Modularity of Vertex Algebras, Deformation of Conformal Structure, and Fusion Rules of Regular Affine Algebras
Modularity of Relatively Rational Vertex Algebras and Fusion Rules of Regular Affine Algebras
Tomoyuki Arakawa^{1}, Jethro van Ekeren^{*}^{*}*email: ^{2}
^{1}Research Institute for Mathematical Sciences, Kyoto, Japan/MIT, Cambridge, USA
^{2}Instituto de Matemática e Estatística (GMA), UFF, Niterói RJ, Brazil
Abstract. We study modularity of the characters of a vertex (super)algebra equipped with a family of conformal structures. Along the way we introduce the notion of rationality and cofiniteness relative to such a family. We apply the results to determine modular transformations of trace functions on admissible modules over affine KacMoody algebras and, via BRST reduction, trace functions on regular affine algebras.
1. Introduction
A striking feature of the representation theory of infinite dimensional Lie algebras and vertex algebras is the appearance of modular functions as normalised graded dimensions of integrable modules. The phenomenon of modularity is in turn the source of important technical tools in the representation theory of these algebras.
Let be a conformal vertex algebra of central charge , assumed to be rational and cofinite. In [Z96] Zhu showed that the normalised graded dimensions of irreducible positive energy modules, and more generally the trace functions
of on such modules (viewed as functions of the upper half complex plane), are jointly invariant. In particular
(1.1) 
where is the neutral Zhu mode (see (2.6)), and the sum here is over the set of irreducible positive energy modules.
The celebrated Verlinde formula [Verlinde88] (proved as a theorem of vertex algebras by Huang [Huang08Contemp]) determines the decomposition multiplicities of the fusion product between modules in terms of the matrix .
The most striking consequence of Zhu’s result is that normalised graded dimensions are modular functions (this fact is the specialisation of (1.1) to ). The insertion of arbitrary is also important because, while the are known to be linearly independent, their restrictions need not be. Hence à priori the matrix only makes sense before specialisation.
In the first half of this paper we study modularity in the context of a vertex algebra together with an infinitesimal variation of its conformal structure . Let be a conformal vertex algebra, and let be a current (i.e., a vector of conformal weight ). It is well known that the formula
defines a family of conformal structures on , indexed by the parameter .
The vertex algebra is said to be rational if its category of positive energy modules is semisimple. Since the positive energy condition depends on a choice of conformal structure, so does the condition of rationality. One is thus presented with the possibility of a family of conformal structures as above for which is “generically rational”, i.e., rational for all in some neighbourhood of , but not necessarily at itself. In fact this situation occurs relatively frequently.
Indeed we may speak of the subcategory of the category of positive energy modules which retain the positive energy condition upon deformation of to for small . We call such modules “stable”, and we say that is rational relative to if its category of stable positive energy modules is semisimple.
In [Z96] Zhu introduced the Poisson algebra canonically associated with the vertex algebra . He also identified the condition as crucial for establishing his modularity theorem. This condition came to be known as cofiniteness. To formulate the correct analogue of cofiniteness in our context, we make the following general definition.
Definition 1.1.
Let be a vertex algebra, decomposed as a direct sum of its subalgebra and the module . The quotient
is a Poisson algebra. We say that is cofinite relative to the decomposition if
We remark that relative cofiniteness is implied by cofiniteness either of itself or of , but the converse is not true. The applications that most interest us involve relatively cofinite but non cofinite vertex algebras.
Let and be as above, and suppose that decomposes under the action of into the sum of the zero eigenspace and a complementary invariant subspace . We say that is cofinite relative to if it is cofinite relative to the decomposition .
The neutral mode of is . Hence trace functions on modules are naturally functions of alongside and .
Before stating our first main theorem, we record that the formula
(1.2) 
defines a representation of on the space of functions of (see Proposition 5.10). Here the endomorphism is the neutral Zhu mode that appeared in (1.1) already, and is Li’s shift operator. The definitions of these are recalled below in Definitions 2.10 and 2.13, respectively.
We prove the following result as Theorem 5.12. In fact we derive it from the stronger but more technical Proposition 5.11.
Theorem 1.2.
Let be a conformal vertex (super)algebra graded by integer conformal weights. Let be a current satisfying the OPE relation
and such that acts semisimply on . Assume to be rational relative to and cofinite relative to , and write for the set of irreducible stable positive energy modules. Let
be the supertrace function of on . Then

There exists such that converges on the domain

The relation
where is some representation of , is satisfied for all if it is satisfied for .
We make some remarks on the theorem and its proof. The essential idea of the proof is to apply Zhu’s modularity theorem to the vertex algebra . However equips with noninteger conformal weights, and Zhu’s theorem actually fails in this case. This situation is rectified in the reference [JVE13], where it is shown that modular transformations map the trace functions to trace functions on particular twisted modules. The task becomes to relate trace functions on twisted and untwisted modules. This is achieved by the use of Li’s shift operators (which appear explicitly in (1.2) above). The condition of relative cofiniteness is inspired by the work [DLMadmiss], and was used in [JVE13].
The transformation (1.2) was uncovered in the case of superconformal vertex algebras in [HVE14, Theorem 9.13 (b)], with equal to the current of the algebra. There the functions are shown to be flat sections of the bundle of conformal blocks over the universal elliptic curve, and (1.2) is derived from the geometry of this bundle.
We also note that a result closely related to Theorem 1.2 was recently and independently obtained in [Krauel.C2.case] in the case of rational and cofinite (see also [KM15]).
An important class of vertex algebras that are relatively cofinite and generically rational in the sense discussed above is afforded by the simple affine vertex algebras at admissible level.
Let be a finite dimensional simple Lie algebra over , and the corresponding affine KacMoody algebra. In [KWPNAS] Kac and Wakimoto identified the notion of admissible weight and initiated the study of the characters
of the irreducible modules of admissible highest weight .
We recall that is said to be an admissible number for if is an admissible weight. If is an admissible number then it is either principal or else coprincipal. Roughly speaking these cases distinguish whether the sub root system of is equivalent to that of or else to that of the Langlands dual of the affine algebra associated with , respectively (see Section 3 for precise definitions). We denote by (resp. ) the set of principal (resp. coprincipal) weights of level .
In [KW89] Kac and Wakimoto showed that if is a principal admissible number for and then
for some representation of . They also explicitly computed the matrix
In Section 4 we extend this result to the coprincipal case, and we compute the matrix explicitly.
Now let be the universal affine vertex algebra at admissible level , and its simple quotient. A smooth module is naturally a module. Consider the subcategory of the BGG category consisting of modules that descend to modules. It was proved in [A12catO] that this category is semisimple, i.e., that is rational in the category . Furthermore if is principal (resp. coprincipal) then the simple objects are precisely the irreducible modules for (resp. ).
For an admissible weight, we introduce the trace function
(1.3) 
of on . The KacWakimoto character is recovered from as the specialisation. As an application of Theorem 1.2 we prove the following.
Theorem 1.3.
Finally we apply Theorem 1.3 to solve a problem in the representation theory of affine algebras.
Recall that from the data of and as above, plus a choice of nilpotent element , the universal affine algebra is defined as the quantized DrinfeldSokolov reduction [FeiginFrenkel], [KRW03]. We focus on the case that be a regular nilpotent element and a principal admissible number, and we omit from the notation.
It was proved in [A12rational] and [A.assoc.var] that the simple quotient of is a rational and cofinite vertex algebra. Zhu’s theorem therefore asserts modularity for . The matrix of can be deduced from that of using the EulerPoincaré principle. With the matrix in hand one may use the Verlinde formula to compute the fusion rules of .
The fusion rules of , for simply laced, were worked out by Frenkel, Kac and Wakimoto in [FKW] by carrying out the calculation outlined above at the level of the characters , i.e., at the level of graded dimensions of modules. As noted above the graded dimensions are not linearly independent. However Theorem 1.3 upgrades the result to an identity between trace functions of arbitrary , and the result of [FKW] is confirmed.
Acknowledgements The first author is partially supported by JSPS KAKENHI Grants (#25287004 and #26610006). The second author was supported by an Alexander von Humboldt Foundation grant and later by CAPESBrazil. The second author would like to thank Victor G. Kac for several ideas which go back to discussions had with him in 2011. The work has been presented at conferences “Lie and Jordan Algebras IV”, Bento Gonçalves, Brazil, December 2015, “Quántum 2016”, Córdoba, Argentina, February 2016, and “Vertex Algebras and Quantum Groups” Banff, Canada, March 2016. The authors would like to thank the organisers of these conferences.
Notation Implicitly tensor products are taken over the ground field of complex numbers. The domain of the complex variable is the upper half complex plane, denoted , and . The letter is used for the central charge, and in the matrix . We trust that no confusion will arise.
2. Preliminaries on Vertex Algebras
2.1. Vertex Algebras
For background on vertex algebras we refer the reader to the book [KacVA]. Note that ‘vertex algebra’ implicitly includes the super case.
Definition 2.1.
A vertex algebra consists of a vector superspace with a distinguished vacuum vector and a vertex operation, which is an even linear map , written , such that the following are satisfied:

(Unit axioms) and for all .

(Borcherds identity)
(2.1) for all , .
The operator is called the translation operator and it satisfies . The operators are called modes.
A useful special case of Borcherds identity is
(2.2) 
or, in the more compact bracket notation,
Definition 2.2.
A conformal structure on the vertex algebra is a vector such that furnishes with an action of the Virasoro algebra, i.e.,
for some constant . This action is required to satisfy , and that act semisimply on with non negative rational eigenvalues, bounded below. The constant is called the central charge of . A conformal vertex algebra is a vertex algebra together with a choice of conformal structure.
After fixing a conformal structure on a vertex algebra , we call the eigenvalue of a vector its conformal weight, which we denote , and we denote by the subspace of vectors with conformal weight . The conformal indexing of modes (relative to ) is defined by
In terms of the conformal indexing Borcherds identity becomes
(2.3) 
Definition 2.3.
Let be a vertex algebra. A (weak) module is a vector superspace together with an even map , written , such that , and (2.1) holds for all , , and for all . Now let be a conformal vertex algebra. A positive energy module is a weak module with grading by finite dimensional eigenspaces , with eigenvalues bounded below.
An automorphism of the vertex algebra is such that . An automorphism of a conformal vertex algebra is one that fixes .
Definition 2.4.
Let be an automorphism of the vertex algebra of finite order , and write for its eigenspace (so is defined modulo ). A (weak) twisted module is a vector superspace together with an even map , written for , such that , and (2.3) holds for all , and , and for all , and .
Remark 2.5.
For integer graded our definition coincides with that used in [DLM98] and in [Li95]. In [DLM00] a different convention is used which exchanges the notions of  and twisted modules. The operator to be defined in equation 2.4 below is the inverse of used in [DLM00, Equation (8.1)]. Note that in the present setting a vertex algebra is an twisted module.
Definition 2.6.
A vertex algebra is said to be rational if it has finitely many irreducible positive energy modules, and every positive energy module decomposes into a direct sum of irreducible positive energy modules with finite dimensional graded pieces.
2.2. Relative Cofiniteness
We introduce a notion which we call relative cofiniteness, generalising the well known cofiniteness condition of Zhu [Z96].
Definition 2.7.
Let be a vertex algebra extension of by its module . Put
Then we say is cofinite relative to the decomposition if .
Note that , so is naturally a quotient of . The case recovers cofiniteness of . On the other hand if is cofinite then it is cofinite relative to any decomposition .
In this paper we mainly use splittings of the following form: is the fixed point subalgebra of with respect to a finite order automorphism , and is the sum of the remaining eigenspaces.
Lemma 2.8.
Let and be vertex algebras carrying automorphisms of equal order, with and the corresponding splittings. If and are relatively cofinite, then so is the tensor product with its natural vertex algebra structure and splitting induced by the product automorphism.
Proof.
Recall the tensor product vertex algebra structure is . We have
Hence is a quotient of which is finite dimensional. ∎
Lemma 2.9.
The quotient is naturally a Poisson algebra with commutative product and Poisson bracket .
Proof.
In [Z96, Section 4.4] Zhu proved the case, i.e., that the quotient is a Poisson algebra with product and bracket as indicated.
In the general case is the quotient of by the image of , so it suffices to show that the latter subspace is a Poisson ideal. Let and , then on the one hand
and on the other, putting , in (2.1),
∎
2.3. Trace Functions and Modular Invariance
Let be a conformal vertex algebra, and let be an irreducible positive energy module graded by finite dimensional eigenspaces for . We define the supertrace function of on to be
wherever the right hand side converges.
More generally let be commuting finite order automorphisms of , and let be an irreducible twisted module. The “twisted” action
of on defines a new structure of twisted module, which we denote . If then we say is invariant, and we are then able to choose an equivalence of twisted modules. In other words
(2.4) 
We define the twisted supertrace function of on the invariant twisted positive energy module to be
wherever the right hand side converges.
In order to describe modular invariance of supertrace functions we must recall the definition of Zhu’s modes.
Definition 2.10.
Let be a conformal vertex algebra (with rational conformal weights), and let . Then
We also write .
Explicitly
(2.5) 
and
(2.6) 
If has integer conformal weights, then is again a conformal vertex algebra. Indeed the two conformal vertex algebra structures are seen to be isomorphic because of Huang’s change of coordinate formula, which we now recall. With we associate the linear endomorphism of defined by
For the series is defined by . Huang [HuangConfBook] proved the formula
which is basic to the geometric approach to vertex algebras explained in [FBZ04]. If we take as in Definition 2.10 and put then
So indeed is isomorphic to via . One easily checks .
We now recall the main theorem of [JVE13], which is a generalisation of Dong, Li, and Mason’s [DLM00, Theorem 1.3] to the case of vertex (super)algebras graded by rational conformal weights.
Theorem 2.11 ([Jve13, Theorem 1.3 and Remark 5.2]).
Let be a graded conformal vertex algebra and let be a cyclic group of automorphisms of of finite order .

Let denote the invariant subalgebra of , and the direct sum of the nontrivial eigenspaces of . Suppose is relatively cofinite. Let and let be a invariant irreducible positive energy twisted module. Then the series defining converges absolutely to a holomorphic function of .

Suppose further that is rational for each . For with , let denote the (finite) set of irreducible invariant twisted modules, and let denote the vector space spanned by as ranges over . If where , then under the action
the vector space is mapped isomorphically to .
Remark 2.12.
If is graded (as will be the case in this article), then the action of on twisted supertrace functions is a representation.
2.4. Li’s Operators
Let be a graded conformal vertex algebra. Let be an even vector satisfying the Heisenberg bracket relation
(2.7) 
(the value of the constant is unimportant at the moment). Suppose further that acts semisimply on with eigenvalues in a lattice, i.e., a discrete subset, .
We now recall the operator series used by Li to ‘shift’ between differently twisted modules.
Definition 2.13.
For as above, let
(2.8) 
This expression makes sense on untwisted modules, and more generally on twisted modules whenever . The shifted module of a module is defined to be as a vector space, equipped with the vertex operation
The following theorem is due to Li (under conditions weaker than (2.7) actually).
Theorem 2.14 ([Li95], Proposition 5.4).
Let be a finite order automorphism of , let be as above, and let be a twisted module. Then is a twisted module.
For later use we recall some special cases of the action of shift operators. Suppose satisfy . Then we have
(2.9)  
(2.10) 
Let us write
One easily verifies that
(2.11) 
whenever . Hence
(2.12)  
(2.13) 
Lemma 2.15.
Let be as above, and let be an twisted module. Then

The module may be written as for some untwisted module ,

Let considered as an automorphism of , and let considered as an autmorphism of via the identification above. Then (2.4) is satisfied. In particular is a invariant module.
Proof.
Part (1) is an immediate consequence of Theorem 2.14. Part (2) is a simple computation. Indeed commutes with , so we have
Thus provides the intertwining map that we need. ∎
3. Preliminaries on Lie Algebras
3.1. Lie Algebras and Affine Vertex Algebras
Let be a finite dimensional simple Lie algebra over of rank . We fix a Cartan subalgebra and a triangular decomposition , with Borel subalgebra . We then have the set of roots, and its subsets of positive roots, and of simple roots. We denote by the root lattice .
There is a unique up to scaling nondegenerate invariant bilinear form on , which induces a form on . The roots come in one or two norms, and the lacing number is the ratio between these norms. We denote by (resp. ) the set of long (resp. short) roots, and we normalise the form so that the long roots have norm . We denote by the corresponding identification .
We let denote the highest root with respect to the height function on . It is a long root. We similarly denote by the highest of the short roots.
The simple coroots are by definition . We denote by the coroot lattice . The coroots come in one or two norms, namely and . We denote by (resp. ) the set of long (resp. short) coroots. We let denote the highest coroot with respect to the height function on , and the highest of the short coroots. In fact and .
The weight lattice is the natural dual of . The fundamental weights , which form a basis of , are by definition dual to the simple coroots, similarly the fundamental coweights are dual to the simple roots . We put .
The marks and comarks () are defined by , and . The dual Coxeter number is . We have the relation . The Weyl vector is , and dual Weyl vector . Clearly