Modular Data and Verlinde Formulae
for Fractional Level WZW Models II
Abstract.
This article gives a complete account of the modular properties and Verlinde formula for conformal field theories based on the affine KacMoody algebra at an arbitrary admissible level . Starting from spectral flow and the structure theory of relaxed highest weight modules, characters are computed and modular transformations are derived for every irreducible admissible module. The culmination is the application of a continuous version of the Verlinde formula to deduce nonnegative integer structure coefficients which are identified with Grothendieck fusion coefficients. The Grothendieck fusion rules are determined explicitly. These rules reproduce the wellknown “fusion rules” of Koh and Sorba, negative coefficients included, upon quotienting the Grothendieck fusion ring by a certain ideal.
1. Introduction
This is the sequel to the article [1] devoted to solving the longstanding problem of determining the (Grothendieck) fusion coefficients, for admissible level WessZuminoWitten models, from a formula of Verlinde type. The main issue here is that initial attempts to do so, using the standard Verlinde formula for highest weight modules [2], led to certain “fusion coefficients” being negative integers [3] (we refer to [1] for further historical detail). The mechanism responsible for these negative coefficients was only obtained recently [4] for the admissible level . There, it was pointed out that this negativity resulted from assuming that the irreducible modules of the spectrum were all highest weight and from not properly accounting for the regions of convergence of the highest weight modules’ characters (see [5, 6] for a more detailed discussion).
While this mechanism accounts for what goes wrong in applying the standard Verlinde formula, the problem of how to modify this formula so as to obtain nonnegative integer fusion coefficients remained. This was addressed in [1] wherein the modular properties of the models at levels and were analysed. The main result was that a continuous version of the Verlinde formula may be applied to each of these theories and that the results were consistent with the known fusion rules (which have only been computed for these levels [7, 6]). In particular, the continuum Verlinde formula yielded nonnegative integers that precisely reproduced the Grothendieck fusion coefficients. The aim of this article is to generalise the continuum Verlinde computations to all admissible levels, for at least, and show that the mechanism identified to generate the negative “fusion coefficients” when is also responsible in this greater generality.
The methodology employed here to tame the modular properties of fractional level WessZuminoWitten models is but one instance of a general programme we are developing (see [8] for a review) to deal with Verlinde formulae for logarithmic conformal field theories. Indeed, it is known that the models with and are necessarily logarithmic [7, 9, 5] and this is surely the case more generally. This programme is, in some respects, a farreaching extension to general logarithmic conformal field theories of ideas which were originally developed in the string theory literature to deal with supersymmetric and noncompact spacetimes (see [10, 11, 12, 13] for example). Besides the theories considered here, this programme has already been successfully applied to the Grothendieck fusion rules of [14], its extended algebras [15] and its Takiff version [16], the singlet and triplet models [8, 17] and even the Virasoro algebra [18].
We begin, as always, with notation and conventions. Section 2 describes this for and its highest weight modules before introducing the conjugation and spectral flow automorphisms which play such a vital role in what follows. Section 3 defines the notion of admissibility, first for the level and then for modules. Theorems of Adamović and Milas are then quoted [19] giving the irreducible admissibles in the category of highest weight modules and the category of relaxed highest weight modules. We then introduce an analogue of the Kac table familiar from the Virasoro minimal models to organise the admissible irreducibles. Finally, we extend our collection of admissibles using spectral flow and catalogue the relationships between spectral flow versions of irreducible admissibles. At this point, we define appropriate notions (following [8]) of “standard”, “typical” and “atypical” modules. In this setting, all highest weight admissibles are atypical and a standard module is typical if and only if it is irreducible.
Our first main result is the character formula for a general standard module. Unlike the characters of the highest weight modules, the standard characters do not converge anywhere and must be represented as distributions. The result, given in Section 4 (Proposition 4 and Corollary 5), describes the character as a sum of delta functions weighted by Virasoro minimal model characters. This is surely a manifestation of quantum hamiltonian reduction [20, 21] and it lifts the observation of [22], where it was noticed that residues of admissible highest weight characters involved minimal model characters, to a much more elegant setting. The modular transformation rules of the standard characters are then computed in Section 5 (Theorem 6) and we verify that one obtains a (projective) representation of the modular group of uncountablyinfinite dimension. Moreover, the “Smatrix” is seen to be symmetric and unitary.
Section 6 then addresses the atypical characters. We wish to determine them as distributions so as to avoid the convergence issues that stymied progress for so long, so we derive resolutions for each atypical module in terms of reducible but indecomposable (atypical) standard modules. The resulting character formulae then allow us to compute the modular transformation rules of (certain) atypical characters in Section 7 (Theorem 11). In particular, we obtain the Stransformation of the vacuum character (the vacuum module is highest weight, hence atypical). These atypical computations rely on a rather ungainly identity (Lemma 10) whose representationtheoretic significance is not yet apparent to us. Presumably, generalising these results to higher rank affine KacMoody algebras will clear this up.
In any case, we now have all the ingredients to apply the obvious continuum analogue of the Verlinde formula. Assuming that this does yield the Grothendieck fusion coefficients, we then compute the complete set of Grothendieck fusion rules explicitly. This is detailed in Section 8 (see Propositions 13, 14, 15 and 18). When we can be sure that the corresponding fusion products are completely reducible, these results can be immediately lifted to the fusion ring itself. In this way, we prove (Theorem 16) that the fusion ring of an admissible level theory always contains a subring isomorphic to that of a particular nonnegative integer level theory. One consequence is that one obtains, for almost all admissible levels, a nontrivial simple current generalising that which gives the ghosts in the theory [4].
Another consequence of our explicit computations is that all the Grothendieck fusion coefficients, as computed by the continuum Verlinde formula, are nonnegative integers (Theorem 19). Because the resolutions we have used lead to alternating sums for atypical characters in terms of standard ones, this nonnegativity result is highly nontrivial and represents a very strong endorsement of our claim that the continuum Verlinde formula does indeed give the Grothendieck fusion coefficients correctly. A second strong endorsement is discussed in Section 9 where we recover the “fusion rules” of [3], negative coefficients and all, for all admissible levels , by applying the mechanism explained in [4] to our Grothendieck fusion rules. These two endorsements give us complete confidence that we have solved the longstanding problem of modular properties and Verlinde formulae for fractional level WessZuminoWitten models.
Throughout the text, we illustrate our results by applying them to the levels and , thereby checking against what was reported in [1]. Section 10 concludes the article by discussing three other admissible level theories which are also of independent interest. In each case, we exhaustively describe the Grothendieck fusion rules and compute the extended algebra defined by the simple current guaranteed by Theorem 16. When , we obtain in this way a conformal embedding of into . When , the extended algebra is the reduced superconformal algebra at . Finally, yields an interesting simple current extension that we tentatively identify with the quantum hamiltonian reduction of .
Of course, there are many points that remain to be addressed. First, it is clear that one should be able to generalise our results to higher rank fractional level affine KacMoody algebras and superalgebras and it would be extremely interesting to do so. Moreover, the relationship (if any) between these fractional level models and the WessZuminoWitten models on noncompact Lie groups requires clarification. Even at the level of , there are many fascinating questions still to consider, for example, that of classifying modular invariant partition functions for the admissible level theories. Mathematically, one should also ask after homological characterisations of the spectrum: What is the physical category of modules? Which modules are projective in this category? Which are rigid? Can we characterise admissible staggered modules as was done for the Virasoro algebra in [23]? Even more interesting, and perhaps more relevant for comparison with noncompact target space models, what happens if we relax the irreducibility of the vacuum module? It is clear that the study of logarithmic theories with affine symmetries will remain rich and rewarding. We hope to report further on this study in the future.
2. and its Representations
Consider the simple complex Lie algebra and its standard basis elements
(2.1) 
This basis is tailored to a triangular decomposition respecting the adjoint (conjugate transpose) that picks out the real form . Indeed, the Cartan element is clearly selfadjoint and the raising and lowering operators and are swapped by the adjoint. In what follows, we want to study conformal field theories whose symmetry algebras are the affine KacMoody algebras at levels which are not nonnegative integers. The wellknown quantisation of the level for the WessZuminoWitten model on suggests that one should not lift the adjoint to . Instead, the absence of levelquantisation for leads us to propose lifting the adjoint that picks out the other real form .
The adjoint simply negates the basis elements , and , hence may be described as negation followed by complex conjugation: . This means that this basis is not suited to triangular decompositions that respect the adjoint. For this reason, we choose a new basis of :
(2.2) 
Because and with respect to the adjoint, this basis is suited to the desired triangular decomposition. Note that the nonvanishing commutation relations in this basis are
(2.3) 
Similarly, the trace form in this basis attracts an unfamiliar sign:
(2.4) 
We remark that choosing the adjoint correctly is not just mathematical sophistry — this choice plays a subtle, but vital, role in many aspects of the representation theory, unitarity being the most obvious. An example of this subtlety appears in the theory which has a simple current extension which fails to be associative when the adjoint is chosen [4]. The associative extension one obtains with the adjoint is, of course, the ghost system (see Section 10).
The commutation relations of the affine KacMoody algebra are therefore
(2.5) 
where is central. We will habitually replace by its common eigenvalue , the level, when acting upon the modules comprising each theory.^{1}^{1}1Technically, we should do this in the universal enveloping algebra by quotienting by the ideal generated by . Doing this at the level of the Lie algebra is a standard sloppiness which leads to no harm. With this replacement, the Sugawara construction gives the standard energymomentum tensor
(2.6) 
at least when . The modes of then generate a copy of the Virasoro algebra of central charge
(2.7) 
Here, we take the opportunity to introduce the notation .
The triangular decomposition that we have chosen for lifts, in the standard manner, to one for . The notions of highest weight states and Verma modules are then available. An easy consequence of (2.6) is that a highest weight state of weight (eigenvalue) will have conformal dimension (eigenvalue)
(2.8) 
We will denote the Verma module generated by a highest weight state of weight by . The irreducible quotient of will be denoted by if , and by otherwise. The notation here is chosen to reflect the nature of the zerograde subspace (the states of minimal conformal dimension) of the irreducible as an module. When , this subspace forms a finitedimensional irreducible module, whereas it forms an infinitedimensional irreducible of the discrete series type otherwise. We will refer to as the vacuum module and its highest weight state as the vacuum in what follows.
The subgroup of automorphisms of which leave the span of the zeromodes , and invariant is isomorphic to . We take the order two generator to be the conjugation automorphism which is the Weyl reflection corresponding to the finite simple root. The infinite order generator is the spectral flow automorphism which may be regarded as a square root of the affine Weyl translation by the (finite) simple coroot (in fact, is translation by the dual of the finite simple root). These automorphisms fix , hence the level is preserved, and otherwise act as follows:
(2.9) 
The normality of the subgroup generated by follows from .
One important use for these automorphisms is to modify the action of on any module , thereby obtaining new modules and . The first is precisely the module conjugate to — its weights are the negatives of the weights of , though the conformal dimensions remain unchanged. The second is called the spectral flow image of — its weights have been shifted by a fixed amount, but its conformal dimensions also change. Explicitly, the modified algebra action defining these new modules is given by
(2.10) 
It is easy to check that if is a state of weight and conformal dimension , then the state satisfies
(2.11) 
In what follows, we will usually omit the superscript “” which distinguishes the induced spectral flow maps between modules from the spectral flow algebra automorphisms. Which is meant should be clear from the context.
3. Admissible Levels and Modules
Recall that when the level is a nonnegative integer, the (chiral) spectrum of any conformal field theory with symmetry and an irreducible vacuum module may only contain the irreducible modules with . This is the spectrum of the WessZuminoWitten model on . The reason boils down to the following fact: Let denote the highest weight state of the vacuum Verma module . As , possesses a nontrivial singular vector , meaning that it is not descended from the trivial singular vector , which has to be set to zero in order to form the irreducible vacuum module . Setting this singular vector to zero is only consistent with the statefield correspondence of conformal field theory if the spectrum is restricted as above. This seems to have been first explained in [24], though the argument has since been modified and made rigorous within the formalism of vertex algebras by Zhu [25].
It is natural to ask if there are other levels at which an irreducible vacuum module similarly constrains the spectrum. To have such constraints, one needs to know when the corresponding vacuum Verma module has a nontrivial singular vector. This question may be answered using the KacKazhdan formula [26] for the determinant of the Shapovalov form in each (affine) weight space. The result is that such a nontrivial singular vector exists precisely when
(3.1) 
Moreover, the singular vector will have weight and conformal dimension . Levels satisfying the above conditions are called admissible. Equivalently, is said to be admissible if the universal vertex algebra corresponding to is not simple.
Determining the constraints that this singular vector imposes on the spectrum is not quite as easy. One has a semiexplicit formula for the singular vector due to Malikov, Feigin and Fuchs [27]. However, this formula involves rational powers of the affine modes which must be massaged using analytic continuations of the commutation rules in order to arrive at an explicit expression (see [28, 29, 30] for concrete examples of such massaging).
Example (see [7]).
The level has , hence and . This level is therefore admissible. KacKazhdan tells us that the nontrivial singular vector in the vacuum Verma module has weight and conformal dimension . It is given, in the MalikovFeiginFuchs form, by
(3.2) 
For deriving constraints, it is in fact more convenient to consider the descendant whose weight is . This state will also be set to in the irreducible vacuum module. Massaging the above expression appropriately leads to the (renormalised) explicit form
(3.3) 
The field , and so its zeromode , must therefore act as on the spectrum. But, applying to a highest weight state of weight gives
(3.4) 
hence we conclude that the only highest weight states allowed are those with weights , and . It follows that the only highest weight modules in the spectrum are the irreducibles , and .
Example (see [9, 4]).
For , we have and , so this level is also admissible. The nontrivial singular vector has weight and conformal dimension , and its zeroweight descendant takes the form
(3.5) 
The zeromode of the field then acts on a highest weight state as
(3.6) 
so the allowed highest weight modules are the irreducibles , , and .
As these examples show, unpacking the MalikovFeiginFuchs formula for the nontrivial vacuum singular vector is extremely cumbersome. It is therefore rather remarkable that the constraints upon the spectrum have been worked out for arbitrary admissible levels. This result is due to Adamović and Milas [19] who determined Zhu’s algebra using an explicit formula of Fuchs [31] for a projection of the nontrivial singular vector onto the universal enveloping algebra of . Modules which are allowed in the spectrum of an admissible level theory are also said to be admissible.^{2}^{2}2The original definition of admissibility is that of Kac and Wakimoto [32] who defined admissible weights in order to derive a generalisation of the WeylKac character formula for integrable modules. Their admissible weights are precisely the highest weight modules of the admissible modules, as they have been defined here. We just prefer to arrive at the definition from consideration of the vertex algebra. The spectrum of admissible highest weight modules is as follows:
Theorem 1 (Adamović–Milas).
Let be an admissible level and let
(3.7) 
The admissible highest weight modules are then exhausted by the following irreducibles:

, for ,

, for and .
Mathematically, admissibility just means that the highest weight module is a module for the (simple) vertex algebra associated with at the admissible level . It is convenient to extend this definition of admissibility beyond the highest weight category — from now on, any vertex algebra module will be termed admissible. Note that when , so , the set of type modules is empty and the admissible highest weight modules are precisely the with .
It is convenient to collect the admissible highest weights into a table, analogous to the Kac table which gives the allowed conformal dimensions for the highest weight states of a Virasoro minimal model. We present some of these tables, both for admissible highest weights and their conformal dimensions
(3.8) 
in Figure 1. We note that, if one ignores the leftmost column () which describes the type admissibles, then these tables have symmetries similar to Kac tables. In particular, we have
(3.9) 
This similarity between the table of type admissibles and the Kac table for the minimal model is more than just analogy. In particular, note that if we take to define a Virasoro central charge and Virasoro conformal dimensions by
(3.10) 
then one finds that
(3.11) 
This relation is the key upon which a large proportion of the following analysis rests.