Vertex operator algebras, the Verlinde conjecture, and modular tensor categories

Abstract

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V_(n) = 0 for n < 0, , and the contragredient module V' is isomorphic to V as a V-module; (ii) every weak V-module is completely reducible; (iii) V is C₂-cofinite. We announce a proof of the Verlinde conjecture for V, that is, of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation τ → –1/τ on the space of characters of irreducible V-modules. We discuss some consequences of the Verlinde conjecture, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of τ → –1/τ, and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of V-modules when V satisfies in addition the condition that irreducible V-modules not equivalent to V have no nonzero elements of weight 0. In particular, the category of V-modules has a natural structure of modular tensor category.

In 1987, by comparing fusion algebras with certain algebras obtained in the study of conformal field theories on genus-one Riemann surfaces, Verlinde (1) conjectured that the matrices formed by the fusion rules are diagonalized by the matrix given by the action of the modular transformation τ → –1/τ on the space of characters of a rational conformal field theory. From this conjecture, Verlinde obtained the famous Verlinde formulas for the fusion rules and, more generally, for the dimensions of the spaces of conformal blocks on Riemann surfaces of arbitrary genera. In the particular case of the conformal field theories associated to affine Lie algebras (the Wess–Zumino–Novikov–Witten models), the Verlinde formulas give a surprising formula for the dimensions of the spaces of sections of the “generalized theta divisors,” which has given rise to a great deal of excitement and new mathematics. See the works by Tsuchiya et al. (2), Beauville and Laszlo (3), Faltings (4), and Kumar et al. (5) for details and proofs of this particular case of the Verlinde formulas.

In 1988, Moore and Seiberg (6) showed on a physical level of rigor that the Verlinde conjecture is a consequence of the axioms for rational conformal field theories. This result of Moore and Seiberg is based on certain polynomial equations that they derived from the axioms for rational conformal field theories (6, 7). Moore and Seiberg further demonstrated that these polynomial equations are actually conformal-field-theoretic analogues of the tensor category theory for group representations. Their work greatly advanced our understanding of the structure of conformal field theories. In particular, the notion of modular tensor category was later introduced to summarize the properties of the Moore–Seiberg polynomial equations and has played a central role in the development of conformal field theories and 3D topological field theories. See, for example, refs. 8 and 9 for the theory of modular tensor categories, their applications, and references to many important works done by mathematicians and physicists.

The work of Moore and Seiberg gave a conceptual understanding of the Verlinde conjecture and the modular tensor categories arising in conformal field theories. However, it is a very hard problem to mathematically construct theories satisfying the axioms for rational conformal field theories. In fact, these axioms for rational conformal field theories are much stronger than the Verlinde conjecture and the modular tensor category structures. In the general theory of vertex operator algebras, introduced and studied first by Borcherds (10) and Frenkel et al. (11), a mathematical version of the notion of fusion rule was introduced and studied by Frenkel et al. (12) using intertwining operators, and the modular transformations were given by Zhu's modular invariance theorem (13). By using these notions and some natural conditions, including in particular Zhu's C₂-cofiniteness condition, one can formulate a general version of the Verlinde conjecture in the framework of the theory of vertex operator algebras. Additional results on intertwining operators and modular invariance were obtained by Huang and Lepowsky (14–17), Huang (18–20), Dong et al. (21), and Miyamoto (29). However, these results were still not enough for the proof of this general version of the Verlinde conjecture. The main obstructions were the duality and modular invariance properties for genus-zero and genus-one multipoint correlation functions constructed from intertwining operators for a vertex operator algebra satisfying the conditions mentioned above. These properties were proven recently (22, 23).

In this article, we announce a proof of the general version of the Verlinde conjecture above. Our theorem assumes only that the vertex operator algebra that we consider satisfies certain natural grading, finiteness, and reductivity properties (see Section 2). We also discuss some consequences of our theorem, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of τ → –1/τ, and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of modules for such a vertex operator algebra constructed by Huang and Lepowsky (14–18, 22) when V satisfies in addition the condition that irreducible modules not equivalent to the algebra (as a module) have no nonzero elements of weight 0. In particular, the category of modules for such a vertex operator algebra has a natural structure of modular tensor category.

This article is organized as follows: In section 1 we give the definitions of fusion rule, the fusing and braiding isomorphisms in terms of matrix elements, and the corresponding action of the modular transformation. These are the basic ingredients needed in the formulations of the main results given in sections 2 and 3, and they are in fact based on substantial mathematical results in refs. 18–20 and 13 and 20, respectively. Our main theorems on the Verlinde conjecture, on the Verlinde formula for the fusion rules, on the formula for the matrix given by the action of τ → –1/τ, and on the symmetry of this matrix are stated in Section 2. A very brief sketch of the proof of the Verlinde conjecture is given in this section. In section 3 our main theorem on the modular tensor category structure is stated, and a sketch of the proof is given.

1. Fusion Rules, Fusing and Braiding Isomorphisms, and Modular Transformations

We assume that the reader is familiar with the basic definitions and results in the theory of vertex operator algebras as introduced and presented in refs. 10 and 11. We shall use the notations, terminology, and formulations in refs. 11, 12, and 24.

Let V be a simple vertex operator algebra, V' the contragredient module of V, and C₂(V) the subspace of V spanned by u_–2v for u, v ∈ V. In this article, we shall always assume that V satisfies the following conditions:

1. V_(n) = 0 for n < 0, , and V' is isomorphic to V as a V-module.

2. Every weak V-module is completely reducible.

3. V is C₂-cofinite, that is, dim V/C₂(V) < ∞.

We recall that an weak V-module is a vector space that admits an , equipped with a vertex operator map

satisfying all axioms for V-modules except that the condition L(0)w = nw for w ∈ W_(n) is replaced by u_kw ∈ W_{[m–k–1 + n]} for u ∈ V_(m) and w ∈ W_[n]. Condition 2 is equivalent to the statement that every finitely generated weak V-module is a V-module, and every V-module is completely reducible.

From ref. 21 we know that there are only finitely many inequivalent irreducible V-modules. Let 𝒜 be the set of equivalence classes of irreducible V-modules. We denote the equivalence class containing V by e. For each , we choose a representative W^a of a. Note that the contragredient module of an irreducible module is also irreducible (see ref. 12). Thus, we have a map

From refs. 21 and 25 we know that irreducible V-modules are in fact graded by rational numbers. Thus, for , there exist such that .

Let for be the space of intertwining operators of type and for the fusion rule, that is, the dimension of the space of intertwining operators of type . For any , we know from ref. 12 that for w_a1 ∈ W^a₁ and w_a2 ∈ W^a₂

[1.1]

where

From refs. 23 and 26–28, we also know that the fusion rules for are all finite. For , let be the matrix whose entries are for , that is,

We also need matrix elements of fusing and braiding isomorphisms. In the proof of the Verlinde conjecture, we need to use several bases of one space of intertwining operators. We shall use p = 1, 2, 3, 4, 5, 6,... to label different bases. For p = 1, 2, 3, 4, 5, 6,... and , let be a basis of . For , w_a1 ∈ W^a₁, w_a2 ∈ W^a₂, w_a3 ∈ W^a₃, and , using the differential equations satisfied by the series

and

it was proved in ref. 23 that these series are convergent in the regions |z₁| > |z₂| > 0 and |z₂| > |z₁ – z₂| > 0, respectively. Note that for any , , and are a bases of and respectively. The associativity of intertwining operators proved and studied in refs. 18, 20, and 22 says that there exist

for , , , , , such that

[1.2]

when |z₁| > |z₂| > |z₁ – z₂| > 0, for , w_a1 ∈ W^a₁, w_a2 ∈ W^a₂, w_a3 ∈ W^a₃, , , and . The numbers

together give a matrix that represents a linear isomorphism

called the fusing isomorphism, such that these numbers are the matrix elements.

By the commutativity of intertwining operators proved and studied in refs. 19, 20, and 22, for any fixed , there exist

for , , , , , such that the analytic extension of the single-valued analytic function

on the region |z₁| > |z₂| > 0, 0 ≤ arg z₁, arg z₂ < 2π along the path

to the region |z₂| > |z₁| > 0, 0 ≤ arg z₁, arg z₂ < 2π is

The numbers

together give a linear isomorphism,

called the braiding isomorphism, such that these numbers are the matrix elements.

We need an action of S₃ on the space

For , , consider the isomorphisms and given in equations 7.1 and 7.13 in ref. 15. For , , we define

We have the following:

Proposition. The actions σ₁₂ and σ₂₃ of (ref. 12) and (ref. 22) on 𝒱 generate a left action of S₃ on 𝒱.

We now choose a basis , , of for each triple . For , we choose to be the vertex operator Y_Wa defining the module structure on W^a, and we choose to be the intertwining operator defined using the action of σ₁₂,

for u ∈ V and w_a ∈ W^a. Because V' as a V-module is isomorphic to V, we have e' = e. From ref. 12, we know that there is a nondegenerate invariant bilinear form (·,·) on V such that (1, 1) = 1. We choose to be the intertwining operator defined using the action of σ₂₃ by

that is,

for u ∈ V, w_a ∈ W^a and . Because the actions of σ₁₂ and σ₂₃ generate the action of S₃ on 𝒱, we have

for any . When a₁, a₂, a₃ ≠ e, we choose , to be an arbitrary basis of . Note that for each element σ ∈ S₃, is also a basis of .

We now discuss modular transformations. Let q_τ = e^2πiτ for . We consider the q_τ traces of the vertex operators Y_W^a for on the irreducible V-modules W^a of the following form,

[1.3]

for u ∈ V. In ref. 13, under some conditions slightly different from (mostly stronger than) those we assume in this article, Zhu proved that these q traces are independent of z, are absolutely convergent when 0 < |q_τ| < 1, and can be analytically extended to analytic functions of τ in the upper-half plane. We shall denote the analytic extension of Eq. 1.3 by

In ref. 13, under conditions alluded to above, Zhu also proved the following modular invariance property: For

let τ' = aτ + b/cτ + d. Then there exist unique for a₁, such that

for u ∈ V. Dong et al. (21), among many other things, improved Zhu's results by showing that they also hold for vertex operator algebras satisfying the conditions (slightly weaker than what) we assume in this article. In particular, for

there exist unique for such that

for u ∈ V. When u = 1, we see that the matrix actually acts on the space spanned by the vacuum characters .

2. Verlinde Conjecture and Consequences

I (30) proved the following general version of the Verlinde conjecture in the framework of vertex operator algebras (compare with section 3 in ref. 1 and section 4 in ref. 6).

Theorem 2.1. Let V be a vertex operator algebra satisfying the following conditions:

1. V_(n) = 0 for n < 0, , and V' is isomorphic to V as a V-module.

2. Every weak V-module is completely reducible.

3. V is C₂-cofinite, that is, dim V/C₂(V) < ∞.

Then for ,

and

[2.1]

where are matrix elements of the square of the braiding isomorphism. In particular, the matrix S diagonalizes the matrices for all .

Sketch of the proof: Moore and Seiberg (6) showed that the conclusions of the theorem follow from the following formulas (which they derived by assuming the axioms of rational conformal field theories): For ,

[2.2]

and

[2.3]

Thus, the main work is to prove these two formulas. The proofs of these formulas are based in turn on the proofs of a number of other formulas and on nontrivial applications of a number of results in the theory of vertex operator algebras; thus, here we can only outline what is used in the proofs.

The proof of the first formula (Eq. 2.2) uses mainly the works of Huang and Lepowsky (14–17) and Huang (refs. 18–20 and 22) on the tensor product theory, intertwining operator algebras, and the construction of genus-zero chiral conformal field theories. The main technical result used is the associativity for intertwining operators proved in refs. 18 and 22 for vertex operator algebras satisfying the three conditions stated in Theorem 2.1. Using the associativity for intertwining operators repeatedly to express the correlation functions obtained from products of three suitable intertwining operators as linear combinations of the correlation functions obtained from iterates of three intertwining operators in two ways, we obtain a formula for the matrix elements of the fusing isomorphisms. Then, using certain properties of the matrix elements of the fusing isomorphisms and their inverses, we obtain the first formula (Eq. 2.2).

The proof of the second formula (Eq. 2.3) heavily uses the results obtained in ref. 23 on the convergence and analytic extensions of the q_τ traces of products of what we call “geometrically modified intertwining operators,” the genus-one associativity, and the modular invariance of these analytic extensions of the q_τ traces, where q_τ = e^2πiτ. These results allow us to (rigorously) establish a formula that corresponds to the fact that the modular transformation τ → –1/τ changes one basic Dehn twist on the Teichmüller space of genus-one Riemann surfaces to the other. Calculating the matrices corresponding to the Dehn twists and substituting the results into this formula, we obtain Eq. 2.3.

As in ref. 6, the conclusions of Theorem 2.1 follow immediately from Eqs. 2.2 and 2.3.

Remark. Note that finitely generated weak V-modules are what naturally appear in the proofs of the theorems on genus-zero and genus-one correlation functions. Thus, condition 2 is natural and necessary because the Verlinde conjecture concerns V-modules, not finitely generated weak V-modules. Condition 3 would be a consequence of the finiteness of the dimensions of genus-one conformal blocks if the conformal field theory had been constructed and is thus natural and necessary. For vertex operator algebras associated with affine Lie algebras (Wess–Zumino–Novikov–Witten models) and vertex operator algebras associated with the Virasoro algebra (minimal models), condition 2 can be verified easily by reformulating the corresponding complete reducibility results in the representation theory of affine Lie algebras and the Virasoro algebra. For these vertex operator algebras, condition 3 can also be verified easily by using results in the representation theory of affine Lie algebras and the Virasoro algebra. In fact, condition 3 was stated to hold for these algebras by Zhu (13) and verified by Dong et al. (21) (see also ref. 28 for the case of minimal models).

Using the fact that for , we can easily derive the following formulas from Theorem 2.1 (compare with section 3 in ref. 1).

Theorem 2.2. Let V be a vertex operator algebra satisfying the conditions in Section 1. Then we have for and

[2.4]

Theorem 2.3.

[2.5]

Using Eq. 2.5 and certain properties of the matrix elements of the fusing and braiding isomorphisms, we can prove the following theorem.

Theorem 2.4. The matrix () is symmetric.

3. Rigidity, Nondegeneracy Property, and Modular Tensor Categories

A tensor category with tensor product bifunctor ⊠ and unit object V is rigid if for every object W in the category there are right and left dual objects W* and *W together with morphisms e_W: W* ⊠ W → V, i_W: V → W ⊠ W*, e'_W: W ⊠ *W → V, and i'_W: V → *W ⊠ W such that the compositions of the morphisms in the sequence

and three similar sequences are equal to the identity I_W on W. Rigidity is a standard notion in the theory of tensor categories. A rigid braided tensor category together with a twist (a natural isomorphism from the category to itself) satisfying natural conditions (see refs. 8 and 9 for the precise conditions) is called a ribbon category. A semisimple ribbon category with finitely many inequivalent irreducible objects is a modular tensor category if it has the following nondegeneracy property: The m χ m matrix formed by the traces of the morphism c_Wiw_j ○ c_Ww_i in the ribbon category for the irreducible modules W₁,..., W^j_m is invertible. The term “modular tensor category” was first suggested by Frenkel to summarize the Moore–Seiberg theory of polynomial equations. See refs. 8 and 9 for details of the theory of modular tensor categories.

The results in the proceeding section give the following:

Theorem 3.1. Let V be a simple vertex operator algebra satisfying the following conditions:

1. V_(n) = 0 for n < 0, , and W₍₀₎ = 0 for any irreducible V-module W which is not equivalent to V.

2. Every weak V-module is completely reducible.

3. V is C₂-cofinite, that is, dim V/C₂(V) < ∞.

Then the braided tensor category structure on the category of V-modules constructed in refs. 14–18 and 22 is rigid, has a natural structure of ribbon category, and has the nondegeneracy property. In particular, the category of V-modules has a natural structure of modular tensor category.

Sketch of the proof: Note that condition 1 implies that V' is equivalent to V as a V-module. Thus, condition 1 is stronger than condition 1 in the preceding section. In particular, we can use all the results in the proceeding section. This slightly stronger condition 1 is needed in the proof of the rigidity and nondegeneracy property.

We take both the left and right dual of a V-module W to be the contragredient module W' of W. Because our tensor category is semisimple, to prove the rigidity we need only discuss irreducible modules. For any V-module , we use to denote its algebraic completion . For , using the universal property (see definition 3.1 in ref. 16 and definition 12.1 in ref. 17) for the tensor product module (W^a)' ⊠ W^a, we know that there exists a unique module map ê_a: (W^a)' ⊠ W^a → V such that

for w_a ∈ W^a and , where is the tensor product of w₁ and is the natural extension of to . Similarly, we have a module map from W^a ⊠ (W^a)' to V. Because W^a ⊠ (W^a)' is completely reducible and the fusion rule is 1, there is a V-submodule of W^a ⊠ (W^a)' that is isomorphic to V under the module map from W^a ⊠ (W^a)' to V. Thus, we obtain a module map i_a: V → W^a ⊠ (W^a)' that maps V isomorphically to this submodule of W^a ⊠ (W^a)'. Now,

[3.1]

is a module map from an irreducible module to itself, so it must be the identity map multiplied by a number. One can calculate this number explicitly, and it is equal to

From Theorem 2.1, this number is not 0. Let

Then the map obtained from Eq. 3.1 by replacing ê_a by e_a is the identity. Similarly, we can prove that all the other maps in the definition of rigidity are also equal to the identity. Thus, the tensor category is rigid.

For any , we define the twist on W^a to be e^2πiha. Then, it is easy to verify that the rigid braided tensor category with this twist is a ribbon category.

To prove the nondegeneracy property, we use Eq. 2.5. Now it is easy to calculate in the tensor category the trace of c_W^a2, _W^a1○ c_W^a1, _W^a2 for , where c_W^a1, _W^a2: W^a₁ ⊠ W^a₂ → W^a₂ ⊠ W^a₁ is the braiding isomorphism. The result is

By Eq. 2.5, this is equal to

and these numbers form an invertible matrix. The other data and axioms for modular tensor categories can be given or proved trivially. Thus, the tensor category is modular.