R ab ( g B ). We shall assume the bosonic WZNW model to be solved and hencethat all these bosonic correlators are known. Let us comment, however, that the dependence ofsuch correlation functions on the insertion points of R ab ( g B ) is controlled by null vector decouplingequations. As usual, these can be exploited to derive integral formulas for the required correlationfunctions. We shall not go into any more detail here.Instead, let us now comment on the second part of the computation that deals with the fermionicsector. Since we are dealing with r chiral bc systems at central charge c = −
2, the evaluation israther standard. According to the usual rules, non-vanishing correlators on the sphere must satisfy θ a − p a = 1, i.e. the number of insertions of a fixed field θ a must exceed the number of insertionsof p a by one. In an N -point correlator, any given component θ a can appear at most N times. Thefields p a , on the other hand, only emerge from the s insertions of the interaction term. Hence, weconclude that all contributions to our correlation function with s ≥ N · r insertions of S int vanish.The non-vanishing terms can be evaluated using that D n Y ν =1 p a ( z ν ) n +1 Y µ =1 θ a ( x µ ) E = Q ν<ν ′ ( z ν − z ν ′ ) Q µ<µ ′ ( x µ − x µ ′ ) Q ν Q µ ( z ν − x µ ) (5.24)and a similar formula applies to ¯ θ a and ¯ p a . These expressions can be inserted into the expansion(5.22). Thereby we obtain a formula for the N -point functions of the WZNW model which presentsit as a sum of at most N · r terms labeled by an integer s . Each individual summand involvesan integration over s insertion points w i . The corresponding integrand factorizes into free fieldcorrelators of the form (5.24) multiplied with a non-trivial ( N + s )-point function in the bosonicWZNW model for the group G B .Let us point out that for a given choice of N fields, the perturbative evaluation of the correlatormay truncate way before we reach s max . An extreme example appears when all the fields Φ µ i =ˆΞ V µ i , i = 2 , . . . , N, are images of purely bosonic fields V µ i while the first field contains the maximalnumber of fermionic factors, both for left and right movers. In that case, only the term with s = 0 contributes and hence these fields of the WZNW model on the supergroup possess the samecorrelation functions as in the bosonic WZNW model, i.e. (cid:10) Φ , ,...,rµ ;1 , ,...,r ( z , ¯ z ) Φ µ ( z , ¯ z ) · · · Φ µ N ( z N , ¯ z N ) (cid:11) = (cid:10) V µ ( z , ¯ z ) V µ ( z , ¯ z ) · · · V µ N ( z N , ¯ z N ) (cid:11) (5.25)33here the correlation function on the right hand side is to be evaluated in the bosonic WZNWmodel. The result is a direct analogue of the corresponding formula (4.26) in the minisuperspacetheory. During the course of the previous sections we frequently assumed that the bosonic subgroup G B ⊂ G was compact and simply-connected. On a technical level, this condition is required in order torender the matrix R ( g B ) well-defined which entered the expression for the differential operatorsimplementing the isometries of G on the function space F ( G ). On the other hand this choiceautomatically limited our considerations to WZNW models with (the analogue of a) charge conju-gate modular invariant. In this subsection we would like to sketch how such a restriction may beovercome.Let us recall the situation for bosonic WZNW models first. It is well-known that a non-simply-connected group manifold G can be described geometrically as an orbifold ˜ G / Γ where ˜ G is theuniversal covering group and Γ ∼ = π ( G ) ⊂ Z ( ˜ G ) is a subgroup of its center. The simplest exampleis SO (3) = SU (2) / Z . In conformal field theory, orbifolds of the previous type are implementedby means of a simple current extension of the theory with charge conjugate modular invariant [65](see also [66]). This construction of the G WZNW model rests on the fact that the ˜ G modelcontains sufficiently many simple currents, one for each element in the center Z ( ˜ G ). Incidently,these are in one-to-one correspondence with (spectral flow) automorphisms of the current algebraˆ g . Such simple current extensions exhaust all modular invariants related to the current algebraˆ g , apart from some exceptional cases at low levels.Now it has been shown in [67] that the global topology of a Lie super group is completelyinherited from that of its bosonic subgroup. Consequently, given a supergroup G with bosonicsubgroup G = ˜ G / Γ, there exists a covering supergroup ˜ G with bosonic subgroup ˜ G , and one has G = ˜ G/ Γ. Note that central elements in ˜ G are also central in ˜ G . Having constructed the WZNWmodel on the covering supergroup ˜ G , we would like to divide by Γ. But, as we have just stated,elements of Γ can all be identified with elements in the center of the bosonic subgroup ˜ G . Therefore,they label certain simple currents of the ˜ G WZNW model. As indicated in the previous paragraph,we may think of these simple currents as (equivalence classes of) spectral flow automorphisms of ˆ g .According to the results of subsection 5.2, all such spectral flow automorphisms may be extendedfrom ˆ g to the current Lie superalgebra ˆ g , in a way that is even consistent with the free fermionconstruction. Consequently, the elements of our designated orbifold group Γ label a certain set ofspectral flow automorphisms of ˆ g . It is the action of these spectral flow automorphisms that onehas to use in order to construct the orbifold CFT belonging to the supergroup G = ˜ G/ Γ.Our discussion so far has been fairly abstract and we would like to flesh it out a bit more.Actually, the details of the orbifold construction are not much different from what is done inbosonic models. For simplicity, let us assume that Γ is cyclic and of finite order. We shall denotethe generating element by γ . In order to illustrate the relation between orbifolds and spectral flowautomorphisms, we depart from the conventional orbifold approach. Namely, we include (chiral)twisted sectors on which the supercurrents X satisfy boundary conditions of the form X ( e πi z ) = (cid:0) γ ( X ) (cid:1) ( z ) . (5.26)There exists a basis X σ , on which γ acts diagonally as a multiplication with some phase exp(2 πiγ σ ).If γ σ is an integer, then X σ has integer moding in the twisted sector, otherwise its modes are34ational. All these twisted sectors emerge by acting with certain (meromorphic or not) spectralflow automorphisms on the untwisted representations (see subsection 5.2). The discussion of theprevious paragraph supplied us with the relevant set of spectral flow automorphisms and hence witha list of chiral sectors to be incorporated in the construction of the G = ˜ G/ Γ orbifold theory. Sectorsof the full non-chiral theory are obtained by independent action of spectral flow automorphisms onleft and right-movers in the parent theory on ˜ G . Therefore, even meromorphic spectral flows leadto new non-chiral sectors, though these are put together from untwisted representations of the leftand right movers. All this has been worked out for many interesting bosonic models, such as e.g.the SO (3) = SU (2) / Z WZNW model. WZNW models on non-simply-connected supergroups areno harder to deal with. Let us finally comment on the connection of the algebraic orbifolds with the Lagrangian picture.Looking at our free field resolution (3.5) one might have had the naive idea to replace the bosonicmodel by its orbifold and then to add fermions and interaction terms in the same way as before. Butthis is not at all what we suggest to do. In particular, the orbifold group Γ need not be a symmetryof the interaction term if there is no action on the fermions. Even worse, the vertex operator R ab ( g B ) occuring in the interaction may not be part of the spectrum of the purely bosonic WZNWmodel. As a consequence, the perturbed correlation functions with insertions of this operatorare not well-defined. This happens, for example, if we try to supersymmetrize the bosonic group SO (3) × U (1). The fermions of the extended model with su (2 |
1) symmetry transform in the spin1 / SU (2) which does not descend to a representation of SO (3) = SU (2) / Z .Hence, it is absolutely crucial to depart from the full SU (2 |
1) WZNW model and to divide the fullorbifold action on both bosonic and fermionic variables.
Various logarithmic conformal field theories have been considered in the literature. The beststudied examples are the triplet models in which the conformal symmetry is extended by a tripletof currents, each having spin h = 2 p − p = 2, Gaberdiel and Kausch have been able to come up witha consistent local theory [11]. The extended chiral symmetry of the triplet models is denoted by W ,p . The latter are believed to be part of a family of more general W -algebras W q,p where p and q are co-prime. All of these possess interesting indecomposable representations. Their representationtheory is particularly well understood for q = 1, see [14] and references therein.This final section has two aims. First of all we would like to illustrate that the existing resultson the representation theory of W ,p -algebras and the local triplet model (for p = 2) fit very nicelyinto one common picture with the logarithmic WZNW models on type I supergroups. But giventhe remarkable progress with the latter, and in particular with the construction of infinitely manyfamilies of new local non-chiral models, our results lead to a number of interesting predictions on W q,p -algebras and the associated local logarithmic conformal field theories. The SO (3) theory also shows that the orbifold construction might suffer from obstructions, depending on thechoice of the level. A more detailed treatment of such issues for supergroup orbifolds is left for future work. .1 Chiral representation theory Let us begin this subsection by reviewing some results on the representation theory of W ,p = W ( p )(see [14] and references therein). This chiral algebra is known to admit 2 p irreducible highestweight representations V ± s where s = 1 , . . . , p . While V ± p do not admit non-split extensions, allother 2( p −
1) representations appear in the head of the following indecomposables, R ± s : V ± s → V ∓ p − s → V ± s (6.1)where s runs from s = 1 to s = p −
1. Hence the representations V ± p can be considered typicalwhereas all others are atypical. Moreover, the indecomposables R ± s are the projective covers ofthe atypicals V ± s and play the role of the representations P in section 2.2.3. The typical modules V ± p are projective as well, in agreement with results on the fusion for W (2) representations, see[69]. The fusion rules of W p,q -models have recently been addressed in [16].The representation theory of W ( p )-algebras also contains analogues of our Kac modules foratypical representations. These have the form K ± s : V ± s → V ∓ p − s (6.2)where s = 1 , . . . , p −
1. In view of the role they are going to play we will simply refer to therepresentations K ± s as “Kac modules” as well. They are obtained as quotients of the projectivecovers R ± s . For the typical representations V ± p , the associated irreducibles, “Kac modules” andprojective covers all coincide. In this sense, we shall also write K ± p = V ± p = R ± p , just as fortypical representations of Lie superalgebras. Furthermore, among the quotients of the projectivecovers one can also find 4( p −
1) “zig-zag” modules, containing three irreducible representationseach. It seems likely, that these are just the first few examples among an infinite series of zig-zagrepresentations of W ( p ), in close analogy to representations of the Lie superalgebra gl (1 |
1) (see e.g.[71]). The main difference between gl (1 |
1) and W ( p ) zig-zag modules is that the constituents ofthe former are pairwise inequivalent. Zig-zag modules of W ( p ), on the other hand, are built froma pair of irreducibles, each appearing with some multiplicity. This opens the possibility to closezig-zag modules of W ( p ) into rings. Representations of all these different shapes were found andinvestigated for the quantum groups [72, 73] which are dual to W ( p ), in the sense of Kazhdan-Lusztig duality.Let us also compare some further properties of W ( p )-modules with those we discussed for Liesuperalgebras of type I. For example, we have pointed out that all projective modules of type Isuperalgebras possess a “Kac composition series”. The same is true for the projective covers R ± s , R ± s : K ± s → K ∓ p − s (for s < p ) , R ± p = K ± p . (6.3)Moreover, we also observe that the multiplicities in the “Kac composition series” of indecompos-able projective covers (reducible and irreducible) and those of irreducible representations in thecomposition series of “Kac modules” are related by( R µ : K ν ) = [ K ν : V µ ] . (6.4) These diagrams have to be read as follows: To the right we write the maximal fully reducible submodule.Everything left of the rightmost arrow describes the quotient module of the original module with respect to thesubmodule mentioned before. One can then proceed iteratively to define the whole diagram. We use the qualifiers “atypical” and “typical” only to clarify the analogy to the supergroup WZNW models. Incontrast to the latter, the atypical representations are obviously the generic ones for the algebra W ( p ). Using the analogy to the Kazhdan-Lusztig dual quantum group, they have been called Verma modules in [70]. W -algebra representations. While irreducible representations and their projective covers are certainlycentral objects for all Lie superalgebras, some of their properties may differ considerably fromwhat we have seen in the case of type I. We have pointed out already that the existence of a “Kaccomposition series” (or a similar flag ) for projectives and the reciprocity property (2.11) do nothold for more general Lie superalgebras. Hence, these features of W ( p )-modules should not beexpected to carry over to more general W -algebras either. In fact, numerical results of [74] mayindicate that violations even occur for W q,p with p, q = 1. Furthermore, the tensor products forirreducible representations of Lie superalgebras can develop a remarkable complexity. In this sense,the Lie superalgebra gl (1 |
1) is rather well-behaved. Representations of psl (2 | gl (1 |
1) and W ,p suggest that the latter may also be rather unusual creatures in the zoo of W -algebras. In fact,when it comes to the features of fusion, the algebras W q,p may have much more generic properties,resembling very closely those of psl (2 | Regarding the construction of local field theories, the progress with WZNW models on supergroupshas been significantly faster than for minimal logarithmic CFTs. In fact, only the minimal tripletmodel associated with W (2) has been constructed in all detail [11]. Imposing locality constraintson correlation functions, the state space H of this model was shown to have the form H = I ⊕ (cid:0) V +2 ⊗ ¯ V +2 (cid:1) ⊕ (cid:0) V − ⊗ ¯ V − (cid:1) . (6.5)Here, V ± are the typical modules of W (2), in view of their conformal dimensions previously alsodenoted by V − / and V / , and I is a complicated non-chiral indecomposable (denoted by R in[11]) which was obtained originally as a certain quotient of the space (cid:0) R +1 ⊗ ¯ R +1 (cid:1) ⊕ (cid:0) R − ⊗ ¯ R − (cid:1) .The module I is known to possess the following composition series I : (cid:0) V +1 ⊗ ¯ V +1 (cid:1) ⊕ (cid:0) V − ⊗ ¯ V − (cid:1) → (cid:0) V +1 ⊗ ¯ V − (cid:1) ⊕ (cid:0) V − ⊗ ¯ V +1 (cid:1) → (cid:0) V +1 ⊗ ¯ V +1 (cid:1) ⊕ (cid:0) V − ⊗ V − (cid:1) , (6.6)where we used the correspondence V +1 = V and V − = V for the atypical irreducibles of W (2).When acting with elements of either the left or right chiral algebra only, H decomposes into a sumof projectives, each appearing with infinite multiplicity. The individual multiplicity spaces cannotbe promoted to representation spaces of the commuting chiral algebra, but they come equippedwith a grading that is given by the (generalized) eigenvalues of L or ¯ L . When considered asgraded vector spaces, they coincide with the graded carrier spaces of irreducible representations.All this is very reminiscent of what we found in eq. (4.6) while studying the harmonic analysis onsupergroups. A certain similarity between the representation theory of W q,p (or rather its dual quantum group) and psl (2 | (cid:0) K +1 ⊗ ¯ K +1 (cid:1) ⊕ (cid:0) K − ⊗ ¯ K − (cid:1) . Hence, the partition function of the triplet model can be expressed as Z ( q, ¯ q ) = X i =1 , X η = ± χ K ηi ( q ) ¯ χ K ηi (¯ q ) . (6.7)This result is reminiscent of what we found for supergroup WZNW models in section 5.3. Note thatthe modular transformation behaviour for characters of Kac modules is rather simple which makesit easy to check that Z ( q, ¯ q ) is modular invariant. In comparison, the transformation behaviour ofcharacters belonging to atypical irreducible representations of W (2) is rather involved [69], just asfor current superalgebras.The striking similarities between the local triplet theory and the harmonic analysis on super-groups suggest some far reaching generalizations, in particular concerning the state space of a wideclass of local logarithmic conformal field theories. Let us denote the irreducible representationsof some chiral algebra W by V a and their projective covers by P a . For typical representationsthe latter agree (by definition) with the irreducibles. We also introduce the symbol V a when V a is considered merely as an L -graded vector space. Given this notation, we propose that a locallogarithmic conformal field theory with symmetry W can be constructed on the state space H = M a V a ⊗ ¯ P a . (6.8)Our proposal describes the state space of the conjectured local theory as a graded representationspace for W . The extension to the full W ⊗ ¯ W is severely constrained by requiring symmetry withrespect to an exchange of left and right chiral algebras. Concerning the implications for W ( p )-models it is interesting to observe that the same structures were found in the regular representationof the dual quantum group, see [70], page 24, and compare with eq. (2.9) in [37]. Let us point outthat local theories may probably also be built on other state spaces. Examples are given by theorbifold models we described in section 5.4 or by some exceptional truncations of WZNW modelson simply connected Lie supergroups (see [37] for a few examples).Before we conclude we would like to go one step beyond the previous analogy and to proposea more detailed conjecture for the natural state space of the W ( p ) triplet models for arbitrary p .In a straightforward extension of the result (6.5) for p = 2 we believe that a local theory may bebuilt on the space H = M s = p I s ⊕ M η = ± V ηp ⊗ ¯ V ηp . (6.9)The non-chiral indecomposable representations occurring here have the composition series I s : (cid:0) V + s ⊗ ¯ V + s (cid:1) ⊕ (cid:0) V − p − s ⊗ ¯ V − p − s (cid:1) → (cid:0) V + s ⊗ ¯ V − p − s (cid:1) ⊕ (cid:0) V − p − s ⊗ ¯ V + s (cid:1) → (cid:0) V − p − s ⊗ ¯ V − p − s (cid:1) ⊕ (cid:0) V + s ⊗ V + s (cid:1) which coincides with the composition series of (cid:0) K + s ⊗ K + s (cid:1) ⊕ (cid:0) K − p − s ⊗ K − p − s (cid:1) . Consequently, ourproposal is manifestly modular invariant since the partition function can be written as a sum overall “Kac modules”, just as in eq. (6.7). Figure 1 provides an alternative 2-dimensional picture ofthe indecomposables I s . In this form the similarities with analogous pictures for gl (1 |
1) and sl (2 | W ( p )-models [70] are clearly displayed.38 + s ⊗ V + s ( ( QQQQQQQ , , YYYYYYYYYYYYYYYYYYYYYY - - \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ v v mmmmmmm V − p − s ⊗ V − p − s ( ( QQQQQQ v v mmmmmm r r eeeeeeeeeeeeeeeeeeee q q bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb V − p − s ⊗ V + s ( ( QQQQQQ - - \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ V + s ⊗ V − p − s v v mmmmmm , , YYYYYYYYYYYYYYYYYYYYYY V + s ⊗ V − p − s r r eeeeeeeeeeeeeeeeeeee ( ( QQQQQQQ V − p − s ⊗ V + s v v mmmmmmm q q bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb V − p − s ⊗ V − p − s V + s ⊗ V + s Figure 1: The structure of the non-chiral representation I s .The particular relevance of projective modules for local bulk theories is one of the main outcomesfrom the study of WZNW models on supergroups, see also [36, 24, 37]. Their role for logarithmicextensions of minimal models was also emphasized in [75, 77], mostly based on studies of the dualquantum group. It seems worth pointing out, though, that for quotients of supergroups, projectivemodules might not play such a prominent role, even though some of them are likely to be logarithmicas well. Similarly, boundary spectra in logarithmic conformal field theories are known to involveatypical irreducibles as well as projectives. For the triplet model, boundary conditions with anatypical irreducible spectrum of boundary operators were exhibited in the recent work of Gaberdieland Runkel [18]. Studies of branes on supergroups confirm the existence of such boundary spectraand they provide a beautiful geometric explanation [78]. In our paper we presented the main ingredients for a complete solution of arbitrary supergroupWZNW models based on basic Lie superalgebras of type I. All our results relied on a free fermionresolution of the underlying current superalgebra which allowed to keep the bosonic subsymmetrymanifest in all expressions we encountered, i.e. in action functionals, representations, correlationfunctions and other quantities. On the level of the Lagrangian we showed that the original WZNWLagrangian could be written as a sum of a WZNW model for the bosonic subgroup with renor-malized metric and possibly a dilaton, the action for a set of free fermions and an interaction termwhich couples the fermions to a vertex operator of the bosonic model. The usefulness of this con-struction has also been demonstrated in the full quantum theory, e.g. when we reconstructed thecurrent superalgebra in terms of the corresponding bosonic current algebra and free fermions.In order to solve the WZNW model we first focused on its semi-classical, or small curvature limitwhich allowed to reduce the construction of the space of ground states to a problem in harmonicanalysis on a supergroup. We could confirm previous observations [58, 36, 24, 37] that the space offunctions splits into two qualitative very different sectors. First of all, there exists a typical sectorwhich decomposes into a tensor product of irreducible typical representations under the action ofthe supergroup isometry g ⊕ g . On this subspace, the Laplacian is fully diagonalizable and itseigenvalues are determined by a specific quadratic Casimir of g . In addition, the space of functionson a supergroup always exhibits an atypical sector consisting of projective covers entangled in acomplicated way such that the resulting non-chiral modules cannot be written as (a direct sum of)tensor product representations. In this sector the Laplacian is not diagonalizable and the necessityfor a non-trivial entanglement may eventually be traced back to the fact that the left and right39egular action lead to the same expression for the Laplacian. We wish to emphasize that ourderivation of the spectrum has been very general and just relied on the validity of a reciprocitytheorem proven by Zou and Brundan [48, 49].Starting from this semi-classical truncation it has been argued that all its interesting featurespersist in the full quantum theory. In particular, the full state space of the WZNW model isstill composed of a typical and an atypical sector. Again, the representations in the latter donot factorize and the dilatation operators L and ¯ L may not be diagonalized. Since the vacuumrepresentation is always atypical this automatically implies the existence of a logarithmic partner ofthe identity field and makes supergroup WZNW models genuine examples of logarithmic conformalfield theories.It should be noted that, in comparison to ordinary free field constructions [38, 39, 40, 41, 42]which are based on a choice of an abelian subalgebra, our free fermion resolution is much easierto deal with. In particular, the representations of the current superalgebra obtained from thegeneralized Fock spaces (5.1) are typically irreducible. This observation lets us suggest that theserepresentations are the proper generalization of Kac modules in the infinite dimensional setting.Furthermore, there was no need of introducing various screening charges and BRST operators, asimplifying feature that reflects itself in the calculation of correlation functions. The latter couldbe reduced to a perturbative but finite expansion in terms of correlation functions in the productof a purely bosonic WZNW model with renormalized metric and a theory of free fermions.Finally, we commented on possible partition functions and we explained how they are con-structed as a product of partition functions for the constituents in our free fermion resolution.This rather simple behavior is rooted in the fact that traces are insensitive to the compositionstructure of representations. Hence, the full WZNW theory possesses the same partition functionas the decoupled free fermion theory in which products of (reducible) Kac modules appear insteadof projective covers. Taking this assertion for granted, the torus modular invariance of our theoryis satisfied automatically. It might be helpful to add that torus partition functions of many non-rational bosonic conformal field theories, e.g. of Liouville theory or of the H model, are equallyinsensitive to the interaction. This does certainly not imply that the theories are trivial, neither incase of non-rational conformal field theories, nor for WZNW models on supergroups.In the last section of this work we placed our new results on chiral and non-chiral aspects ofsupergroup WZNW models in the context of previous and ongoing work on other logarithmic con-formal field theories, in particular on logarithmic extensions of minimal models. The similarities areremarkable and provide some novel insight that helps to separate generic properties of logarithmicconformal field theories from rather singular coincidences. As an application of the analogies weconjectured a precise formula for the state space of a fully consistent local theory based on an arbi-trary chiral algebra. It adopts a particularly nice shape for the minimal logarithmic W ( p )-theories.Working with supergroup WZNW models has two important advantages over the considerationof non-geometric logarithmic conformal field theories. Concerning the study of chiral aspects, theclose link between the current superalgebra ˆ g and its horizontal subsuperalgebra g provides a ratherstrong handle on the representation theory of W = ˆ g . In fact, since the representation theory of g is under good control, the same is true for its affine extension ˆ g . Even though we have not reallypushed this to the level of mathematical theorems, there is no doubt that rigorous results can beestablished along the lines of our discussion. For some particular examples, this has been carriedout already [36, 24, 37]. The second advantage of supergroup WZNW models is the existenceof an action principle. The latter is particularly powerful when it comes to the construction of40ocal logarithmic field theories, a subject that has been notoriously hard to address for logarithmicextensions of minimal models. In fact, we have seen in section 5.3 that the action leads to arigorous tool for constructing bulk correlation functions. As such, it has already been exploited inthe construction of correlation functions for the GL (1 |
1) WZNW model [36].The present work admits natural extensions in several directions. Among these, the problemof finding concrete expressions for the full correlation functions or, at least, conformal blocks isprobably the most urgent. Another issue of considerable significance is the extension of our ideasto world-sheets with boundaries or, in string theory language, the discussion of D-branes. Inthis context it seems necessary to obtain a better handle on modular transformation properties ofcharacters, including those of irreducible atypical representations [13]. We hope that our work willbe helpful in deriving new character formulas along the lines of [24, 37]. It would also be interestingto work out in greater detail the solution of WZNW models with non-trivial modular invariants.In order to acquire more experience with supersymmetric σ -models and for various applicationsit would be desirable to extend our study to supercoset models. In contrast to bosonic models,there is considerably more freedom in choosing how to gauge. Besides gauging the standard adjointaction, as is done in [79, 80, 81, 82] there are many cases in which purely one-sided cosets areknown or believed to be conformally invariant [19, 20, 21, 22]. Those latter cases are relevant forthe description of AdS-spaces, projective superspaces and even flat Minkowski space. It is worthnoting that the harmonic analysis on coset models G/H with H a bosonic subgroup acting fromthe right, g ∼ gh , can easily be obtained from our results, see section 4.2 and especially eq. (4.8)(cmp. also [83]). All the additional input required is the branching of g -modules into h -modules.Even before carrying out any such decomposition explicitly, we may conclude from eq. (4.8) thatthe resulting g -modules are all projective. This particularly applies to all generalized symmetricspaces which are relevant for the description of AdS-spaces. Let us stress, however, that cosets G/H by some non-trivial super group H may behave differently. In fact, some simple examplesshow how even atypical irreducibles may emerge in their spectrum.Apart from these structural and conceptual issues we also expect our work to have concreteimplications, e.g. in string theory. Let us recall that it is not difficult to write down classical σ -models which can be used to describe string theory on AdS-spaces with various types of backgroundfluxes for instance [19, 84]. But for a long time it has not been clear how to quantize these fieldtheories while keeping the target space supersymmetry manifest. It was only recently that the purespinor approach closed this gap to some extent [85, 86]. Although substantial progress has beenmade on certain aspects of the pure spinor formulation, there exist a variety of open conceptualissues, in particular when curved backgrounds are involved. It was proposed to overcome some ofthem through a reformulation in terms of supergroup WZNW models [87]. The ideas presentedabove may help to gain more control over the relevant models.For a complete picture we also need to solve WZNW models beyond Lie supergroups of type I.These include, in particular, supergroups of type II where the fermions occur in a single multipletof the bosonic subgroup. Structurally, type II implies that there is no natural Z -grading anymorewhich is consistent with the intrinsic Z -grading of the underlying Lie superalgebra. This issueconcerns the two series B ( m, n ) = osp (2 m + 1 | n ) and D ( m, n ) = osp (2 m | n ) of Kac’ classification[45] which e.g. constitute the isometries of superspheres S n + m − | n [5, 88]. Moreover, these seriesinclude the special cases D (2 , α ) and D ( n + 1 , n ) which have been shown [22] to have similarly ex-citing properties as A ( n, n ) = psl ( n +1 | n +1) [23]. Let us note that the WZNW models based on thefamily of exceptional Lie superalgebras D (2 , α ) are also relevant for a manifestly supersymmetric41escription of string backgrounds involving AdS × S × S . Since type II superalgebras do notadmit a canonical (covariant) split of fermionic coordinates into holomorphic and antiholomorphicdegrees of freedom, they require a rather significant extension of the above analysis. Incidently,the same is true for the representation theory of type II superalgebras which is considerably morecomplicated than in the type I case [50]. We hope to return to these issues in future work. Acknowledgements
It is a great pleasure to thank Nathan Berkovits, Thomas Creutzig, Matthias Gaberdiel, Ger-hard G¨otz, Andreas Ludwig, Eric Opdam, Jørgen Rasmussen, Alice Rogers, Ingo Runkel, HubertSaleur, Alexei Semikhatov, Vera Serganova, Arvind Subramaniam and Anne Taormina for usefuland inspiring discussions. Thomas Quella acknowledges the warm hospitality at the KITP in SantaBarbara during an intermediate stage of this work. This research was supported in part by theEU Research Training Network grants “Euclid” (contract number HPRN-CT-2002-00325), “Force-sUniverse” (contract number MRTN-CT-2004-005104), “Superstring Theory” (contract numberMRTN-CT-2004-512194), by the PPARC rolling grant PP/C507145/1, and by the National ScienceFoundation under Grant no. PHY99-07949. Part of this work has been performed while ThomasQuella was working at King’s College London, funded by a PPARC postdoctoral fellowship underreference PPA/P/S/2002/00370.
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