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

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 C ₂(V) the subspace of V spanned by u _–2 v 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 C ₂-cofinite, that is, dim V/C ₂(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_{a 1} ∈ W ^a₁ and w_{a 2} ∈ W ^a₂ 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 |z ₁| > |z ₂| > 0 and |z ₂| > |z ₁ – z ₂| > 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 when |z ₁| > |z ₂| > |z ₁ – z ₂| > 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 |z ₁| > |z ₂| > 0, 0 ≤ arg z ₁, arg z ₂ < 2π along the path to the region |z ₂| > |z ₁| > 0, 0 ≤ arg z ₁, arg z ₂ < 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 S ₃ 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 S ₃ 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 S ₃ on 𝒱, we have for any . When a ₁, a ₂, a ₃ ≠ e, we choose , to be an arbitrary basis of . Note that for each element σ ∈ S ₃, 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, 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 a ₁, 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 C ₂ -cofinite, that is, dim V/C ₂(V) < ∞.

Then for , and 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 , and 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

Theorem 2.3.

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 W ₁,..., 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 C ₂-cofinite, that is, dim V/C ₂(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 w ₁ 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, 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.