Vertex operator algebras and the Verlinde conjecture

Yi-Zhi Huang

Abstract

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V_{(n)} = 0 for n < 0, V_{(0)} = ℂ1 and V′ is isomorphic to V as a V -module. (ii) Every ℕ-gradable weak V -module is completely reducible. (iii) V is C_{2}-cofinite. We prove that the Verlinde conjecture holds for V, that is, the matrices formed by 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. Using this result, we obtain the Verlinde formula for the fusion rules. We also prove that the matrix associated to the modular transformation τ ↦ −1/τ is symmetric.

0 Introduction

The Verlinde conjecture [V] in conformal field theory states that the action of the modular transformation τ ↦ −1/τ on the space of characters of a rational conformal field theory diagonalizes the fusion rules. Combined with axioms for higher-genus rational conformal field theories, the Verlinde conjecture lead to a Verlinde formula [V] for the dimensions of the spaces of conformal blocks on higher-genus Riemann surfaces. In the particular case of the conformal field theories associated to affine Lie algebras (the Wess-Zumino-Novikov-Witten models), this Verlinde formula gives a surprising formula for the dimensions of the spaces of sections of the ”generalized theta divisors” and has given rise to a great deal of excitement and new mathematics. See the works [TUY] by Tsuchiya-Ueno-Yamada, [BL] by Beauville-Laszlo, [F] by Faltings and [KNR] by Kumar-Narasimhan-Ramanathan for details and proofs of this particular case of the Verlinde formulas.

Moore and Seiberg showed in [MS1] on a physical level of rigor that this conjecture indeed follows from the axioms for rational conformal field theories (see [MS2] for more detailed discussions of axioms and assumptions for rational conformal field theories on a physical level of rigor). However, it is a very hard problem to construct theories satisfying these axioms mathematically and therefore the existence of rational conformal field theories is a very strong assumption. In this paper, using the results on the duality and modular invariance of genus-zero and genus-one correlation functions, especially those obtained recently in [H5] and [H6], we formulate and prove a general version of the Verlinde conjecture in the framework of the theory of vertex operator algebras, which were first introduced and studied in mathematics by Borcherds [B] and Frenkel-Lepowsky-Meurman [FLM]. Our theorem assumes only that the vertex operator algebra we consider satisfies certain natural grading, finiteness and reductivity properties (see below). This result is part of a program to construct mathematically conformal field theories in the sense of Kontsevich and Segal [S1] [S2] from representations of vertex operator algebras. It is a main step towards a mathematical proof of the Verlinde formula for the dimensions of conformal blocks on higher-genus Riemann surfaces for conformal field theories other than the Wess-Zumino-Novikov-Witten models.

Here we state our main result. Let V be a simple vertex operator algebra satisfying the following conditions: (i) V_{(n)} = 0 for n < 0, V_{(0)} = ℂ1 and V′ is isomorphic to V as a V -module. (ii) Every ℕ-gradable weak V -module is completely reducible. (iii) V is C_{2}-cofinite (that is, dim V/C_{2}(V) < ∞ where C_{2}(V) is the subspace of V spanned by u_{−2}v for u, v ∈ V. Let A be the set of equivalence classes of irreducible V -modules. For a ∈ A, we choose a representative W^{a} in a such that (W^{a})′ = W^{a′} and W^{e} = V where e is the equivalence class containing V. For a_{1}, a_{2}, a_{3} ∈ A, let N_{a_{1}a_{2}}^{a_{3}} = N_{W^{a_{1}}W^{a_{2}}}^{W^{a_{3}}} be the corresponding fusion rules (see [FHL]). For a ∈ A, let N (a) be the matrix whose entries are N_{aa_{1}}^{a_{2}} for a_{1}, a_{2} ∈ A, that is,

N (a) = (N_{aa_{1}}^{a_{2}}).

For a ∈ A, we have the characters or shifted graded dimensions Tr_{W^{a}}q_{τ}^{L(0)− \frac{c}{24}} where q_{τ} = e^{2πτ} and τ ∈ ℍ. In [Z], Zhu proved that these characters are a basis of a modular invariant vector space. In particular, there exist S_{a_{1}}^{a_{2}} ∈ ℂ for a_{1}, a_{2} ∈ A such that

Tr_{W^{a_{1}}}q_{− \frac{1}{τ}}^{L(0)− \frac{c}{24}} = ∑_{a_{2}∈A} S_{a_{1}}^{a_{2}}Tr_{W^{a_{2}}}q_{τ}^{L(0)− \frac{c}{24}} .

Then our main result says that the matrix (S_{a_{1}}^{a_{2}}) disgonalizes N (a) for a ∈ A. See Theorem 5.2 for a more complete and precise statement. Using this result, we obtain the Verlinde formula for the fusion rules for such a vertex operator algebra. We also prove that the matrix (S_{a_{1}}^{a_{2}}) is symmetric.

Note that fusion rules and modular transformations of characters were already defined mathematically for suitable vertex operator algebras in [FHL] by Frenkel-Huang-Lepowsky using intertwining operators and [Z] by Zhu using his modular invariance theorem, respectively. The C_{2}-cofiniteness condition was also introduced in [Z]. So the general version of the Verlinde conjecture proved in this paper could have been mathematically formulated at that time. Further results on intertwining operators and modular invariance were obtained in [HL1]–[HL4] by Huang-Lepowsky, in [H1], [H2] and [H4] by the author, in [DLM] by Dong-Li-Mason and in [M] by Miyamoto. But these results are still not enough for the proof of this general version of the Verlinde conjecture. The main obstructions are the duality and modular invariance properties for genus-zero and genus-one multi-point correlation functions constructed from intertwining operators for a vertex operator algebra satisfying the conditions above. These properties were proved recently in [H5] and [H6] by the author so that the proof of the Verlinde conjecture in the present paper becomes possible.

The main work of the present paper is to establish mathematically some formulas needed in the proof of the Verlinde conjecture. Most of the formulas were first obtained on a physical level of rigor by Moore and Seiberg [MS1]

[MS2] using axioms for rational conformal field theories. In the present paper, our formulations and proofs are based on the results obtained in the theory of vertex operator algebras. We use only those results which have been established mathematically and we do not assume that all axioms for rational conformal field theories hold. In [MS1] and [MS2], Moore and Seiberg discussed many of the subtle technical details using examples such as the minimal models. Many of these discussions cannot be generalized to general cases. For example, in [MS1] and [MS2], spaces of chiral vertex operators (intertwining operators) are identified with tensor products of spaces of lowest weight vectors in modules. This identification allows them to give an S_{3} action easily on the direct sum of these spaces and the formulas obtained in [MS1] and [MS2] depend heavily on this action. It is known that in general spaces of intertwining operators cannot be identified with tensor products of lowest weight vectors and thus, even if we assume that all axioms for rational conformal field theories hold, the method based on particular examples in [MS1] and [MS2] cannot be directly adopted to establish the formulas we need. In the present paper, we define this action of S_{3} using the skew-symmetry for intertwining operators and the contragredient intertwining operators in [FHL] and [HL2] and prove all the results and formulas needed. There are other examples similar to this one in the present paper. In this sense, even if we assume that all axioms for rational conformal field theories hold, the formulas and results stated in [MS1] and [MS2] are actually conjectures and in the present paper we give mathematical proofs.

We assume that the reader is familiar with the basic theory of vertex operator algebras as presented in [FLM], [FHL] and [LL]. We also assume that the reader has some basic knowledge in the theories of intertwining operators, tensor products, composition-invertible formal series and the Virasoro algebra, and the modular invariance, as developed in, for example, [DLM], [HL1]–[HL4], [H1]–[H6], [M] and [Z].

The present paper is organized as follows: In Section 1, we state our basic assumptions and we discuss intertwining operators and genus-zero correlation functions constructed from them. An action of S_{3} on the direct sum of spaces of intertwining operators among irreducible modules are also given in this section. In section 2, we discuss geometrically-modified intertwining operators and genus-one correlation functions constructed from these operators. The modular transformation associated to τ ↦ −1/τ is recalled in this section. Section 3 is devoted to the proof of three formulas for braiding and fusing matrices. Using all the results obtained in Sections 1, 2 and 3, we prove two formulas derived first by Moore and Seiberg from the axioms for rational conformal field theories in Section 4. Finally in Section 5, we prove the Verlinde conjecture, the Verlinde formula for fusion rules and that the matrix associated to the modular transformation τ ↦ −1/τ is symmetric.

Notations In this paper, i is either \sqrt{−1} or an index, and it should be easy to tell which is which. The symbols ℕ, ℤ_{+}, ℤ, ℚ, ℂ, ℂ^{×} and ℍ denote the nonnegative integers, positive integers, integers, rational numbers, complex numbers, nonzero complex numbers and the upper half plane, respectively. We shall use x, , y, ... to denote commuting formal variables and z, z_{1}, z_{2}, ... to denote complex numbers or complex variables.

Acknowledgment This research is supported in part by NSF grant DMS-0070800. I am grateful to L. Kong for reading the paper and for finding a number of typos.

1 Intertwining operators and genus-zero correlation functions

Let V be a simple vertex operator algebra and C_{2}(V) the subspace of V spanned by u_{−2}v for u, v ∈ V. In the present paper, we shall always assume that V satisfies the following conditions:

1. V_{(n)} = 0 for n < 0, V_{(0)} = ℂ1 and V′ is isomorphic to V as a V -module.

2. Every ℕ-gradable weak V -module is completely reducible.

3. V is C_{2}-cofinite, that is, dim V/C_{2}(V) < ∞.

Note that when V_{(n)} = 0 for n < 0, V_{(0)} = ℂ1, V′ is isomorphic to V as a V -module if for any irreducible V -module not isomorphic to V, W_{(0)} = 0. So the results of the present paper still hold if Condition 1 above is replaced by the condition V_{(n)} = 0 for n < 0, V_{(0)} = ℂ1, and W_{(0)} = 0 for any irreducible V -module not isomorphic to V.

From [DLM], 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. Note that the contragredient module of an irreducible module is also irreducible (see [FHL]). So we have a map

′ : A → A
a ↦ a′

and it is possible to choose a representative W^{a} of a for each a ∈ A such that W^{e} = V and (W^{a})′ = W^{a′}. We fix such a choice. From [AM] and [DLM], we know that irreducible V -modules are in fact graded by rational numbers. Thus for a ∈ A, there exist h_{a} ∈ ℚ such that W^{a} = ∐_{n∈h_{a}+ℕ} W_{(n)}.

Let V_{a_{1}a_{2}}^{a_{3}} for a_{1}, a_{2}, a_{3} ∈ A be the spaces of intertwining operators of type \binom{W^{a_{3}}}{W^{a_{1}}W^{a_{2}}}. For any Y ∈ V_{a_{1}a_{2}}^{a_{3}}, we know from [FHL] that for w_{a_{1}} ∈ W^{a_{1}} and w_{a_{2}} ∈ W^{a_{2}}

Y(w_{a_{1}}, x)w_{a_{2}} ∈ x^{Δ(Y)}W^{a_{3}}[[x, x^{−1}]],  (1.1)

where

Δ(Y) = h_{a_{3}} − h_{a_{1}} − h_{a_{2}}.

In the present paper, we shall use the following conventions: For z ∈ ℂ^{×}, log z = log |z| + i arg z, where 0 ≤ arg z < 2π. For z ∈ ℂ^{×} and r ∈ ℂ, z^{r} = e^{r log z} while for r, s ∈ ℂ, (e^{r})^{s} = e^{rs}. (Note that according to our convention here, (e^{r})^{s} is in general not equal to e^{s log e^{r}}.) Similarly, for z ∈ ℂ^{×} and O an linear operator, z^{O} = e^{(log z)O}, if e^{(log z)O} is well-defined, while for r ∈ ℂ and O an linear operator, (e^{r})^{O} = e^{rO}, if e^{rO} is well defined. Using this convention, we see that for a_{1}, a_{2}, a_{3} ∈ A, z ∈ ℂ^{×}, r ∈ ℂ, Y ∈ V_{a_{1}a_{2}}^{a_{3}}, w_{a_{1}} ∈ W^{a_{1}} and w_{a_{2}} ∈ W^{a_{2}},

Y(w_{a_{1}}, z)w_{a_{2}} = Y(w_{a_{1}}, x)w_{a_{2}}|_{x^{n}=e^{n log z}, n∈ℂ}

and

Y(w_{a_{1}}, e^{r}x)w_{a_{2}} = Y(w_{a_{1}}, y)w_{a_{2}}|_{y^{n}=e^{nr}x^{n}, n∈ℂ}.

We also know that the fusion rules N_{a_{1}a_{2}}^{a_{3}} = N_{W^{a_{1}}W^{a_{2}}}^{W^{a_{3}}} for a_{1}, a_{2}, a_{3} ∈ A are all finite (see [GN], [L], [AN], [H5]). For a_{1}, a_{2}, a_{3} ∈ A, we have isomorphisms Ω_{−r} : V_{a_{1}a_{2}}^{a_{3}} → V_{a_{2}a_{1}}^{a_{3}} and A_{−r} : V_{a_{1}a_{2}}^{a_{3}} → V_{a_{1}a′_{3}}^{a′_{2}} for r ∈ ℤ (see [HL2]). Using these isomorphisms, we define a left action of the symmetric group S_{3} on

V = ∐_{a_{1},a_{2},a_{3}∈A} V_{a_{1}a_{2}}^{a_{3}}

as follows: For a_{1}, a_{2}, a_{3} ∈ A, Y ∈ V_{a_{1}a_{2}}^{a_{3}}, we define

σ_{12}(Y) = e^{πiΔ(Y)}Ω_{−1}(Y)
= e^{−πiΔ(Y)}Ω_{0}(Y),
σ_{23}(Y) = e^{πih_{a_{1}}}A_{−1}(Y)
= e^{−πih_{a_{1}}}A_{0}(Y).

Proposition 1.1 The actions σ_{12} and σ_{23} of (12) and (23) on V defined above generate an action of S_{3} on V.

Proof. We need only prove the relations which must be satisfied by σ_{12} and σ_{23}. We prove only the relation

σ_{12}σ_{23}σ_{12}σ_{23} = σ_{23}σ_{12}.  (1.2)

The other relations can be proved similarly.

Let Y be an element of V_{a_{1}a_{2}}^{a_{3}}. Then for w_{a_{1}} ∈ W^{a_{1}}, w_{a_{2}} ∈ W_{a_{2}} and w′_{a_{3}} ∈ W_{a′_{3}},

⟨σ_{23}(σ_{12}(Y))(w_{a_{2}}, x)w′_{a_{3}}, w_{a_{1}}⟩ = e^{πih_{a_{2}}}⟨w′_{a_{3}}, σ_{12}(Y)(e^{xL(1)}(e^{−πi}x^{−2})^{L(0)}w_{a_{2}}, x^{−1})w_{a_{1}}⟩ = e^{πi(h_{a_{3}}−h_{a_{1}})}⟨w′_{a_{3}}, e^{x^{−1}L(−1)}Y(w_{a_{1}}, e^{−πi}x^{−1})e^{xL(1)}(e^{−πi}x^{−2})^{L(0)}w_{a_{2}}⟩.

(1.3)

On the other hand, ⟨σ_{12}(σ_{23}(σ_{12}(σ_{23}(Y))))(w_{a_{2}}, x)w′_{a_{3}}, w_{a_{1}}⟩

= e^{πi(h_{a_{1}}−h_{a_{2}}−h_{a_{3}})}⟨e^{xL(−1)}σ_{23}(σ_{12}(σ_{23}(Y)))(w′_{a_{3}}, e^{−πi}x)w_{a_{2}}, w_{a_{1}}⟩ = e^{πi(h_{a_{1}}−h_{a_{2}})} ·

·⟨w_{a_{2}}, σ_{12}(σ_{23}(Y))(e^{−xL(1)}(e^{−πi}(e^{−πi}x)^{−2})^{L(0)}w′_{a_{3}}, e^{πi}x^{−1})e^{xL(1)}w_{a_{1}}⟩ = e^{−πih_{a_{3}}}⟨w_{a_{2}}, e^{−x^{−1}L(−1)}σ_{23}(Y)(e^{xL(1)}w_{a_{1}}, x^{−1})e^{−xL(1)}(e^{πi}x^{−2})^{L(0)}w′_{a_{3}}⟩ = e^{πi(h_{a_{1}}−h_{a_{3}})}⟨Y(e^{x^{−1}L(1)}(e^{−πi}(x^{−1})^{−2})^{L(0)}e^{xL(1)}w_{a_{1}}, x)e^{−x^{−1}L(1)}w_{a_{2}},

e^{−xL(1)}(e^{πi}x^{−2})^{L(0)}w′_{a_{3}}⟩

= e^{πi(h_{a_{1}}−h_{a_{3}})}⟨w′_{a_{3}}, (e^{πi}x^{−2})^{L(0)}e^{−xL(−1)} ·

·Y(e^{x^{−1}L(1)}(e^{−πi}x^{2})^{L(0)}e^{xL(1)}w_{a_{1}}, x)e^{−x^{−1}L(1)}w_{a_{2}}⟩

= e^{πi(h_{a_{1}}−h_{a_{3}})}⟨w′_{a_{3}}, e^{x^{−1}L(−1)}(e^{πi}x^{−2})^{L(0)} ·

·Y(e^{x^{−1}L(1)}(e^{−πi}x^{2})^{L(0)}e^{xL(1)}w_{a_{1}}, x)e^{−x^{−1}L(1)}w_{a_{2}}⟩

= e^{πi(h_{a_{1}}−h_{a_{3}})}⟨w′_{a_{3}}, e^{x^{−1}L(−1)} ·

·Y((e^{πi}x^{−2})^{L(0)}e^{x^{−1}L(1)}(e^{−πi}x^{2})^{L(0)}e^{xL(1)}w_{a_{1}}, e^{πi}x^{−1}) ·

·(e^{πi}x^{−2})^{L(0)}e^{−x^{−1}L(1)}w_{a_{2}}⟩
= e^{πi(h_{a_{3}}−h_{a_{1}})}⟨w′_{a_{3}}, e^{x^{−1}L(−1)}Y(w_{a_{1}}, e^{−πi}x^{−1})e^{xL(1)}(e^{−πi}x^{−2})^{L(0)}w_{a_{2}}⟩. (1.4)

From (1.3) and (1.4), we obtain ⟨σ_{23}(σ_{12}(Y))(w_{a_{2}}, x)w′_{a_{3}}, w_{a_{1}}⟩ = ⟨σ_{12}(σ_{23}(σ_{12}(σ_{23}(Y))))(w_{a_{2}}, x)w′_{a_{3}}, w_{a_{1}}⟩.

Since a_{1}, a_{2}, a_{3}, w_{a_{1}}, w_{a_{2}}, w′_{a_{3}} and Y are arbitrary, we see that (1.2) holds.

For p = 1, 2, 3, 4, 5, 6, … and a_{1}, a_{2}, a_{3} ∈ A, let Y_{a_{1}a_{2};i}^{a_{3};(p)}, i = 1, …, N_{a_{1}a_{2}}^{a_{3}}, be basis of V_{a_{1}a_{2}}^{a_{3}}. By the associativity of intertwining operators, there exist

F (Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{6}a_{3};l}^{a_{4};(3)} ⊗ Y_{a_{1}a_{2};k}^{a_{6};(4)}) ∈ ℂ

for a_{1}, …, a_{6} ∈ A, i = 1, …, N_{a_{1}a_{5}}^{a_{4}}, j = 1, …, N_{a_{2}a_{3}}^{a_{5}}, k = 1, …, N_{a_{1}a_{2}}^{a_{6}}, l = 1, …, N_{a_{6}a_{3}}^{a_{4}}, such that

⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩

= ∑_{a_{6}∈A} ∑_{k=1}^{N_{a_{1}a_{2}}^{a_{6}}} ∑_{l=1}^{N_{a_{6}a_{3}}^{a_{4}}} F (Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{6}a_{3};l}^{a_{4};(3)} ⊗ Y_{a_{1}a_{2};k}^{a_{6};(4)}) ·

·⟨w_{a′_{4}}, Y_{a_{6}a_{3};l}^{a_{4};(3)}(Y_{a_{1}a_{2};k}^{a_{6};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩  (1.5)

when |z_{1}| > |z_{2}| > |z_{1} − z_{2}| > 0, for a_{1}, …, a_{5} ∈ A, w_{a_{1}} ∈ W^{a_{1}}, w_{a_{2}} ∈ W^{a_{2}}, w_{a_{3}} ∈ W^{a_{3}}, w_{a′_{4}} ∈ W^{a′_{4}} = (W^{a_{4}})′, i = 1, …, N_{a_{1}a_{5}}^{a_{4}} and j = 1, …, N_{a_{2}a_{3}}^{a_{5}}. Note that here we have used our convention on the choices of values of intertwining operators. The numbers

F (Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{6}a_{3};l}^{a_{4};(3)} ⊗ Y_{a_{1}a_{2};k}^{a_{6};(4)})

are matrix elements of the fusing isomorphism. In [H4], when these four basis are chosen to be the same, these matrix elements are denoted as F_{a_{5};a_{6}}^{i,j;k,l}(a_{1}, a_{2}, a_{3}; a_{4}). In the present paper, since we want to emphasize their dependence on the basis and since we shall need matrix elements of the fusing isomorphism under different basis, we have to instead use the notation above.

The fusing isomorphism are invertible. Thus for any basis as above, there exist

F^{−1}(Y_{a_{6}a_{3};l}^{a_{4};(1)} ⊗ Y_{a_{1}a_{2};k}^{a_{6};(2)}; Y_{a_{1}a_{5};i}^{a_{4};(3)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(4)}) ∈ ℂ

for a_{1}, …, a_{6} ∈ A, i = 1, …, N_{a_{1}a_{5}}^{a_{4}}, j = 1, …, N_{a_{2}a_{3}}^{a_{5}}, k = 1, …, N_{a_{1}a_{2}}^{a_{6}}, l = 1, …, N_{a_{6}a_{3}}^{a_{4}}, such that

⟨w_{a′_{4}}, Y_{a_{6}a_{3};l}^{a_{4};(1)}(Y_{a_{1}a_{2};k}^{a_{6};(2)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩

= ∑_{a_{5}∈A} ∑_{i=1}^{N_{a_{1}a_{5}}^{a_{4}}} ∑_{j=1}^{N_{a_{2}a_{3}}^{a_{5}}} F^{−1}(Y_{a_{6}a_{3};l}^{a_{4};(1)} ⊗ Y_{a_{1}a_{2};k}^{a_{6};(2)}; Y_{a_{1}a_{5};i}^{a_{4};(3)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(4)}) ·

·⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(3)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(4)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩  (1.6)

when |z_{1}| > |z_{2}| > |z_{1}−z_{2}| > 0, for a_{1}, …, a_{4}, a_{6} ∈ A, w_{a_{1}} ∈ W^{a_{1}}, w_{a_{2}} ∈ W^{a_{2}}, w_{a_{3}} ∈ W^{a_{3}}, w_{a′_{4}} ∈ W^{a′_{4}} = (W^{a_{4}})′, k = 1, …, N_{a_{1}a_{2}}^{a_{6}} and l = 1, …, N_{a_{6}a_{3}}^{a_{4}}. These numbers are matrix elements of the inverse of the fusing isomorphism.

From [H4], we know that

⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩,  (1.7)
⟨w_{a′_{4}}, Y_{a_{6}a_{3};l}^{a_{4};(3)}(Y_{a_{1}a_{2};k}^{a_{6};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩⟩,  (1.8)
⟨w_{a′_{4}}, Y_{a_{2}a_{6};k}^{a_{4};(5)}(w_{a_{2}}, z_{2})Y_{a_{1}a_{3};l}^{a_{6};(6)}(w_{a_{1}}, z_{1})w_{a_{3}}⟩  (1.9)

can all be analytically extended to multi-valued analytic functions on

M^{2} = {(z_{1}, z_{2}) ∈ ℂ^{2} | z_{1}, z_{2} ≠ 0, z_{1} ≠ z_{2}}.

We can lift multi-valued analytic functions on M^{2} to single-valued analytic functions on the universal covering M̃^{2} of M^{2}. However, these single-valued liftings are not unique. To obtain a unique lifting, we have to give singlevalued branches of the multi-valued analytic functions in simply-connected subsets of M^{2}.

We use

M^{2}(|z_{1}| > |z_{2}| > 0, 0 ≤ arg z_{1}, arg z_{2} < 2π),  (1.10)
M^{2}(|z_{2}| > |z_{1}| > 0, 0 ≤ arg z_{1}, arg z_{2} < 2π),  (1.11)
M^{2}(|z_{2}| > |z_{1} − z_{2}| > 0, 0 ≤ arg z_{2}, arg(z_{1} − z_{2}) < 2π)  (1.12)

to denote the region |z_{1}| > |z_{2}| > 0 with cuts along the real lines in the z_{1}- and z_{2}-planes, the region |z_{2}| > |z_{1}| > 0 with cuts along the real lines in the z_{1}- and z_{2}-planes, the region |z_{2}| > |z_{1} − z_{2}| > 0 with cuts along the real lines in the z_{2}- and z_{1} −z_{2}-planes, respectively. The multi-valued analytic extensions of (1.7) and (1.8) have single-valued branches (1.7) and (1.8), respectively, on (1.10) and (1.12), respectively. Thus we have the corresponding single-valued analytic functions on M̃^{2}. We denote these analytic functions by

E(⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, x_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, x_{2})w_{a_{3}}⟩)

and

E(⟨w_{a′_{4}}, Y_{a_{6}a_{3};l}^{a_{4};(3)}(Y_{a_{1}a_{2};k}^{a_{6};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩),

respectively.

Then from (1.5) and (1.6), we immediately obtain

E(⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩)

= ∑_{a_{6}∈A} ∑_{k=1}^{N_{a_{1}a_{2}}^{a_{6}}} ∑_{l=1}^{N_{a_{6}a_{3}}^{a_{4}}} F (Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{6}a_{3};l}^{a_{4};(3)} ⊗ Y_{a_{1}a_{2};k}^{a_{6};(4)}) ·

·E(⟨w_{a′_{4}}, Y_{a_{6}a_{3};l}^{a_{4};(3)}(Y_{a_{1}a_{2};k}^{a_{6};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩) (1.13)

and

E(⟨w_{a′_{4}}, Y_{a_{6}a_{3};l}^{a_{4};(1)}(Y_{a_{1}a_{2};k}^{a_{6};(2)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩)

= ∑_{a_{5}∈A} ∑_{i=1}^{N_{a_{1}a_{5}}^{a_{4}}} ∑_{j=1}^{N_{a_{2}a_{3}}^{a_{5}}} F^{−1}(Y_{a_{6}a_{3};l}^{a_{4};(1)} ⊗ Y_{a_{1}a_{2};k}^{a_{6};(2)}; Y_{a_{1}a_{5};i}^{a_{4};(3)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(4)}) ·

·E(⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(3)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(4)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩).  (1.14)

From a single-valued analytic function f on M̃^{2}, we can construct other single-valued analytic functions on M̃^{2} such that they and f correspond to the same multi-valued analytic functions on M^{2}. One particularly interesting construction is given by a braiding operation. To give this construction, we need only give a single-valued branch of the multi-valued analytic function on M^{2} corresponding to f on a simply-connected region in M^{2}. We can view the region (1.11) as a simply-connected region in M̃^{2}. Thus f gives a singlevalued function on this region. For any r ∈ ℤ, we now give a single-valued analytic functions on the region (1.10) in the following way: Consider the path

t ↦ (\frac{3}{2} − \frac{e^{(2r+1)πit}}{2} , \frac{3}{2} + \frac{e^{(2r+1)πit}}{2} )

from the point (1, 2) in the region (1.11) to the point (2, 1) in the region (1.10). This path and the restriction of f to the region (1.11) gives a unique single-valued analytic function on the region (1.10). This single-valued analytic function on this simply-connected region determines uniquely a singlevalued analytic function B^{(r)}(f) on M̃^{2} such that f and B^{(r)}(f) correspond to the same multi-valued analytic function on M^{2}.

We now consider (1.9). From the discussion above, we have a singlevalued analytic function

E(⟨w_{a′_{4}}, Y_{a_{2}a_{6};k}^{a_{4};(5)}(w_{a_{2}}, z_{2})Y_{a_{1}a_{3};l}^{a_{6};(6)}(w_{a_{1}}, z_{1})w_{a_{3}}⟩)

on M̃^{2}. Apply our construction above, we obtain another single-valued analytic function

B^{(r)}(E(⟨w_{a′_{4}}, Y_{a_{2}a_{6};k}^{a_{4};(3)}(w_{a_{2}}, z_{2})Y_{a_{1}a_{3};l}^{a_{6};(4)}(w_{a_{1}}, z_{1})w_{a_{3}}⟩))

on M̃^{2}. In terms of these functions, the commutativity for intertwining operators (see [H2] and [H4]) can be written as

B^{(r)}(E(⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{2})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{1})w_{a_{3}}⟩))

= ∑_{a_{6}∈A} ∑_{k=1}^{N_{a_{2}a_{6}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{3}}^{a_{6}}} B^{(r)}(Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{2}a_{6};k}^{a_{4};(3)} ⊗ Y_{a_{1}a_{3};l}^{a_{6};(4)}) ·

·E(⟨w_{a′_{4}}, Y_{a_{2}a_{6};k}^{a_{4};(3)}(w_{a_{2}}, z_{1})Y_{a_{1}a_{3};l}^{a_{6};(4)}(w_{a_{1}}, z_{2})w_{a_{3}}⟩),

where the numbers

B^{(r)}(Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{2}a_{6};k}^{a_{4};(3)} ⊗ Y_{a_{1}a_{3};l}^{a_{6};(4)})

are the matrix elements of the braiding isomorphism.

From the construction above, it is easy to see that the square (B^{(r)})^{2} of B^{(r)} is actually the map sending a single-valued analytic function f on M̃^{2} to another one (B^{(r)})^{2}(f) corresponding to the same multi-valued analytic functions on M^{2} in the following way: Consider the path

t ↦ (\frac{3}{2} + \frac{e^{2(2r+1)πit}}{2} , \frac{3}{2} − \frac{e^{2(2r+1)πit}}{2} )

from the point (2, 1) to itself. This path and the restriction of f to the region (1.10) gives another single-valued analytic function on the same region. This new single-valued analytic function on the region (1.10) gives a single-valued analytic function (B^{(r)})^{2}(f) on M̃^{2}. From this description of (B^{(r)})^{2}(f), we see that (B^{(r)})^{2}(f) is in fact the monodromy of the multi-valued function corresponding to f given by log(z_{1} − z_{2}) ↦ log(z_{1} − z_{2}) + 2(2r + 1)πi.

We shall also use similar notations to denote the matrix elements of the square (B^{(r)})^{2} of B^{(r)} under the basis above as

(B^{(r)})^{2}(Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{1}a_{6};i}^{a_{4};(3)} ⊗ Y_{a_{2}a_{3};j}^{a_{6};(4)}),

that is,

(B^{(r)})^{2}(E(⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩))

= ∑_{a_{6}∈A} ∑_{k=1}^{N_{a_{2}a_{6}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{3}}^{a_{6}}}(B^{(r)})^{2}(Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{1}a_{6};i}^{a_{4};(3)} ⊗ Y_{a_{2}a_{3};j}^{a_{6};(4)}) ·

·E(⟨w_{a′_{4}}, Y_{a_{1}a_{6};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{6};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩).  (1.15)

Similarly to correlation functions constructed from products and iterates of intertwining operators above, if we have a series φ which is absolutely convergent in a region in M^{2} and can be analytically extended to a multi-valued analytic function on M^{2}, then we obtain a unique single-valued analytic function E(φ) on M̃^{2}. For example, we know that for z ∈ ℂ,

⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})e^{zL(−1)}Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2} − z)e^{−zL(−1)}w_{a_{3}}⟩

is absolutely convergent in the region given by |z_{1}| > |z_{2}| > 0 and |z_{2}−z| > |z| and it can be analytically extended to a multi-valued analytic function on M^{2}. Thus we have a single-valued analytic function

E(⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})e^{zL(−1)}Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2} − z)e^{−zL(−1)}w_{a_{3}}⟩)

on M̃^{2}. All the other properties of intertwining operators, for example, the skew-symmetry, the contragredient intertwining operators and so on, can all be expressed using equalities for such single-valued analytic functions on M̃^{2}.

We shall need the following result:

Proposition 1.2 For a_{1}, a_{2}, a_{3}, a_{4} ∈ A, the maps from W^{a′_{4}} ⊗ W^{a_{1}} ⊗ W^{a_{2}} ⊗ W^{a_{3}} to the space of single-valued analytic functions on M̃^{2} given by

w_{a′_{4}} ⊗ w_{a_{1}} ⊗ w_{a_{2}} ⊗ w_{a_{3}} ↦ E(⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩),

a_{5} ∈ A, i = 1, …, N_{a_{1}a_{5}}^{a_{4}}, j = 1, …, N_{a_{2}a_{3}}^{a_{5}}, are linearly independent. Similarly, for a_{1}, a_{2}, a_{3}, a_{4} ∈ A, the maps from W^{a′_{4}} ⊗ W^{a_{1}} ⊗ W^{a_{2}} ⊗ W^{a_{3}} to the space of single-valued analytic functions on M̃^{2} given by

w_{a′_{4}} ⊗ w_{a_{1}} ⊗ w_{a_{2}} ⊗ w_{a_{3}} ↦ E(⟨w_{a′_{4}}, Y_{a_{6}a_{3};k}^{a_{4};(3)}(Y_{a_{1}a_{2};l}^{a_{6};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩),

a_{6} ∈ A, k = 1, …, N_{a_{6}a_{3}}^{a_{4}}, l = 1, …, N_{a_{1}a_{2}}^{a_{6}}, are linearly independent.

Proof. We prove only the linear independence of the maps obtained from products of intertwining operators. For iterates, the proof is similar.

Since analytic extensions are unique, we need only prove that the maps from W^{a′_{4}} ⊗ W^{a_{1}} ⊗ W^{a_{2}} ⊗ W^{a_{3}} to the space of the single-valued analytic functions on the region (1.10) given by

w_{a′_{4}} ⊗ w_{a_{1}} ⊗ w_{a_{2}} ⊗ w_{a_{3}} ↦ ⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}, ⟩,

a_{5} ∈ A, i = 1, …, N_{a_{1}a_{5}}^{a_{4}}, j = 1, …, N_{a_{2}a_{3}}^{a_{5}}, are linearly independent. Assume that

∑_{a_{5}∈A} ∑_{i=1}^{N_{a_{1}a_{5}}^{a_{4}}} ∑_{j=1}^{N_{a_{2}a_{3}}^{a_{5}}} λ_{a_{5},i,j}⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩ = 0 (1.16)

Since (1.16) holds for all z_{1} and z_{2} satisfying |z_{1}| > |z_{2}| > 0, we obtain the following equation in formal variables:

∑_{a_{5}∈A} ∑_{i=1}^{N_{a_{1}a_{5}}^{a_{4}}} ∑_{j=1}^{N_{a_{2}a_{3}}^{a_{5}}} λ_{a_{5},i,j}⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, x_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, x_{2})w_{a_{3}}⟩ = 0 (1.17)

We want to show that λ_{a_{5},i,j} = 0 for a_{5} ∈ A, i = 1, …, N_{a_{1}a_{5}}^{a_{4}} and j = 1, …, N_{a_{2}a_{3}}^{a_{5}}.

From the tensor product theory in [HL1] and [HL4], we know that the tensor product module W^{a_{2}} ⊠_{P (z_{2})} W^{a_{3}} is isomorphic to ⊕_{a_{5}∈A}N_{a_{2}a_{3}}^{a_{5}}W^{a_{5}}. For a_{5} ∈ A and j = 1, …, N_{a_{2}a_{3}}^{a_{5}}, let π_{a_{5};j} be the projections from ⊕_{a_{5}∈A}N_{a_{2}a_{3}}^{a_{5}}W^{a_{5}} to the j-th copy of W^{a_{5}}. Let f : W^{a_{2}} ⊠_{P (z_{2})} W^{a_{3}} → ⊕_{a_{5}∈A}N_{a_{2}a_{3}}^{a_{5}}W^{a_{5}} be the isomorphism such that

\overline{π}_{a_{5};j}(\overline{f}(w_{a_{2}} ⊠_{P (z_{2})} w_{a_{3}})) = Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}

for w_{a_{2}} ∈ W^{a_{2}} and w_{a_{3}} ∈ W^{a_{3}}, where

\overline{π}_{a_{5};j} : \overline{⊕_{a_{5}∈A}N_{a_{2}a_{3}}^{a_{5}}W^{a_{5}}} → \overline{W}^{a_{5}}

and

\overline{f} : \overline{W^{a_{2}} ⊠_{P (z_{2})} W^{a_{3}}} → \overline{⊕_{a_{5}∈A}N_{a_{2}a_{3}}^{a_{5}}W^{a_{5}}}

are the natural extensions of π_{a_{5};j} and f. By the universal property for the tensor product module, such f indeed exists. Let Y_{2} be the intertwining operator corresponding to the intertwining map ⊠_{P (z_{2})} : W^{a_{2}} ⊗ W^{a_{3}} → W^{a_{2}} ⊠_{P (z_{2})} W^{a_{3}} (see [HL1] and [HL4]). Then we have

π_{a_{5};j}(f(Y_{2}(w_{a_{2}}, x)w_{a_{3}})) = Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, x)w_{a_{3}}  (1.18)

for w_{a_{2}} ∈ W^{a_{2}} and w_{a_{3}} ∈ W^{a_{3}}. Let Y_{1} be the intertwining operator of type \binom{W^{a_{4}}}{W^{a_{1}} (W^{a_{2}}⊠_{P (z_{2})}W^{a_{3}})} given by

Y_{1}(w_{a_{1}}, x)w = ∑_{a_{5}∈A} ∑_{i=1}^{N_{a_{1}a_{5}}^{a_{4}}} ∑_{j=1}^{N_{a_{2}a_{3}}^{a_{5}}} λ_{a_{5},i,j}Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, x)π_{a_{5};j}(f(w))  (1.19)

for w_{a_{1}} ∈ W^{a_{1}} and w ∈ W^{a_{2}} ⊠_{P (z_{2})} W^{a_{3}}.

By (1.18) and (1.19), the left-hand side of (1.17) is equal to

∑_{a_{5}∈A} ∑_{i=1}^{N_{a_{1}a_{5}}^{a_{4}}} ∑_{j=1}^{N_{a_{2}a_{3}}^{a_{5}}} λ_{a_{5},i,j}⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, x_{1})π_{a_{5};j}(f(Y_{2}(w_{a_{2}}, x_{2})w_{a_{3}}))⟩

= ⟨w_{a′_{4}}, Y_{1}(w_{a_{1}}, x_{1})Y_{2}(w_{a_{2}}, x_{2})w_{a_{3}}⟩.

Thus we have

⟨w_{a′_{4}}, Y_{1}(w_{a_{1}}, x_{1})Y_{2}(w_{a_{2}}, x_{2})w_{a_{3}}⟩ = 0  (1.20)

for w_{a′_{4}} ∈ W^{a′_{4}}, w_{a_{1}} ∈ W^{a_{1}}, w_{a_{2}} ∈ W^{a_{2}}, w_{a_{3}} ∈ W^{a_{3}}. Since the homogeneous components of w_{a_{2}} ⊠_{P (z_{2})} w_{a_{3}} or equivalently the x_{2}-coefficients of Y_{2}(w_{a_{2}}, x_{2})w_{a_{3}} for w_{a_{2}} ∈ W^{a_{2}} and w_{a_{3}} ∈ W^{a_{3}} span W^{a_{2}} ⊠_{P (z_{2})} W^{a_{3}} (see [H1]), we obtain from (1.20) that

⟨w_{a′_{4}}, Y(w_{a_{1}}, x_{1})w⟩ = 0.

for w_{a′_{4}} ∈ W^{a′_{4}}, w_{a_{1}} ∈ W^{a_{1}} and w ∈ W^{a_{2}} ⊠_{P (z_{2})} W^{a_{3}}.

Now take w to be an element of W^{a_{2}} ⊠_{P (z_{2})} W^{a_{3}} such that f(w) is in the j-th copy of W^{a_{5}} in ⊕_{a_{5}∈A}N_{a_{2}a_{3}}^{a_{5}}W^{a_{5}}, that is, take w such that π_{a_{5};j}(f(w)) = f(w), π_{a_{5};m}(f(w)) = 0 for m ≠ j and π_{a;m}(f(w)) = 0 for a ≠ a_{5}. Then we have

∑_{i=1}^{N_{a_{1}a_{5}}^{a_{4}}} λ_{a_{5},i,j}⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, x_{1})f(w)⟩ = 0.

Since w_{a′_{4}}, w_{a_{1}}, w are arbitrary elements of W^{a′_{4}}, W^{a_{1}} and the j-th copy of W^{a_{5}} in ⊕_{a_{5}∈A}N_{a_{2}a_{3}}^{a_{5}}W^{a_{5}}, respectively, we obtain

∑_{i=1}^{N_{a_{1}a_{5}}^{a_{4}}} λ_{a_{5},i,j}Y_{a_{1}a_{5};i}^{a_{4};(1)} = 0.

Since Y_{a_{1}a_{5};i}^{a_{4};(1)} for i = 1, …, N_{a_{1}a_{5}}^{a_{4}} are linearly independent, we obtain λ_{a_{5},i,j} = 0 for a_{5} ∈ A, i = 1, …, N_{a_{1}a_{5}}^{a_{4}} and j = 1, …, N_{a_{2}a_{3}}^{a_{5}}.

Since

⟨w_{a′_{2}}, Y_{a_{2}e;1}^{a_{2}}(w_{a_{2}}^{1}, z_{1})Y_{a′_{3}a_{3};1}^{e}(w_{a′_{3}}, z_{2})Y_{a_{1}a_{2};i}^{a_{3}}(w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩  (1.21)

and

⟨w_{a′_{2}}, Y_{a_{4}a_{3};k}^{a_{2}}(Y_{a_{2}a′_{3};j}^{a_{4}}(w_{a_{2}}^{1}, z_{1} − z_{2})w_{a′_{3}}, z_{2})Y_{a_{1}a_{2};i}^{a_{3}}(w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩  (1.22)

satisfy a system of differential equations of regular singular points with coefficients in

ℂ[z_{1}, z_{2}^{−1}, z_{2}, z_{2}^{−1}, z_{3}, z_{3}^{−1}, (z_{1} − z_{2})^{−1}, (z_{1} − z_{3})^{−1}, (z_{2} − z_{3})^{−1}]

(see [H5]), they are absolutely convergent in the region given by |z_{1}| > |z_{2}| > |z_{3}| > 0 and in the region given by |z_{2}| > |z_{1} − z_{2}| > 0, |z_{2}| > |z_{3}| > 0, |z_{2} − z_{3}| > |z_{1} − z_{2}| > 0, respectively. Using the same method as in [H4], we see that (1.21) and (1.22) can also be analytically extended to multi-valued analytic functions on

M^{3} = {(z_{1}, z_{2}, z_{3}) ∈ ℂ^{3} | z_{1}, z_{2}, z_{3} ≠ 0, z_{1} ≠ z_{2}, z_{2} ≠ z_{3}, z_{1} ≠ z_{3}}.

We can also choose single valued branches of these multi-valued analytic functions on the region given by |z_{1}| > |z_{2}| > |z_{3}| > 0 with cuts along the real lines in the z_{1}-, z_{2}- and z_{3}-planes and on the region |z_{2}| > |z_{1} − z_{2}| > 0, |z_{2}| > |z_{3}| > 0, |z_{2} − z_{3}| > |z_{1} − z_{2}| > 0 with cuts along the real lines in the z_{2}-, z_{1} − z_{2}- and z_{3}-planes.

Similarly to the case of two intertwining operators above, we also have the single-valued analytic functions

E(⟨w_{a′_{2}}, Y_{a_{2}e;1}^{a_{2}}(w_{a_{2}}^{1}, z_{1})Y_{a′_{3}a_{3};1}^{e}(w_{a′_{3}}, z_{2})Y_{a_{1}a_{2};i}^{a_{3}}(w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩)

and

E(⟨w_{a′_{2}}, Y_{a_{4}a_{3};k}^{a_{2}}(Y_{a_{2}a′_{3};j}^{a_{4}}(w_{a_{2}}^{1}, z_{1} − z_{2})w_{a′_{3}}, z_{2})Y_{a_{1}a_{2};i}^{a_{3}}(w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩)

on the universal covering M̃^{3} of M^{3}. The associativity can also be written using these single valued analytic functions. Similarly we have other single valued analytic functions on M̃^{3}, for example,

E(⟨w_{a′_{2}}, Y_{a_{4}a_{3};k}^{a_{2}}(Y_{a_{2}a′_{3};j}^{a_{4}}(w_{a_{2}}^{1}, z_{1} − z_{2})w_{a′_{3}}, z_{2})Y_{a_{1}a_{2};i}^{a_{3}}(w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩),
B_{12}^{(r)}(E(⟨w_{a′_{2}}, Y_{a_{2}e;1}^{a_{2}}(w_{a_{2}}^{1}, z_{1})Y_{a′_{3}a_{3};1}^{e}(w_{a′_{3}}, z_{2})Y_{a_{1}a_{2};i}^{a_{3}}(w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩)),
B_{23}^{(r)}(E(⟨w_{a′_{2}}, Y_{a_{2}e;1}^{a_{2}}(w_{a_{2}}^{1}, z_{1})Y_{a′_{3}a_{3};1}^{e}(w_{a′_{3}}, z_{2})Y_{a_{1}a_{2};i}^{a_{3}}(w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩))

and so on, where the subscripts 12 and 23 in B_{12}^{(r)} and B_{23}^{(r)}, respectively, mean that they corresponding to braiding isomorphisms which braid the first two and the last two intertwining operators, respectively.

Similarly to the case of two intertwining operators, we also have the following result:

Proposition 1.3 For a_{1}, a_{2}, a_{3}, a_{4}, a_{5} ∈ A, the maps from W^{a′_{5}} ⊗ W^{a_{1}} ⊗ W^{a_{2}} ⊗ W^{a_{3}} ⊗ W^{a_{4}} to the space of single-valued analytic functions on M̃^{3} given by

w_{a′_{5}} ⊗ w_{a_{1}} ⊗ w_{a_{2}} ⊗ w_{a_{3}} ⊗ w_{a_{4}} ↦

E(⟨w_{a′_{4}}, Y_{a_{6}a_{4};k}^{a_{5};(1)}(Y_{a_{7}a_{3};l}^{a_{6};(2)}(Y_{a_{1}a_{2};m}^{a_{7};(3)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2} − z_{3})w_{a_{3}}, z_{3})w_{a_{4}}⟩)

for a_{6}, a_{7} ∈ A, k = 1, …, N_{a_{6}a_{4}}^{a_{5}}, l = 1, …, N_{a_{7}a_{3}}^{a_{6}} and m = 1, …, N_{a_{1}a_{2}}^{a_{7}} are linearly independent. Similarly, a_{1}, a_{2}, a_{3}, a_{4}, a_{5} ∈ A, the maps from W^{a′_{5}} ⊗W^{a_{1}} ⊗W^{a_{2}} ⊗W^{a_{3}} ⊗W^{a_{4}} to the space of single-valued analytic functions on M̃^{3} given by the lifting to the universal covering of analytic extensions of products or combinations of products and iterates of basis of intertwining operators are linearly independent.

The proof of this proposition is similar to the proof of Proposition 1.2 and is omitted.

2 Geometrically-modified intertwining operators and genus-one correlation functions

We now discuss genus-one correlation functions. In the present paper, we need only one or two point functions. Genus-one correlation functions are defined using what we called geometrically-modified intertwining operators (see [H6]). We shall first discuss these operators and prove some important properties needed in this paper.

Let A_{j}, j ∈ ℤ_{+}, be the complex numbers defined by

\frac{1}{2πi} log(1 + 2πiy) = (exp ( ∑_{j∈ℤ_{+}} A_{j}y^{j+1} \frac{∂}{∂y} )) y.

For any V -module W, let

U(x) = (2πix)^{L(0)}e^{−L^{+}(A)} ∈ (End W){x}  (2.1)

where L_{+}(A) = ∑_{j∈ℤ_{+}} A_{j}L(j) and L(j) for j ∈ ℤ are the Virasoro operators on W. Given an intertwining operator Y of type \binom{W^{a_{3}}}{W^{a_{1}}W^{a_{2}}}, the map W^{a_{1}} ⊗ W^{a_{2}} → W^{a_{3}}{x} defined by w_{a_{1}} ⊗ w_{a_{2}} ↦ Y(U(x)w_{a_{1}}, x)w_{a_{2}} is the corresponding geometrically-modified intertwining operator. We need the following property of geometrically-modified intertwining operators:

Proposition 2.1 Let Y be an intertwining operator of type \binom{W^{a_{3}}}{W^{a_{1}}W^{a_{2}}}. Then for w_{a_{1}} ∈ W^{a_{1}}, w_{a_{1}} ∈ W^{a_{1}} and w′_{a_{3}} ∈ W^{a′_{3}}, we have

⟨w′_{a_{3}}, σ_{23}(Y)(U(x)w_{a_{1}}, x)w_{a_{2}}⟩ = e^{πih_{a_{1}}}⟨Y(U(x^{−1})e^{−πiL(0)}w_{a_{1}}, x^{−1})w′_{a_{3}}, w_{a_{2}}⟩
= e^{−πih_{a_{1}}}⟨Y(U(x^{−1})e^{πiL(0)}w_{a_{1}}, x^{−1})w′_{a_{3}}, w_{a_{2}}⟩.

(2.2)

Proof. We first prove the first equality in (2.2). By the definition of σ_{23}(Y), we have

⟨w′_{a_{3}}, σ_{23}(Y)(U(x)w_{a_{1}}, x)w_{a_{2}}⟩
= e^{πih_{a_{1}}}⟨Y(e^{xL(1)}(e^{−πi}x^{−2})^{L(0)}U(x)w_{a_{1}}, x^{−1})w′_{a_{3}}, w_{a_{2}}⟩.  (2.3)

We now calculate e^{xL(1)}(e^{−πi}x^{−2})^{L(0)}U(x)w. The equality

\frac{1}{2πi} log (1 − \frac{x}{y^{−1} + x}) = −\frac{1}{2πi} log(1 + xy)

gives

e−xy^{2} \frac{∂}{∂y} (e^{−πi}x^{−2})−y \frac{∂}{∂y} (2πix)−y \frac{∂}{∂y} e ∑_{j∈ℤ_{+}} A_{j}y^{j+1} \frac{∂}{∂y} y

= (2πix^{−1})−y \frac{∂}{∂y} e ∑_{j∈ℤ_{+}} A_{j}y^{j+1} \frac{∂}{∂y} eπiy \frac{∂}{∂y} y.

Using the method developed in Chapters 4 and 5 of [H3], we obtain

e^{xL(1)}(e^{−πi}x^{−2})^{L(0)}(2πix)^{L(0)}e^{−L_{+}(A)} = (2πix^{−1})^{L(0)}e^{−L_{+}(A)}e^{−πiL(0)},

that is,

e^{xL(1)}(e^{−πi}x^{−2})^{L(0)}U(x) = U(x^{−1})e^{−πiL(0)},

Thus the right-hand side of (2.3) is equal to

e^{πih_{a_{1}}}⟨Y(U(x^{−1})e^{−πiL(0)}w_{a_{1}}, x^{−1})w′_{a_{3}}, w_{a_{2}}⟩,

proving the first equality in (2.2).

The second equality in (2.2) follows immediately from the first equality and the equality

e^{−πL(0)}w_{a_{1}} = e^{πL(0)}e^{−2πL(0)}w_{a_{1}}
= e^{−2πh_{a_{1}}}e^{πL(0)}w_{a_{1}}.

As in the preceding section, For a_{1}, a_{2}, a_{3} ∈ A and p = 1, 2, 3, 4, 5, 6, …, let Y_{a_{1}a_{2};i}^{a_{3};(p)}, i = 1, …, N_{a_{1}a_{2}}^{a_{3}}, be basis of V_{a_{1}a_{2}}^{a_{3}}. Let q_{τ} = e^{2πiτ} for τ ∈ ℍ. We consider q_{τ}-traces of geometrically-modified intertwining operators of the following form:

Tr_{W^{a_{2}}}Y_{a_{1}a_{2};i}^{a_{2};(1)}(U(e^{2πiz})w_{a_{1}}, e^{2πiz})q_{τ}^{L(0)− \frac{c}{24}}  (2.4)

for a_{1}, a_{2} ∈ A, i = 1, …, N_{a_{1}a_{2}}^{a_{2}}. In [M] and [H6], it was shown 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 (2.4) by

E(Tr_{W^{a_{2}}}Y_{a_{1}a_{2};i}^{a_{2};(1)}(U(e^{2πiz})w_{a_{1}}, e^{2πiz})q_{τ}^{L(0)− \frac{c}{24}} ).

These are genus-one one-point correlation functions. In [M] and [H6], the following modular invariance property is also proved: For

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℤ),

let τ′ = \frac{aτ+b}{cτ+d}. Then there exists A_{a_{2}}^{a_{3}} such that

Tr_{W^{a_{2}}}Y_{a_{1}a_{2};i}^{a_{2};(1)} (U(e^{2πi \frac{z}{cτ+d}}) (\frac{1}{cτ + d})^{L(0)} w_{a_{1}}, e^{2πi \frac{z}{cτ+d}} ) q_{τ′}^{L(0)− \frac{c}{24}}

= ∑_{a_{3}∈A} A_{a_{2}}^{a_{3}}Tr_{W^{a_{3}}}Y_{a_{1}a_{3};i}^{a_{2};(1)}(U(e^{2πiz})w_{a_{1}}, e^{2πiz})q_{τ}^{L(0)− \frac{c}{24}} .

This modular invariance property can be written in terms of the analytic extensions as follows:

E (Tr_{W^{a_{2}}}Y_{a_{1}a_{2};i}^{a_{2};(1)} (U(e^{\frac{2πiz}{cτ+d}}) (\frac{1}{cτ + d})^{L(0)} w_{a_{1}}, e^{\frac{2πiz}{cτ+d}} ) q_{τ′}^{L(0)− \frac{c}{24}} )

= ∑_{a_{3}∈A} A_{a_{2}}^{a_{3}}E(Tr_{W^{a_{3}}}Y_{a_{1}a_{3};i}^{a_{3};(1)}(U(e^{2πiz})w_{a_{1}}, e^{2πiz})q_{τ}^{L(0)− \frac{c}{24}} ).

We consider the space F_{1;1} spanned by linear maps of the form

Ψ_{a} : V → 𝔾_{1;1}
u ↦ Ψ_{a}(u; τ)

for a ∈ A, where

Ψ_{a}(u; τ) = E(Tr_{W^{a}}Y_{ea;1}^{a;(1)}(U(e^{2πiz})u, e^{2πiz})q_{τ}^{L(0)− \frac{c}{24}} )

and 𝔾_{1;1} is the space spanned by functions of τ of the form Ψ_{a}(u; τ).

We also need genus-one two-point correlation functions. They are constructed from the q_{τ}-traces of products or iterates of two geometricallymodified intertwining operators as follows: For a_{1}, a_{2}, a_{3}, a_{4} ∈ A, i = 1, …, N_{a_{1}a_{3}}^{a_{4}}, j = 1, …, N_{a_{2}a_{4}}^{a_{3}}, consider

Tr_{W^{a_{4}}}Y_{a_{1}a_{3};i}^{a_{4};(1)}(U(e^{2πiz_{1}})w_{a_{1}}, e^{2πiz_{1}})Y_{a_{2}a_{4};j}^{a_{3};(2)}(U(e^{2πiz_{2}})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} . (2.5)

In [H6], it was shown that these q_{τ}-traces are absolutely convergent when 0 < |q_{τ}| < |e^{2πiz_{1}}| < |e^{2πiz_{2}}| < 1 and can be analytically extended to multivalued analytic functions on

M_{1}^{2} = {(z_{1}, z_{2}, τ) ∈ ℂ^{3} | z_{1} ≠ z_{2} + pτ + q for p, q ∈ ℤ, τ ∈ ℍ}.

(In fact, these multi-valued analytic functions depend only on z_{1} − z_{2} and τ by the L(−1)-derivative property. See [H6].) These multi-valued analytic functions on M_{1}^{2} are genus-one two-point correlation functions. They can be lifted to single-valued analytic functions on the universal covering M̃_{1}^{2}. As in the case of genus-zero correlation functions, these liftings are not unique. To obtain a unique single-valued analytic function on M̃_{1}^{2} from a multi-valued analytic functions f on M_{1}^{2}, we have to give a single valued branch of f on some simply connected region in M_{1}^{2} or values of f on some simplyconnected subset of M_{1}^{2}. In fact, (2.5) gives a single valued branch of its analytic extension in the region given by 0 < |q_{τ}| < |e^{2πiz_{1}}| < |e^{2πiz_{2}}| < 1. We denote this single-valued analytic function on M̃_{1}^{2} by

E(Tr_{W^{a_{4}}}Y_{a_{1}a_{3};i}^{a_{4};(1)}(U(e^{2πiz_{1}})w_{a_{1}}, e^{2πiz_{1}})Y_{a_{2}a_{4};j}^{a_{3};(2)}(U(e^{2πiz_{2}})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ).

(2.6)

Similarly, these genus-one two-point correlation functions can also be constructed from q_{τ}-traces of iterates of two intertwining operators as follows: For a_{1}, a_{2}, a_{5}, a_{6} ∈ A, k = 1, …, N_{a_{3}a_{4}}^{a_{4}}, l = 1, …, N_{a_{1}a_{2}}^{a_{3}}, consider

Tr_{W^{a_{6}}}Y_{a_{5}a_{6};k}^{a_{6};(3)}(U(e^{2πiz_{1}})Y_{a_{1}a_{2};l}^{a_{5};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} .  (2.7)

In [H6], it was shown that these q_{τ}-traces are absolutely convergent when 0 < |q_{τ}| < |e^{2πiz_{2}}| < 1 and 0 < |e^{2πi(z_{1}−z_{2})} −1| < 1 and can be analytically extended to multi-valued analytic functions on M_{1}^{2}. (Again, these multi-valued analytic functions depend only on z_{1} − z_{2} and τ by the L(−1)-derivative property.) These multi-valued analytic functions on M_{1}^{2} are also genus-one two-point correlation functions. The multi-valued analytic extension of (2.7) can also be lifted uniquely to a single-valued analytic function on M̃_{1}^{2} using the single-valued branch (2.7). We denote it by

E(Tr_{W^{a_{6}}}Y_{a_{5}a_{6};k}^{a_{6};(3)}(U(e^{2πiz_{1}})Y_{a_{1}a_{2};l}^{a_{5};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ). (2.8)

In [H6], an associativity property for geometrically-modified intertwining operators is proved. This associativity together with the convergence property of q_{τ}-traces of gives

E(Tr_{W^{a_{4}}}Y_{a_{1}a_{3};i}^{a_{4};(1)}(U(e^{2πiz_{1}})w_{a_{1}}, e^{2πiz_{1}})Y_{a_{2}a_{4};j}^{a_{3};(2)}(U(e^{2πiz_{2}})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

= ∑_{a_{5}∈A} ∑_{l=1}^{N_{a_{1}a_{2}}^{a_{5}}} ∑_{k=1}^{N_{a_{5}a_{4}}^{a_{4}}} F (Y_{a_{1}a_{3};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{4};j}^{a_{3};(2)}; Y_{a_{5}a_{4};k}^{a_{4};(3)} ⊗ Y_{a_{1}a_{2};l}^{a_{5};(4)}) ·

·E(Tr_{W^{a_{4}}}Y_{a_{5}a_{4};k}^{a_{4};(3)}(U(e^{2πiz_{1}})Y_{a_{1}a_{2};l}^{a_{5};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ) (2.9) and

E(Tr_{W^{a_{4}}}Y_{a_{3}a_{4};k}^{a_{4};(3)}(U(e^{2πiz_{1}})Y_{a_{1}a_{2};l}^{a_{3};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}}

= ∑_{a_{5}∈A} ∑_{i=1}^{N_{a_{1}a_{2}}^{a_{6}}} ∑_{j=1}^{N_{a_{6}a_{3}}^{a_{4}}} F^{−1}(Y_{a_{3}a_{4};k}^{a_{4};(3)} ⊗ Y_{a_{1}a_{2};l}^{a_{3};(4)}; Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{4};j}^{a_{5};(2)}) ·

·E(Tr_{W^{a_{4}}}Y_{a_{1}a_{5};i}^{a_{4};(1)}(U(e^{2πiz_{1}})w_{a_{1}}, e^{2πiz_{1}}) ·

·Y_{a_{2}a_{4};j}^{a_{5};(2)}(U(e^{2πiz_{2}})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ). (2.10)

In particular, the space of all single-valued analytic functions on M̃_{1}^{2} spanned by functions of the form (2.6) and the space of all single-valued analytic functions on M̃_{1}^{2} spanned by functions of the form (2.8) are the same. We shall denote this space by 𝔾_{1;2}.

We need the following result:

Proposition 2.2 For a_{1}, a_{2} ∈ A, the maps from W^{a_{1}} ⊗ W^{a_{2}} to 𝔾_{1;2} given by

w_{a_{1}} ⊗ w_{a_{2}} ↦

E(Tr_{W^{a_{4}}}Y_{a_{1}a_{3};i}^{a_{4};(1)}(U(e^{2πiz_{1}})w_{a_{1}}, e^{2πiz_{1}})Y_{a_{2}a_{4};j}^{a_{3};(2)}(U(e^{2πiz_{2}})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ),

a_{3}, a_{4} ∈ A, i = 1, …, N_{a_{1}a_{3}}^{a_{4}}, j = 1, …, N_{a_{2}a_{4}}^{a_{3}}, are linearly independent. Similarly, for a_{1}, a_{2} ∈ A, the maps from W^{a_{1}} ⊗ W^{a_{2}} to 𝔾_{1;2} given by

w_{a_{1}} ⊗ w_{a_{2}} ↦ E(Tr_{W^{a_{4}}}Y_{a_{3}a_{4};k}^{a_{4};(3)}(U(e^{2πiz_{2}})Y_{a_{1}a_{2};l}^{a_{3};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ),

a_{3}, a_{4} ∈ A, k = 1, …, N_{a_{3}a_{4}}^{a_{3}}, l = 1, …, N_{a_{1}a_{2}}^{a_{3}}, are linearly independent.

Proof. We prove only the linear independence of the maps obtained from q_{τ}-traces of iterates of intertwining operators. For the linear independence of the maps obtained from traces of products of intertwining operators, the proof is similar.

Since analytic extensions are unique, we need only prove that the maps given by

w_{a_{1}} ⊗ w_{a_{2}} ↦ Tr_{W^{a_{4}}}Y_{a_{3}a_{4};k}^{a_{4};(3)}(U(e^{2πiz_{1}})Y_{a_{1}a_{2};l}^{a_{3};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} , a_{3}, a_{4} ∈ A, k = 1, …, N_{a_{3}a_{4}}^{a_{3}}, l = 1, …, N_{a_{1}a_{2}}^{a_{3}}, are linearly independent. Assume

∑_{a_{3},a_{4}∈A} ∑_{k=1}^{N_{a_{3}a_{4}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{4}}^{a_{2}}} λ_{a_{3},a_{4},k,l} ·

·Tr_{W^{a_{4}}}Y_{a_{3}a_{4};k}^{a_{4};(3)}(U(e^{2πiz_{2}})Y_{a_{1}a_{2};l}^{a_{3};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}}

= 0  (2.11)

for w_{a_{1}} ∈ W^{a_{1}} and w_{a_{2}} ∈ W^{a_{2}}. Since (2.11) holds for all z_{1} and z_{2} satisfying 0 < |q_{τ}| < |e^{2πiz_{2}}| < 1 and 0 < |e^{2πi(z_{1}−z_{2})} − 1| < 1, we obtain the following equation in which the variable z_{1} − z_{2} is replaced by a formal variable x_{0}:

∑_{a_{3},a_{4}∈A} ∑_{k=1}^{N_{a_{3}a_{4}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{4}}^{a_{2}}} λ_{a_{3},a_{4},k,l} ·

·Tr_{W^{a_{4}}}Y_{a_{3}a_{4};k}^{a_{4};(3)}(U(e^{2πiz_{2}})Y_{a_{1}a_{2};l}^{a_{3};(4)}(w_{a_{1}}, x_{0})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}}
= 0  (2.12)

for w_{a_{1}} ∈ W^{a_{1}} and w_{a_{2}} ∈ W^{a_{2}}. We want to show that λ_{a_{3},a_{4},k,l} = 0 for a_{3}, a_{4} ∈ A, k = 1, …, N_{a_{3}a_{4}}^{a_{3}} and l = 1, …, N_{a_{1}a_{2}}^{a_{3}}.

As in the proof of Proposition 1.2, there exists an isomorphism

f : W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}} → ⊕_{a_{3}∈A}N_{a_{1}a_{2}}^{a_{3}}W^{a_{3}}

such that for w_{a_{1}} ∈ W^{a_{1}} and w_{a_{2}} ∈ W^{a_{2}},

\overline{π}_{a_{3};l}(\overline{f}(w_{a_{1}} ⊠_{P (z_{z_{1}−z_{2}})} w_{a_{2}})) = Y_{a_{1}a_{2};l}^{a_{3};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}},

where π_{a_{3};l} is the projections from ⊕_{a_{3}∈A}N_{a_{1}a_{2}}^{a_{3}}W^{a_{3}} to the l-th copy of W^{a_{3}} and

\overline{π}_{a_{3};l} : \overline{⊕_{a_{3}∈A}N_{a_{1}a_{2}}^{a_{3}}W^{a_{3}}} → \overline{W}^{a_{3}}

and

\overline{f} : \overline{W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}}} → \overline{⊕_{a_{3}∈A}N_{a_{1}a_{2}}^{a_{3}}W^{a_{3}}}

are the natural extension of π_{a_{3};l} and f, respectively. Let Y_{2} be the intertwining operator corresponding to the intertwining map ⊠_{P (z_{2})} : W^{a_{1}} ⊗ W^{a_{2}} → W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}} (see [HL1] and [HL4]). Then we have

π_{a_{3};l}(f(Y_{2}(w_{a_{1}}, x)w_{a_{2}})) = Y_{a_{1}a_{2};j}^{a_{3};(4)}(w_{a_{1}}, x)w_{a_{2}}  (2.13)

for w_{a_{1}} ∈ W^{a_{1}} and w_{a_{2}} ∈ W^{a_{2}}. For a_{4} ∈ A, let Y_{a_{4}} be the intertwining operator of type \binom{W^{a_{4}}}{(W^{a_{1}}⊠_{P (z_{1}−z_{2})}W^{a_{2}}) W^{a_{4}}} given by

Y_{a_{4}}(w, x)w_{a_{4}} = ∑_{a_{3}∈A} ∑_{k=1}^{N_{a_{3}a_{4}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{4}}^{a_{2}}} λ_{a_{3},a_{4},k,l}Y_{a_{3}a_{4};k}^{a_{4};(3)}(π_{a_{3};l}(f(w)), x)w_{a_{4}}

for w ∈ W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}} and w_{a_{4}} ∈ W^{a_{4}}. Then the left-hand side of (2.12) is equal to

∑_{a_{3},a_{4}∈A} ∑_{k=1}^{N_{a_{3}a_{4}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{2}}^{a_{3}}} λ_{a_{3},a_{4},k,l} ·

·Tr_{W^{a_{4}}}Y_{a_{3}a_{4};k}^{a_{4};(3)}(U(e^{2πiz_{2}})π_{a_{3};l}(f(Y_{2}(w_{a_{1}}, x_{0})w_{a_{2}})), e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}}

= ∑_{a_{4}∈A} Tr_{W^{a_{4}}}Y_{a_{4}}(U(e^{2πiz_{2}})Y_{2}(w_{a_{1}}, x_{0})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}}  (2.14)

By (2.12) and (2.14), we have

∑_{a_{4}∈A} Tr_{W^{a_{4}}}Y_{a_{4}}(U(e^{2πiz_{2}})Y_{2}(w_{a_{1}}, x_{0})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} = 0  (2.15)

for w_{a_{1}} ∈ W^{a_{1}} and w_{a_{2}} ∈ W^{a_{2}}. Since the coefficients of Y_{2}(w_{a_{1}}, x_{0})w_{a_{2}} for w_{a_{1}} ∈ W^{a_{1}} and w_{a_{2}} ∈ W^{a_{2}} span W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}}, we obtain from (2.15) that

∑_{a_{4}∈A} Tr_{W^{a_{4}}}Y_{a_{4}}(U(e^{2πiz_{2}})w, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} = 0  (2.16)

for w ∈ W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}} and w_{a_{3}} ∈ W^{a_{3}}.

Since W^{a_{4}} for a_{4} ∈ A are irreducible V -modules, we have T (W^{a_{4}}) = W_{(h_{a_{4}})}^{a_{4}}, where

T (W^{a_{4}}) = {w ∈ W^{a_{4}} | u_{n}w = 0, u ∈ V, wt u − n − 1 < 0}

(see [H6]). Since τ ∈ ℍ is arbitrary, from (2.16), we have

∑_{a_{4}∈A} Tr_{T (W^{a_{4}})}o_{Y_{a_{4}}}(U(1)w) = 0  (2.17)

for w ∈ W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}}, where

o_{Y_{a_{4}}}(w̃) = (Y_{a_{4}})_{wt ŵ−1}(ŵ)

for homogeneous w̃ ∈ W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}} (see Chapter 6 of [H6] for more details). Since W^{a_{4}} for a_{4} ∈ A are inequivalent irreducible V -modules, T (W^{a_{4}}) for a_{4} ∈ A are inequivalent irreducible Ã(V)-modules by Proposition 6.5 in [H6]. Thus Tr_{T (W^{a_{4}})} for a_{4} ∈ A are linearly independent. So from (2.17), we obtain

o_{Y_{a_{4}}}(U(1)w) = 0  (2.18)

for w ∈ W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}}. Thus ρ(Y_{a_{4}}) = 0 where

ρ(Y_{a_{4}}) : Ã(W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}}) ⊗_{Ã(V)} T (W^{a_{4}}) → T (W^{a_{4}})

is given by

ρ(Y_{a_{4}})((w + Õ(W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}})) ⊗ w_{a_{4}}) = o_{Y_{a_{4}}}(U(1)w)w_{a_{4}}

for w ∈ W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}} and w_{a_{4}} ∈ T (W^{a_{4}}) (see chapter 6 of [H6]). By Theorem 6.9 in [H6], ρ(Y_{a_{4}}) = 0 is equivalent to Y_{a_{4}} = 0 for a_{4} ∈ A, that is

Y_{a_{4}}(w, x) = 0  (2.19)

for a_{4} ∈ A and w ∈ W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}}.

For a_{3}, a_{4} ∈ A and l = 1, …, N_{a_{1}a_{2}}^{a_{3}}, take w to be an element of the tensor product module W^{a_{1}} ⊠_{P (z_{1}−z_{2})} W^{a_{2}} such that f(w) is in the l-th copy of W^{a_{3}} in ⊕_{a_{3}∈A}N_{a_{1}a_{2}}^{a_{3}}W^{a_{3}}, that is, π_{a_{3};l}(f(w)) = f(w), π_{a_{3};m}(f(w)) = 0 for m ≠ l and π_{a;m}(f(w)) = 0 for a ≠ a_{3}. Then by (2.19) and the definition of the intertwining operator Y_{a_{4}}, we have

∑_{k=1}^{N_{a_{3}a_{4}}^{a_{4}}} λ_{a_{3},a_{4},k,l}Y_{a_{3}a_{4};k}^{a_{4};(3)}(f(w), x) = 0.

Since f(w) is an arbitrary element of the l-th copy of W^{a_{3}} in ⊕_{a_{3}∈A}N_{a_{1}a_{2}}^{a_{3}}W^{a_{3}}, we obtain

∑_{k=1}^{N_{a_{3}a_{4}}^{a_{4}}} λ_{a_{3},a_{4},k,l}Y_{a_{3}a_{4};k}^{a_{4};(3)} = 0

for a_{3}, a_{4} ∈ A and l = 1, …, N_{a_{1}a_{2}}^{a_{3}}. Since Y_{a_{3}a_{4};k}^{a_{4};(3)} for k = 1, …, N_{a_{3}a_{4}}^{a_{4}} are linearly independent, we obtain λ_{a_{3},a_{4},k,l} = 0 for a_{3}, a_{4} ∈ A, k = 1, …, N_{a_{3}a_{4}}^{a_{4}} and l = 1, …, N_{a_{1}a_{2}}^{a_{3}}.

We now introduce a space F_{1;2} spanned by linear maps of the form

Ψ_{a_{1},a_{2},a_{3}}^{k,l} : ∐_{a∈A} W^{a} ⊗ W^{a′} → 𝔾_{1;2}
w_{a} ⊗ w_{a′} ↦ Ψ_{a_{1},a_{2},a_{3}}(w_{a}, w_{a′}; z_{1}, z_{2}; τ)

for a_{1}, a_{2}, a_{3} ∈ A, k = 1, …, N_{a_{3}a_{1}}^{a_{1}}, l = 1, …, N_{a_{2}a′_{2}}^{a_{3}}, where

Ψ_{a_{1},a_{2},a_{3}}^{k,l}(w_{a}, w_{a′}; z_{1}, z_{2}; τ) = 0

when a ≠ a_{2} and

Ψ_{a_{1},a_{2},a_{3}}^{k,l}(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2}; τ)

= E(Tr_{W^{a_{1}}}Y_{a_{3}a_{1};k}^{a_{1};(1)}(U(e^{2πiz_{2}})Y_{a_{2}a′_{2};l}^{a_{3};(2)}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ).

Let F_{1;2}^{e} be the subspace of F_{1;2} spanned by maps of the form Ψ_{a_{1},a_{2},e}^{1,1} for a_{1}, a_{2} ∈ A and let F_{1;2}^{ne} be the subspace of F_{1;2} spanned by maps of the form Ψ_{a_{1},a_{2},a_{3}}^{k,l} for a_{1}, a_{2}, a_{3} ∈ A, a_{3} ≠ e, k = 1, …, N_{a_{3}a_{1}}^{a_{1}} and l = 1, …, N_{a_{2}a′_{2}}^{a_{3}}. We have:

Proposition 2.3 The intersection of F_{1;2}^{e} and F_{1;2}^{ne} is 0. In particular,

F_{1;2} = F_{1;2}^{e} ⊕ F_{1;2}^{ne}

and there exist a projection π : F_{1;2} → F_{1;2}^{e}.

Proof. By Proposition 2.2, Ψ_{a_{1},a_{2},a_{3}}^{k,l} for a_{1}, a_{2}, a_{3} ∈ A, k = 1, …, N_{a_{3}a_{1}}^{a_{1}} and l = 1, …, N_{a_{2}a′_{2}}^{a_{3}} are linearly independent. Thus the intersection of the space spanned by Ψ_{a_{1},a_{2},e}^{1,1} for a_{1}, a_{2} ∈ A and the space spanned by Ψ_{a_{1},a_{2},a_{3}}^{k,l} for a_{1}, a_{2}, a_{3} ∈ A, a_{3} ≠ e, k = 1, …, N_{a_{3}a_{1}}^{a_{1}} and l = 1, …, N_{a_{2}a′_{2}}^{a_{3}} are 0

We define S : F_{1;1} → F_{1;1} as follows: For a ∈ A, let

(S(Ψ_{a}))(u; τ) = Ψ_{a} ((−\frac{1}{τ})^{L(0)} u; −\frac{1}{τ})

= Tr_{W^{a}}Y_{ea;1}^{a;(1)} (U(e^{−2πi \frac{z_{2}}{τ}}) (−\frac{1}{τ})^{L(0)} u, e^{−2πi \frac{z_{2}}{τ}} ) q_{− \frac{1}{τ}}^{L(0)− \frac{c}{24}} .

Here we have used our convention that

(−\frac{1}{τ})^{L(0)} = e^{(log(− \frac{1}{τ}))L(0)}.

Note that by the modular invariance of genus-one one-point functions proved in [M] and [H6], S(Ψ_{a}) is indeed in F_{1;1}. Thus we do obtain maps S : F_{1;1} → F_{1;1}.

Now we define an action of the map S on the space F_{1;2}^{e} by

(S(Ψ_{a_{1},a_{2},e}^{1,1}))(w_{a}, w_{a′}; z_{1}, z_{2}; τ) = 0

when a ≠ a_{2} and

(S(Ψ_{a_{1},a_{2},e}^{1,1}))(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2}; τ)

= E(Tr_{W^{a_{1}}}Y_{ea_{1};1}^{a_{1};(1)}(U(e^{−2πi \frac{z_{2}}{τ}}) (−\frac{1}{τ})^{L(0)} ·

·Y_{a_{2}a′_{2};1}^{e;(2)}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{−2πi \frac{z_{2}}{τ}} )q_{− \frac{1}{τ}}^{L(0)− \frac{c}{24}} )

= E(Tr_{W^{a_{1}}}Y_{ea_{1};1}^{a_{1};(1)}(U(e^{−2πi \frac{z_{2}}{τ}}) ·

·Y_{a_{2}a′_{2};1}^{e;(2)} ((−\frac{1}{τ})^{L(0)} w_{a_{2}}, −\frac{1}{τ}z_{1} − (−\frac{1}{τ}z_{2}) ) (−\frac{1}{τ})^{L(0)} w_{a′_{2}}, e^{−2πi \frac{z_{2}}{τ}} ) ·

·q_{− \frac{1}{τ}}^{L(0)− \frac{c}{24}} )

= Ψ_{a_{1},a_{2},e}^{1,1} ((−\frac{1}{τ})^{L(0)} w_{a}, (−\frac{1}{τ})^{L(0)} w_{a′}; −\frac{1}{τ}z_{1}, −\frac{1}{τ}z_{2}; −\frac{1}{τ}) .

We shall also need the following result:

Proposition 2.4 For a ∈ A, u ∈ V, we have

Tr_{W^{a}}Y_{W^{a}}(U(e^{2πiz})u, e^{2πiz})q_{τ}^{L(0)− \frac{c}{24}}
= Tr_{W^{a′_{4}}}Y_{W^{a′}}(U(e^{−2πiz})e^{πiL(0)}u, e^{−2πiz})q_{τ}^{L(0)− \frac{c}{24}} ).  (2.20)

For a_{3}, a_{4} ∈ A, i = 1, …, N_{a_{1}a_{3}}^{a_{4}}, j = 1, …, N_{a_{2}a_{4}}^{a_{3}}, w_{a_{1}} ∈ W^{a_{1}}, w_{a_{2}} ∈ W^{a_{2}}, we have

E(Tr_{W^{a_{4}}}σ_{23}(Y_{a_{1}a′_{4};i}^{a′_{3};(1)})(U(e^{2πiz_{1}})w_{a_{1}}, e^{2πiz_{1}})·

·σ_{23}(Y_{a_{2}a′_{3};j}^{a′_{4};(2)})(U(e^{2πiz_{2}})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )
= e^{−πi(h_{a_{1}}+h_{a_{2}})}E(Tr_{W^{a′_{4}}}Y_{a_{2}a′_{3};j}^{a′_{4};(2)}(U(e^{−2πiz_{2}})e^{πiL(0)}w_{a_{2}}, e^{−2πiz_{2}}) ·

·Y_{a_{1}a′_{4};i}^{a′_{3};(1)}(U(e^{−2πiz_{1}})e^{πiL(0)}w_{a_{1}}, e^{−2πiz_{1}})q_{τ}^{L(0)− \frac{c}{24}} )

(2.21)

and

E(Tr_{W^{a′_{4}}}σ_{23}(Y_{a_{3}a_{4};k}^{a_{4};(3)})(U(e^{2πiz_{2}})Y_{a_{1}a_{2};l}^{a_{3};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

= e^{πih_{a_{3}}}E(Tr_{W^{a_{4}}}Y_{a_{3}a_{4};k}^{a_{4};(3)}(U(e^{2πiz_{2}})e^{−πiL(0)} ·

·Y_{a_{1}a_{2};l}^{a_{3};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ). (2.22)

Proof. These formulas follows immediately from (2.2).

3 Properties of fusing and braiding matrices

In the present section, we prove some properties of the fusing and braiding matrices. These properties play important role in the proofs of Moore-Seiberg formulas in the next section and in the proof of the symmetry of the matrix associated to the modular transformation τ ↦ −1/τ in Section 5.

In this section, for p = 1, 1, 3, 4, 5, 6 and a_{1}, a_{2}, a_{3} ∈ A, Y_{a_{1}a_{2};i}^{a_{3};(p)}, i = 1, …, N_{a_{1}a_{2}}^{a_{3}}, are basis of V_{a_{1}a_{2}}^{a_{3}}.

Proposition 3.1 The following equality expressing the squares of braiding matrices in terms of the fusing matrices and the inverses of fusing matrices holds:

∑_{a_{7}∈A} ∑_{k=1}^{N_{a_{1}a_{2}}^{a_{7}}} ∑_{l=1}^{N_{a_{7}a_{3}}^{a_{4}}} F (Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{7}a_{3};l}^{a_{4};(5)} ⊗ Y_{a_{1}a_{2};k}^{a_{7};(6)})·

·e^{2(2r+1)πi(h_{a_{7}}−h_{a_{1}}−h_{a_{2}})}F^{−1}(Y_{a_{7}a_{3};l}^{a_{4};(5)} ⊗ Y_{a_{1}a_{2};k}^{a_{7};(6)}; Y_{a_{1}a_{6};p}^{a_{4};(3)} ⊗ Y_{a_{2}a_{3};q}^{a_{6};(4)})

= (B^{(r)})^{2}(Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{1}a_{6};p}^{a_{4};(3)} ⊗ Y_{a_{2}a_{3};q}^{a_{6};(4)}).  (3.1)

Proof. Let w_{a_{1}} ∈ W^{a_{1}}, w_{a_{2}} ∈ W^{a_{2}}, w_{a_{3}} ∈ W^{a_{3}} and w_{a′_{4}} ∈ W^{a′_{4}}. Then we have

E(⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩)

= ∑_{a_{7}∈A} ∑_{k=1}^{N_{a_{1}a_{2}}^{a_{7}}} ∑_{l=1}^{N_{a_{7}a_{3}}^{a_{4}}} F (Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{7}a_{3};l}^{a_{4};(5)} ⊗ Y_{a_{1}a_{2};k}^{a_{7};(6)}) ·

·E(⟨w_{a′_{4}}, Y_{a_{7}a_{3};l}^{a_{4};(5)}(Y_{a_{1}a_{2};k}^{a_{7};(6)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩).

Applying (B^{(r)})^{2} to both sides of the above formula, we obtain

(B^{(r)})^{2}(E(⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩))

= ∑_{a_{7}∈A} ∑_{k=1}^{N_{a_{1}a_{2}}^{a_{7}}} ∑_{l=1}^{N_{a_{7}a_{3}}^{a_{4}}} F (Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{7}a_{3};l}^{a_{4};(5)} ⊗ Y_{a_{1}a_{2};k}^{a_{7};(6)}) ·

·(B^{(r)})^{2}(E(⟨w_{a′_{4}}, Y_{a_{7}a_{3};l}^{a_{4};(5)}(Y_{a_{1}a_{2};k}^{a_{7};(6)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩)).

(3.2)

In Section 1, we have seen that (B^{(r)})^{2} is the monodromy given by

log(z_{1} − z_{2}) ↦ log(z_{1} − z_{2}) + 2(2r + 1)πi.

Using this fact and

Y_{a_{1}a_{2};k}^{a_{7};(6)}(w_{a_{1}}, x)|_{x^{n}=e^{n(log(z_{1}−z_{2})+2(2r+1)πi)}, n∈ℂ}

= e^{2(2r+1)πi(h_{a_{7}}−h_{a_{1}}−h_{a_{2}})}Y_{a_{1}a_{2};k}^{a_{7};(6)}(w_{a_{1}}, x)|_{x^{n}=e^{n log(z_{1}−z_{2})}, n∈ℂ} ,

we have (B^{(r)})^{2}(E(⟨w_{a′_{4}}, Y_{a_{7}a_{3};l}^{a_{4};(5)}(Y_{a_{1}a_{2};k}^{a_{7};(6)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩))

= e^{2(2r+1)πi(h_{a_{7}}−h_{a_{1}}−h_{a_{2}})}E(⟨w_{a′_{4}}, Y_{a_{7}a_{3};l}^{a_{4};(5)}(Y_{a_{1}a_{2};k}^{a_{7};(6)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩).

(3.3)

Using (3.2), (3.3) and the associativity (1.6) expressed in terms of the matrix elements of the inverse of the fusing isomorphism, we obtain

(B^{(r)})^{2}(E(⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩))

= ∑_{a_{7}∈A} ∑_{k=1}^{N_{a_{1}a_{2}}^{a_{7}}} ∑_{l=1}^{N_{a_{7}a_{3}}^{a_{4}}} F (Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{7}a_{3};l}^{a_{4};(5)} ⊗ Y_{a_{1}a_{2};k}^{a_{7};(6)}) ·

·e^{2(2r+1)πi(h_{a_{7}}−h_{a_{1}}−h_{a_{2}})} ·

·E(⟨w_{a′_{4}}, Y_{a_{7}a_{3};l}^{a_{4};(5)}(Y_{a_{1}a_{2};k}^{a_{7};(6)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩)

= ∑_{a_{7}∈A} ∑_{k=1}^{N_{a_{1}a_{2}}^{a_{7}}} ∑_{l=1}^{N_{a_{7}a_{3}}^{a_{4}}} F (Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{7}a_{3};l}^{a_{4};(5)} ⊗ Y_{a_{1}a_{2};k}^{a_{7};(6)}) ·

·e^{2(2r+1)πi(h_{a_{7}}−h_{a_{1}}−h_{a_{2}})} ·

· ∑_{a_{6}∈A} ∑_{p=1}^{N_{a_{1}a_{6}}^{a_{4}}} ∑_{q=1}^{N_{a_{2}a_{3}}^{a_{6}}} F^{−1}(Y_{a_{7}a_{3};l}^{a_{4};(5)} ⊗ Y_{a_{1}a_{2};k}^{a_{7};(6)}; Y_{a_{1}a_{6};p}^{a_{4};(3)} ⊗ Y_{a_{2}a_{3};q}^{a_{6};(4)}) ·

·E(⟨w_{a′_{4}}, Y_{a_{1}a_{6};p}^{a_{4};(3)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};q}^{a_{6};(4)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩).  (3.4)

Comparing (3.4) with the definition (1.15) of the matrix elements of (B^{(r)})^{2} and using Proposition 1.2, we obtain (3.1).

Proposition 3.2 The following equality expressing the inverses of the fusing matrices in terms of the fusing matrices holds:

F^{−1}(Y_{a_{6}a_{3};l}^{a_{4};(1)} ⊗ Y_{a_{1}a_{2};k}^{a_{6};(2)}; Y_{a_{1}a_{5};i}^{a_{4};(3)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(4)})
= F (σ_{12}(Y_{a_{6}a_{3};l}^{a_{4};(1)}) ⊗ σ_{12}(Y_{a_{1}a_{2};k}^{a_{6};(2)}); σ_{12}(Y_{a_{1}a_{5};i}^{a_{4};(3)}) ⊗ σ_{12}(Y_{a_{2}a_{3};j}^{a_{5};(4)})). (3.5)

Proof. Let w_{a_{1}} ∈ W^{a_{1}}, w_{a_{2}} ∈ W^{a_{2}}, w_{a_{3}} ∈ W^{a_{3}} and w_{a′_{4}} ∈ W^{a′_{4}}. Then using the definition of σ_{12}, the relation σ_{12}^{2} = 1, the associativity (1.5) and the L(−1)-conjugation property (see [FHL]), we have

E(⟨w_{a′_{4}}, Y_{a_{6}a_{3};l}^{a_{4};(1)}(Y_{a_{1}a_{2};k}^{a_{6};(2)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩)

= E(⟨w_{a′_{4}}, σ_{12}^{2}(Y_{a_{6}a_{3};l}^{a_{4};(1)})(σ_{12}^{2}(Y_{a_{1}a_{2};k}^{a_{6};(2)})(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩)
= e^{πi(h_{a_{4}}−h_{a_{6}}−h_{a_{3}})}e^{πi(h_{a_{6}}−h_{a_{1}}−h_{a_{2}})} ·
·E(⟨w_{a′_{4}}, e^{z_{2}L(−1)}σ_{12}(Y_{a_{6}a_{3};l}^{a_{4};(1)})(w_{a_{3}}, e^{−πi}z_{2}) ·

·e^{(z_{1}−z_{2})L(−1)}σ_{12}(Y_{a_{1}a_{2};k}^{a_{6};(2)})(w_{a_{2}}, e^{−πi}(z_{1} − z_{2}))w_{a_{1}}⟩)

= e^{πi(h_{a_{4}}−h_{a_{1}}−h_{a_{2}}−h_{a_{3}})}E(⟨w_{a′_{4}}, e^{z_{1}L(−1)}σ_{12}(Y_{a_{6}a_{3};l}^{a_{4};(1)})(w_{a_{3}}, e^{−πi}z_{1}) ·

·σ_{12}(Y_{a_{1}a_{2};k}^{a_{6};(2)})(w_{a_{2}}, e^{−πi}(z_{1} − z_{2}))w_{a_{1}}⟩)

= e^{πi(h_{a_{4}}−h_{a_{1}}−h_{a_{2}}−h_{a_{3}})} ∑_{a_{5}∈A} ∑_{i=1}^{N_{a_{1}a_{5}}^{a_{4}}} ∑_{j=1}^{N_{a_{2}a_{3}}^{a_{5}}}

·F (σ_{12}(Y_{a_{6}a_{3};l}^{a_{4};(1)}) ⊗ σ_{12}(Y_{a_{1}a_{2};k}^{a_{6};(2)}); σ_{12}(Y_{a_{1}a_{5};i}^{a_{4};(3)}) ⊗ σ_{12}(Y_{a_{2}a_{3};j}^{a_{5};(4)})) ·

·E(⟨w_{a′_{4}}, e^{z_{1}L(−1)}σ_{12}(Y_{a_{1}a_{5};i}^{a_{4};(3)})(σ_{12}(Y_{a_{2}a_{3};j}^{a_{5};(4)})(w_{a_{3}}, e^{−πi}z_{2}) ·

·w_{a_{2}}, e^{−πi}(z_{1} − z_{2}))w_{a_{1}}⟩)

= e^{πi(h_{a_{4}}−h_{a_{1}}−h_{a_{5}})}e^{πi(h_{a_{5}}−h_{a_{2}}−h_{a_{3}})} ∑_{a_{5}∈A} ∑_{i=1}^{N_{a_{1}a_{5}}^{a_{4}}} ∑_{j=1}^{N_{a_{2}a_{3}}^{a_{5}}}

·F (σ_{12}(Y_{a_{6}a_{3};l}^{a_{4};(1)}) ⊗ σ_{12}(Y_{a_{1}a_{2};k}^{a_{6};(2)}); σ_{12}(Y_{a_{1}a_{5};i}^{a_{4};(3)}) ⊗ σ_{12}(Y_{a_{2}a_{3};j}^{a_{5};(4)})) ·

·E(⟨w_{a′_{4}}, e^{z_{1}L(−1)}σ_{12}(Y_{a_{1}a_{5};i}^{a_{4};(3)})(e^{z_{2}L(−1)} ·

·σ_{12}(Y_{a_{2}a_{3};j}^{a_{5};(4)})(w_{a_{3}}, e^{−πi}z_{2})w_{a_{2}}, e^{−πi}z_{1})w_{a_{1}}⟩)

= ∑_{a_{5}∈A} ∑_{i=1}^{N_{a_{1}a_{5}}^{a_{4}}} ∑_{j=1}^{N_{a_{2}a_{3}}^{a_{5}}}

F (σ_{12}(Y_{a_{6}a_{3};l}^{a_{4};(1)}) ⊗ σ_{12}(Y_{a_{1}a_{2};k}^{a_{6};(2)}); σ_{12}(Y_{a_{1}a_{5};i}^{a_{4};(3)}) ⊗ σ_{12}(Y_{a_{2}a_{3};j}^{a_{5};(4)})) ·

·E(⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(3)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(4)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩).  (3.6)

Comparing (3.6) with (1.14) and using Proposition 1.2, we obtain (3.5).

In the proof of the next property of fusing matrices, we need the following lemma:

Lemma 3.3 For any z_{1}, z_{2} ∈ ℂ^{×} satisfying z_{1} ≠ z_{2} and any V -module W, the following equalities for maps from W to \overline{W} holds:

e^{z_{2}L(1)}e^{z_{1}^{−1}L(−1)} = e^{(z_{1}−z_{2})^{−1}L(−1)}e^{z_{1}^{−1}z_{2}(z_{1}−z_{2})L(1)}(z_{1}(z_{1} − z_{2})^{−1})^{2L(0)}, (3.7)

e^{z_{1}^{−1}z_{2}(z_{1}−z_{2})L(1)}(z_{1}(z_{1} − z_{2})^{−1})^{2L(0)}e^{z_{1}L(1)}(e^{−πi}z_{1}^{−2})^{L(0)}

= e^{(z_{1}−z_{2})L(1)}(e^{−πi}(z_{1} − z_{2})^{−2})^{L(0)}.  (3.8)

Proof. From the identity

\frac{1}{x^{−1} + z_{2}} − z_{1}^{−1} = (z_{1}(z_{1} − z_{2})^{−1})^{−2} (\frac{1}{(x − (z_{1} − z_{2})^{−1})^{−1} + z_{1}^{−1}z_{2}(z_{1} − z_{2})}) ,

we obtain

e^{−z_{2}x^{2} \frac{d}{dx}} e^{−z_{1}^{−1} \frac{d}{dx}} x = e^{−(z_{1}−z_{2})^{−1} \frac{d}{dx}} e^{−z_{1}^{−1}z_{2}(z_{1}−z_{2})x^{2} \frac{d}{dx}} (z_{1}(z_{1}−z_{2})^{−1})^{−2x \frac{d}{dx}} x. (3.9)

Using (3.9) and the theory developed in Chapters 4 and 5 of [H3], we obtain (3.7).

To prove (3.8), we note that the weight of L(1) is −1. So we have

(z_{1}(z_{1} − z_{2})^{−1})^{2L(0)}L(1)(z_{1}(z_{1} − z_{2})^{−1})^{−2L(0)} = z_{1}^{−2}(z_{1} − z_{2})^{2}L(1). (3.10)

Using (3.10), we obtain

(z_{1}(z_{1} − z_{2})^{−1})^{2L(0)}e^{z_{1}L(1)} = (z_{1}(z_{1} − z_{2})^{−1})^{2L(0)}e^{z_{1}L(1)}(z_{1}(z_{1} − z_{2})^{−1})^{−2L(0)}(z_{1}(z_{1} − z_{2})^{−1})^{2L(0)}

= e^{z_{1}(z_{1}(z_{1}−z_{2})^{−1})^{2L(0)}L(1)(z_{1}(z_{1}−z_{2})^{−1})^{−2L(0)}}(z_{1}(z_{1} − z_{2})^{−1})^{2L(0)}
= e^{z_{1}^{−1}(z_{1}−z_{2})^{2}L(1)}(z_{1}(z_{1} − z_{2})^{−1})^{2L(0)}.  (3.11)

The equality (3.8) follows immediately from (3.11).

Let σ_{123} = σ_{12}σ_{23} and σ_{132} = σ_{23}σ_{12}. We have:

Proposition 3.4 The following equality between fusing matrices holds:

F (Y_{a_{1}a_{5};i}^{a_{4};(1)} ⊗ Y_{a_{2}a_{3};j}^{a_{5};(2)}; Y_{a_{6}a_{3};l}^{a_{4};(3)} ⊗ Y_{a_{1}a_{2};k}^{a_{6};(4)})
= F (σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)}) ⊗ σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)}); σ_{123}(Y_{a_{1}a_{2};k}^{a_{6};(4)}) ⊗ σ_{132}(Y_{a_{6}a_{3};l}^{a_{4};(3)})).

(3.12)

Proof. Let w_{a_{1}} ∈ W^{a_{1}}, w_{a_{2}} ∈ W^{a_{2}}, w_{a_{3}} ∈ W^{a_{3}} and w_{a′_{4}} ∈ W^{a′_{4}}. Then using the definitions of σ_{12} and σ_{23} and the relations σ_{12}^{2} = σ_{23}^{2} = 1, we obtain

E(⟨w_{a′_{4}}, Y_{a_{1}a_{5};i}^{a_{4};(1)}(w_{a_{1}}, z_{1})Y_{a_{2}a_{3};j}^{a_{5};(2)}(w_{a_{2}}, z_{2})w_{a_{3}}⟩)

= E(⟨w_{a′_{4}}, σ_{23}^{2}(Y_{a_{1}a_{5};i}^{a_{4};(1)})(w_{a_{1}}, z_{1})σ_{12}^{2}(Y_{a_{2}a_{3};j}^{a_{5};(2)})(w_{a_{2}}, z_{2})w_{a_{3}}⟩)
= e^{πi(h_{a_{1}}+h_{a_{5}}−h_{a_{2}}−h_{a_{3}})} ·
·E(⟨σ_{23}(Y_{a_{1}a_{5};i}^{a_{4};(1)})(e^{z_{1}L(1)}(e^{−πi}z_{1}^{−2})^{L(0)}w_{a_{1}}, z_{1}^{−1})w_{a′_{4}},

e^{z_{2}L(−1)}σ_{12}(Y_{a_{2}a_{3};j}^{a_{5};(2)})(w_{a_{3}}, e^{−πi}z_{2})w_{a_{2}}⟩)

= e^{πi(h_{a_{1}}+h_{a_{5}}−h_{a_{2}}−h_{a_{3}})} ·
·E(⟨σ_{12}^{2}(σ_{23}(Y_{a_{1}a_{5};i}^{a_{4};(1)}))(e^{z_{1}L(1)}(e^{−πi}z_{1}^{−2})^{L(0)}w_{a_{1}}, z_{1}^{−1})w_{a′_{4}},

e^{z_{2}L(−1)}σ_{23}^{2}(σ_{12}(Y_{a_{2}a_{3};j}^{a_{5};(2)}))(w_{a_{3}}, e^{−πi}z_{2})w_{a_{2}}⟩)

= e^{πi(2h_{a_{5}}−h_{a_{2}}−h_{a_{4}})} ·

·E(⟨σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)})(e^{−z_{2}L(1)}(e^{−πi}z_{2}^{−2})^{L(0)}w_{a_{3}}, e^{πi}z_{2}^{−1})e^{z_{2}L(1)}e^{z_{1}^{−1}L(−1)} ·

·σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)})(w_{a′_{4}}, e^{−πi}z_{1}^{−1})e^{z_{1}L(1)}(e^{−πi}z_{1}^{−2})^{L(0)}w_{a_{1}}, w_{a_{2}}⟩). (3.13)

By (3.7), (3.8), the L(−1)-, L(0)- and L(1)-conjugation formulas (see [FHL]), (1.1) and the associativity, the right-hand side of (4.5) is equal to

e^{πi(2h_{a_{5}}−h_{a_{2}}−h_{a_{4}})}· ·E(⟨σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)})(e^{−z_{2}L(1)}(e^{−πi}z_{2}^{−2})^{L(0)}w_{a_{3}}, e^{πi}z_{2}^{−1}) ·

·e^{(z_{1}−z_{2})^{−1}L(−1)}e^{z_{1}^{−1}z_{2}(z_{1}−z_{2})L(1)}(z_{1}(z_{1} − z_{2})^{−1})^{2L(0)} ·

·σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)})(w_{a′_{4}}, e^{−πi}z_{1}^{−1})e^{z_{1}L(1)}(e^{−πi}z_{1}^{−2})^{L(0)}w_{a_{1}}, w_{a_{2}}⟩)

= e^{πi(2h_{a_{5}}−h_{a_{2}}−h_{a_{4}})} · ·E(⟨e^{(z_{1}−z_{2})^{−1}L(−1)}σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)})(e^{−z_{2}L(1)}(e^{−πi}z_{2}^{−2})^{L(0)} ·

·w_{a_{3}}, e^{πi}z_{1}z_{2}^{−1}(z_{1} − z_{2}))e^{z_{1}^{−1}z_{2}(z_{1}−z_{2})L(1)} ·

·σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)})((z_{1}(z_{1} − z_{2})^{−1})^{2L(0)}w_{a′_{4}}, e^{−πi}z_{1}^{−1}(z_{1} − z_{2})^{−2}) ·

·(z_{1}(z_{1} − z_{2})^{−1})^{2L(0)}e^{z_{1}L(1)}(e^{−πi}z_{1}^{−2})^{L(0)}w_{a_{1}}, w_{a_{2}}⟩)

= e^{πi(2h_{a_{5}}−h_{a_{2}}−h_{a_{4}})} · ·E(⟨e^{(z_{1}−z_{2})^{−1}L(−1)}σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)})(e^{−z_{2}L(1)}(e^{−πi}z_{2}^{−2})^{L(0)} ·

·w_{a_{3}}, e^{πi}z_{1}z_{2}^{−1}(z_{1} − z_{2}))σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)})(e^{z_{2}L(1)}w_{a′_{4}}, e^{−πi}(z_{1} − z_{2})^{−1}) ·

·e^{z_{1}^{−1}z_{2}(z_{1}−z_{2})L(1)}(z_{1}(z_{1} − z_{2})^{−1})^{2L(0)}e^{z_{1}L(1)}(e^{−πi}z_{1}^{−2})^{L(0)}w_{a_{1}}, w_{a_{2}}⟩) = e^{πi(2h_{a_{5}}−h_{a_{2}}−h_{a_{4}})} ·

·E(⟨e^{(z_{1}−z_{2})^{−1}L(−1)}σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)})(e^{−z_{2}L(1)}(e^{−πi}z_{2}^{−2})^{L(0)} ·

·w_{a_{3}}, e^{2πi}e^{−πi}z_{1}z_{2}^{−1}(z_{1} − z_{2})) ·

·σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)})(e^{z_{2}L(1)}w_{a′_{4}}, e^{−πi}(z_{1} − z_{2})^{−1}) ·

·e^{(z_{1}−z_{2})L(1)}(e^{−πi}(z_{1} − z_{2})^{−2})^{L(0)}w_{a_{1}}, w_{a_{2}}⟩) = e^{πi(h_{a_{2}}−2h_{a_{3}}−h_{a_{4}})} ·

·E(⟨e^{(z_{1}−z_{2})^{−1}L(−1)}σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)})(e^{−z_{2}L(1)}(e^{−πi}z_{2}^{−2})^{L(0)} ·

·w_{a_{3}}, e^{−πi}z_{1}z_{2}^{−1}(z_{1} − z_{2}))σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)})(e^{z_{2}L(1)}w_{a′_{4}}, e^{−πi}(z_{1} − z_{2})^{−1}) ·

·e^{(z_{1}−z_{2})L(1)}(e^{−πi}(z_{1} − z_{2})^{−2})^{L(0)}w_{a_{1}}, w_{a_{2}}⟩)

= e^{πi(h_{a_{2}}−2h_{a_{3}}−h_{a_{4}})} ∑_{a_{6}∈A} ∑_{k=1}^{N_{a_{6}a_{3}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{2}}^{a_{6}}}

F (σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)}) ⊗ σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)}); σ_{123}(Y_{a_{1}a_{2};l}^{a_{6};(4)}) ⊗ σ_{132}(Y_{a_{6}a_{3};j}^{a_{4};(3)})) ·

·E(⟨e^{(z_{1}−z_{2})^{−1}L(−1)}σ_{123}(Y_{a_{1}a_{2};j}^{a_{6};(4)})(σ_{132}(Y_{a_{6}a_{3};i}^{a_{4};(3)})(e^{−z_{2}L(1)}(e^{−πi}z_{2}^{−2})^{L(0)} ·

·w_{a_{3}}, e^{−πi}z_{2}^{−1})e^{z_{2}L(1)}w_{a′_{4}}, e^{−πi}(z_{1} − z_{2})^{−1}) ·
·e^{(z_{1}−z_{2})L(1)}(e^{−πi}(z_{1} − z_{2})^{−2})^{L(0)}w_{a_{1}}, w_{a_{2}}⟩).  (3.14)

Using the definitions of σ_{12} and σ_{23}, the relations σ_{12}^{2} = σ_{23}^{2} = 1, σ_{12}σ_{123} = σ_{23}, σ_{23}σ_{132} = σ_{12}, we see that the right-hand side of (3.14) is equal to

e^{πi(h_{a_{6}}+h_{a_{1}}−2h_{a_{3}}−h_{a_{4}})} ∑_{a_{6}∈A} ∑_{k=1}^{N_{a_{6}a_{3}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{2}}^{a_{6}}}

F (σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)}) ⊗ σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)}); σ_{123}(Y_{a_{1}a_{2};l}^{a_{6};(4)}) ⊗ σ_{132}(Y_{a_{6}a_{3};j}^{a_{4};(3)})) ·

·E(⟨σ_{12}(σ_{123}(Y_{a_{1}a_{2};j}^{a_{6};(4)}))(e^{(z_{1}−z_{2})L(1)}(e^{−πi}(z_{1} − z_{2})^{−2})^{L(0)} ·

·w_{a_{1}}, (z_{1} − z_{2})^{−1})σ_{132}(Y_{a_{6}a_{3};i}^{a_{4};(3)})(e^{−z_{2}L(1)}(e^{−πi}z_{2}^{−2})^{L(0)}w_{a_{3}}, e^{−2πi}e^{πi}z_{2}^{−1}) ·

·e^{z_{2}L(1)}w_{a′_{4}}, w_{a_{2}}⟩)

= e^{πi(h_{a_{1}}+h_{a_{4}}−h_{a_{6}})} ∑_{a_{6}∈A} ∑_{k=1}^{N_{a_{6}a_{3}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{2}}^{a_{6}}}

F (σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)}) ⊗ σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)}); σ_{123}(Y_{a_{1}a_{2};l}^{a_{6};(4)}) ⊗ σ_{132}(Y_{a_{6}a_{3};j}^{a_{4};(3)})) ·

·E(⟨σ_{23}(Y_{a_{1}a_{2};j}^{a_{6};(4)})(e^{(z_{1}−z_{2})L(1)}(e^{−πi}(z_{1} − z_{2})^{−2})^{L(0)}w_{a_{1}}, (z_{1} − z_{2})^{−1}) ·

·σ_{132}(Y_{a_{6}a_{3};i}^{a_{4};(3)})(e^{−z_{2}L(1)}(e^{−πi}z_{2}^{−2})^{L(0)}w_{a_{3}}, e^{πi}z_{2}^{−1})e^{z_{2}L(1)}w_{a′_{4}}, w_{a_{2}}⟩)

= e^{πi(h_{a_{4}}−h_{a_{3}}−h_{a_{6}})} ∑_{a_{6}∈A} ∑_{k=1}^{N_{a_{6}a_{3}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{2}}^{a_{6}}}

F (σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)}) ⊗ σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)}); σ_{123}(Y_{a_{1}a_{2};l}^{a_{6};(4)}) ⊗ σ_{132}(Y_{a_{6}a_{3};j}^{a_{4};(3)})) ·

·E(⟨w_{a′_{4}}, e^{z_{2}L(−1)}σ_{23}(σ_{132}(Y_{a_{6}a_{3};i}^{a_{4};(3)}))(w_{a_{3}}, e^{−πi}z_{2}) ·
·σ_{23}^{2}(Y_{a_{1}a_{2};j}^{a_{6};(4)})(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}⟩)

= e^{πi(h_{a_{4}}−h_{a_{3}}−h_{a_{6}})} ∑_{a_{6}∈A} ∑_{k=1}^{N_{a_{6}a_{3}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{2}}^{a_{6}}}

F (σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)}) ⊗ σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)}); σ_{123}(Y_{a_{1}a_{2};l}^{a_{6};(4)}) ⊗ σ_{132}(Y_{a_{6}a_{3};j}^{a_{4};(3)})) ·

·E(⟨w_{a′_{4}}, e^{z_{2}L(−1)}σ_{12}(Y_{a_{6}a_{3};i}^{a_{4};(3)})(w_{a_{3}}, e^{−πi}z_{2}) ·
·σ_{23}^{2}(Y_{a_{1}a_{2};j}^{a_{6};(4)})(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}⟩)

= ∑_{a_{6}∈A} ∑_{k=1}^{N_{a_{6}a_{3}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{2}}^{a_{6}}}

F (σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)}) ⊗ σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)}); σ_{123}(Y_{a_{1}a_{2};l}^{a_{6};(4)}) ⊗ σ_{132}(Y_{a_{6}a_{3};j}^{a_{4};(3)})) ·

·E(⟨w_{a′_{4}}, σ_{12}^{2}(Y_{a_{6}a_{3};i}^{a_{4};(3)})(σ_{23}^{2}(Y_{a_{1}a_{2};j}^{a_{6};(4)})(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩)

= ∑_{a_{6}∈A} ∑_{k=1}^{N_{a_{6}a_{3}}^{a_{4}}} ∑_{l=1}^{N_{a_{1}a_{2}}^{a_{6}}}

F (σ_{132}(Y_{a_{2}a_{3};j}^{a_{5};(2)}) ⊗ σ_{123}(Y_{a_{1}a_{5};i}^{a_{4};(1)}); σ_{123}(Y_{a_{1}a_{2};l}^{a_{6};(4)}) ⊗ σ_{132}(Y_{a_{6}a_{3};j}^{a_{4};(3)})) ·

·E(⟨w_{a′_{4}}, Y_{a_{6}a_{3};i}^{a_{4};(3)}(Y_{a_{1}a_{2};j}^{a_{6};(4)}(w_{a_{1}}, z_{1} − z_{2})w_{a_{2}}, z_{2})w_{a_{3}}⟩).  (3.15)

Using (3.13)–(3.15), we see that the left-hand side of (3.13) is equal to the right-hand side of (3.15). Comparing this resulting equality with (1.5) and using Proposition 1.2, we obtain (3.12).

4 Moore-Seiberg formulas

In [MS1], Moore and Seiberg proved the Verlinde conjecture by deriving two formulas from the axioms for rational conformal field theories. In this section, we prove these formulas mathematically using the results obtained in [H5] and [H6] and in the preceding sections.

We now want to choose a basis Y_{a_{1}a_{2};i}^{a_{3}}, i = 1, …, N_{a_{1}a_{2}}^{a_{3}}, of V_{a_{1}a_{2}}^{a_{3}} for each triple a_{1}, a_{2}, a_{3} ∈ A. Note that for each element σ ∈ S_{3}, σ(Y)_{a_{1}a_{2};i}^{a_{3}}, i = 1, …, N_{a_{1}a_{2}}^{a_{3}}, is also a basis of V_{a_{1}a_{2}}^{a_{3}}.

For a ∈ A, we choose Y_{ea;1}^{a} to be the the vertex operator Y_{W^{a}} defining the module structure on W^{a} and we choose Y_{ae;1}^{a} to be the intertwining operator defined using the action of σ_{12}, or equivalently the skew-symmetry in this case,

Y_{ae;1}^{a}(w_{a}, x)u = σ_{12}(Y_{ea;1}^{a})(w_{a}, x)u
= e^{xL(−1)}Y_{ea;1}^{a}(u, −x)w_{a}
= e^{xL(−1)}Y_{W^{a}}(u, −x)w_{a}

for u ∈ V and w_{a} ∈ W^{a}. Since V′ as a V -module is isomorphic to V, we have e′ = e. From [FHL], we know that there is a nondegerate invariant bilinear form (·, ·) on V. We choose Y_{aa′;1}^{e} = Y_{aa′;1}^{e′} to be the intertwining operator defined using the action of σ_{23} by

Y_{aa′;1}^{e′} = σ_{23}(Y_{ae;1}^{a}),

that is,

(u, Y_{aa′;1}^{e′}(w_{a}, x)w_{a′}) = e^{πih_{a}}⟨Y_{ae;1}^{a}(e^{xL(1)}(e^{−πi}x^{−2})^{L(0)}w_{a}, x^{−1})u, w_{a′}⟩

for u ∈ V, w_{a} ∈ W^{a} and w_{a′} ∈ W^{a′}. Since the actions of σ_{12} and σ_{23} generate the action of S_{3} on V, we have

Y_{a′a;1}^{e} = σ_{12}(Y_{aa′;1}^{e})

for any a ∈ A. When a_{1}, a_{2}, a_{3} ≠ e, we choose Y_{a_{1}a_{2};i}^{a_{3}}, i = 1, …, N_{a_{1}a_{2}}^{a_{3}}, to be an arbitrary basis of V_{a_{1}a_{2}}^{a_{3}}.

Recall that σ_{123} = σ_{12}σ_{23} and σ_{132} = σ_{23}σ_{12}. The following theorem gives the first Moore-Seiberg formula in [MS1]:

Theorem 4.1 For a_{1}, a_{2}, a_{3} ∈ A, we have

∑_{i=1}^{N_{a_{1}a_{2}}^{a_{3}}} ∑_{k=1}^{N_{a′_{1}a_{3}}^{a_{2}}} F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{3}a_{3};1}^{e}; Y_{a′_{1}a_{3};k}^{a_{2}} ⊗ Y_{a_{2}a′_{3};i}^{a′_{1}})·

·F (Y_{a′_{1}a_{3};k}^{a_{2}} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a′_{1}a_{1};1}^{e})
= N_{a_{1}a_{2}}^{a_{3}}F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a_{2}a′_{2};1}^{e}).  (4.1)

Proof. For a_{1}, a_{2}, a_{3} ∈ A, w_{a_{1}} ∈ W^{a_{1}}, w_{a′_{3}} ∈ W^{a′_{3}}, w_{a_{2}}^{1}, w_{a_{2}}^{2} ∈ W^{a_{2}}, w_{a′_{2}} ∈ W^{a′_{2}}, i = 1, …, N_{a_{1}a_{2}}^{a_{3}}, by the associativity (1.5), we have

E(⟨w_{a′_{2}}, Y_{a_{2}e;1}^{a_{2}}(w_{a_{2}}^{1}, z_{1})Y_{a′_{3}a_{3};1}^{e}(w_{a′_{3}}, z_{2})σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}})(w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩)

= ∑_{a_{4}∈A} ∑_{j=1}^{N_{a_{2}a′_{3}}^{a_{4}}} ∑_{k=1}^{N_{a_{4}a_{3}}^{a_{2}}} F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{3}a_{3};1}^{e}; Y_{a_{4}a_{3};k}^{a_{2}} ⊗ Y_{a_{2}a′_{3};j}^{a_{4}}) ·

·E(⟨w_{a′_{2}}, Y_{a_{4}a_{3};k}^{a_{2}}(Y_{a_{2}a′_{3};j}^{a_{4}}(w_{a_{2}}^{1}, z_{1} − z_{2})w_{a′_{3}}, z_{2})σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}})(w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩)

= ∑_{a_{4}∈A} ∑_{j=1}^{N_{a_{2}a′_{3}}^{a_{4}}} ∑_{k=1}^{N_{a_{4}a_{3}}^{a_{2}}} F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{3}a_{3};1}^{e}; Y_{a_{4}a_{3};k}^{a_{2}} ⊗ Y_{a_{2}a′_{3};j}^{a_{4}}) ·

· ∑_{a_{5}∈A} ∑_{l=1}^{N_{a_{4}a_{1}}^{a_{5}}} ∑_{m=1}^{N_{a_{5}a_{2}}^{a_{2}}} F (Y_{a_{4}a_{3};k}^{a_{2}} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{a_{5}a_{2};m}^{a_{2}} ⊗ Y_{a_{4}a_{1};l}^{a_{5}}) ·

·E(⟨w_{a′_{2}}, Y_{a_{5}a_{2};m}^{a_{2}}(Y_{a_{4}a_{1};l}^{a_{5}}(Y_{a_{2}a′_{3};j}^{a_{4}}(w_{a_{2}}^{1}, z_{1} − z_{2})w_{a′_{3}}, z_{2} − z_{3})w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩)

= ∑_{a_{4},a_{5}∈A} ∑_{j=1}^{N_{a_{2}a′_{3}}^{a_{4}}} ∑_{k=1}^{N_{a_{4}a_{3}}^{a_{2}}} ∑_{l=1}^{N_{a_{4}a_{1}}^{a_{5}}} ∑_{m=1}^{N_{a_{5}a_{2}}^{a_{2}}} F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{3}a_{3};1}^{e}; Y_{a_{4}a_{3};k}^{a_{2}} ⊗ Y_{a_{2}a′_{3};j}^{a_{4}}) ·

·F (Y_{a_{4}a_{3};k}^{a_{2}} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{a_{5}a_{2};m}^{a_{2}} ⊗ Y_{a_{4}a_{1};l}^{a_{5}}) · ·E(⟨w_{a′_{2}}, Y_{a_{5}a_{2};m}^{a_{2}}(Y_{a_{4}a_{1};l}^{a_{5}}(Y_{a_{2}a′_{3};j}^{a_{4}}(w_{a_{2}}^{1}, z_{1} − z_{2})w_{a′_{3}}, z_{2} − z_{3})w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩).

(4.2)

On the other hand, also by the associativity (1.5), we have E(⟨w_{a′_{2}}, Y_{a_{2}e;1}^{a_{2}}(w_{a_{2}}^{1}, z_{1})Y_{a′_{3}a_{3};1}^{e}(w_{a′_{3}}, z_{2})σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}})(w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩)

= ∑_{n=1}^{N_{a′_{3}a_{1}}^{a′_{2}}} F (Y_{a′_{3}a_{3};1}^{e} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{a′_{2}a_{2};1}^{e} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}})) ·

·E(⟨w_{a′_{2}}, Y_{a_{2}e;1}^{a_{2}}(w_{a_{2}}^{1}, z_{1})Y_{a′_{2}a_{2};1}^{e}(σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}})(w_{a′_{3}}, z_{2} − z_{3})w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩)

= ∑_{n=1}^{N_{a′_{3}a_{1}}^{a′_{2}}} F (Y_{a′_{3}a_{3};1}^{e} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{a′_{2}a_{2};1}^{e} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}})) ·

· ∑_{a_{6}∈A} ∑_{r=1}^{N_{a_{2}a′_{2}}^{a_{6}}} ∑_{s=1}^{N_{a_{6}a_{2}}^{a_{2}}} F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{6}a_{2};s}^{a_{2}} ⊗ Y_{a_{2}a′_{2};r}^{a_{6}}) ·

·E(⟨w_{a′_{2}}, Y_{a_{6}a_{2};s}^{a_{2}}(Y_{a_{2}a′_{2};r}^{a_{6}}(w_{a_{2}}^{1}, z_{1} − z_{3}) ·

·σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}})(w_{a′_{3}}, z_{2} − z_{3})w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩)

= ∑_{n=1}^{N_{a′_{3}a_{1}}^{a′_{2}}} F (Y_{a′_{3}a_{3};1}^{e} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{a′_{2}a_{2};1}^{e} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}})) ·

· ∑_{a_{6}∈A} ∑_{r=1}^{N_{a_{2}a′_{2}}^{a_{6}}} ∑_{s=1}^{N_{a_{6}a_{2}}^{a_{2}}} F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{6}a_{2};s}^{a_{2}} ⊗ Y_{a_{2}a′_{2};r}^{a_{6}}) ·

· ∑_{a_{7}∈A} ∑_{p=1}^{N_{a_{2}a′_{3}}^{a_{7}}} ∑_{q=1}^{N_{a_{7}a_{1}}^{a_{6}}} F (Y_{a_{2}a′_{2};r}^{a_{6}} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}}); Y_{a_{7}a_{1};q}^{a_{6}} ⊗ Y_{a_{2}a′_{3};p}^{a_{7}}) ·

·E(⟨w_{a′_{2}}, Y_{a_{6}a_{2};s}^{a_{2}}(Y_{a_{7}a_{1};q}^{a_{6}}(Y_{a_{2}a′_{3};p}^{a_{7}}(w_{a_{2}}^{1}, z_{1} − z_{2})w_{a′_{3}}, z_{2} − z_{3})w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩)

= ∑_{a_{6},a_{7}∈A} ∑_{n=1}^{N_{a′_{3}a_{1}}^{a′_{2}}} ∑_{r=1}^{N_{a_{2}a′_{2}}^{a_{6}}} ∑_{s=1}^{N_{a_{6}a_{2}}^{a_{2}}} ∑_{p=1}^{N_{a_{2}a′_{3}}^{a_{7}}} ∑_{q=1}^{N_{a_{7}a_{1}}^{a_{6}}}

F (Y_{a′_{3}a_{3};1}^{e} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{a′_{2}a_{2};1}^{e} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}})) · ·F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{5}a_{2};m}^{a_{2}} ⊗ Y_{a_{2}a′_{2};r}^{a_{5}}) · ·F (Y_{a_{2}a′_{2};r}^{a_{6}} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}}); Y_{a_{7}a_{1};q}^{a_{6}} ⊗ Y_{a_{2}a′_{3};p}^{a_{7}}) · ·E(⟨w_{a′_{2}}, Y_{a_{6}a_{2};s}^{a_{2}}(Y_{a_{7}a_{1};q}^{a_{6}}(Y_{a_{2}a′_{3};p}^{a_{7}}(w_{a_{2}}^{1}, z_{1} − z_{2})w_{a′_{3}}, z_{2} − z_{3})w_{a_{1}}, z_{3})w_{a_{2}}^{2}⟩).

(4.3)

Using (4.2)–(4.3) and Proposition 1.3, we obtain

∑_{k=1}^{N_{a_{4}a_{3}}^{a_{2}}} F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{3}a_{3};1}^{e}; Y_{a_{4}a_{3};k}^{a_{2}} ⊗ Y_{a_{2}a′_{3};j}^{a_{4}})·

·F (Y_{a_{4}a_{3};k}^{a_{2}} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{a_{5}a_{2};m}^{a_{2}} ⊗ Y_{a_{4}a_{1};l}^{a_{5}})

= ∑_{n=1}^{N_{a′_{3}a_{1}}^{a′_{2}}} ∑_{r=1}^{N_{a_{2}a′_{2}}^{a_{5}}} F (Y_{a′_{3}a_{3};1}^{e} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{a′_{2}a_{2};1}^{e} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}})) ·

·F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{5}a_{2};m}^{a_{2}} ⊗ Y_{a_{2}a′_{2};r}^{a_{5}}) ·
·F (Y_{a_{2}a′_{2};r}^{a_{5}} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}}); Y_{a_{4}a_{1};q}^{a_{5}} ⊗ Y_{a_{2}a′_{3};p}^{a_{4}}).

In particular, for a_{5} = e and a_{4} = a′_{1}, we have

∑_{k=1}^{N_{a′_{1}a_{3}}^{a_{2}}} F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{3}a_{3};1}^{e}; Y_{a′_{1}a_{3};k}^{a_{2}} ⊗ Y_{a_{2}a′_{3};j}^{a′_{1}})·

·F (Y_{a_{4}a_{3};k}^{a_{2}} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{ea_{2};m}^{a_{2}} ⊗ Y_{a′_{1}a_{1};l}^{e})

= ∑_{n=1}^{N_{a′_{3}a_{1}}^{a′_{2}}} F (Y_{a′_{3}a_{3};1}^{e} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{a′_{2}a_{2};1}^{e} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}})) ·

·F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a_{2}a′_{2};1}^{e}) ·
·F (Y_{a_{2}a′_{2};1}^{e} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}}); Y_{a′_{1}a_{1};1}^{e} ⊗ Y_{a_{2}a′_{3};j}^{a′_{1}})

= ( ∑_{n=1}^{N_{a′_{3}a_{1}}^{a′_{2}}} F (Y_{a′_{3}a_{3};1}^{e} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{a′_{2}a_{2};1}^{e} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}})) ·

·F (Y_{a_{2}a′_{2};1}^{e} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}}); Y_{a′_{1}a_{1};1}^{e} ⊗ Y_{a_{2}a′_{3};j}^{a′_{1}})) ·
·F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a_{2}a′_{2};1}^{e}).  (4.4)

On the other hand, by the definition of σ_{12} and σ_{23}, the relations σ_{12}^{2} = σ_{23}^{2} = 1 and the choices of the basis Y_{ea;1}^{a}, Y_{ae;1}^{a}, Y_{aa′}^{e} and Y_{a′a;1}^{e} for a ∈ A, we have

⟨u, Y_{a′_{3}a_{3};1}^{e}(w_{a′_{3}}, x_{1})σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}})(w_{a_{1}}, x_{2})w_{a_{2}}⟩

= ⟨u, σ_{23}^{2}(Y_{a′_{3}a_{3};1}^{e})(w_{a′_{3}}, x_{1})σ_{23}^{2}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(w_{a_{1}}, x_{2})w_{a_{2}}⟩
= e^{πi(h_{a_{1}}+h_{a_{3}})}⟨σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(e^{x_{2}L(1)}(e^{−πi}x_{2}^{−2})^{L(0)}w_{a_{1}}, x_{2}^{−1}) ·
·σ_{23}(Y_{a′_{3}a_{3};1}^{e})(e^{x_{1}L(1)}(e^{−πi}x_{1}^{−2})^{L(0)}w_{a′_{3}}, x_{1}^{−1})u, w_{a_{2}}⟩
= e^{πi(h_{a_{1}}+h_{a_{3}})}⟨σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(e^{x_{2}L(1)}(e^{−πi}x_{2}^{−2})^{L(0)}w_{a_{1}}, x_{2}^{−1}) ·
·Y_{a′_{3}e;1}^{a′_{3}}(e^{x_{1}L(1)}(e^{−πi}x_{1}^{−2})^{L(0)}w_{a′_{3}}, x_{1}^{−1})u, w_{a_{2}}⟩
= e^{πi(h_{a_{1}}+h_{a_{3}})}⟨σ_{12}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(e^{x_{2}L(1)}((e^{−πi}x_{2}^{−2})^{L(0)}w_{a_{1}}, x_{2}^{−1}) ·

·e^{x_{1}^{−1}L(−1)}Y_{ea′_{3};1}^{a′_{3}}(u, −x_{1}^{−1})e^{x_{1}L(1)}((e^{−πi}x_{1}^{−2})^{L(0)}w_{a′_{3}}, w_{a_{2}}⟩

= e^{πi(h_{a_{1}}+h_{a_{3}})} ·
·⟨e^{x_{1}^{−1}L(−1)}σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(e^{x_{2}L(1)}(e^{−πi}x_{2}^{−2})^{L(0)}w_{a_{1}}, x_{2}^{−1} − x_{1}^{−1}) ·
·Y_{W^{a′_{3}}}(u, −x_{1}^{−1})e^{x_{1}L(1)}(e^{−πi}x_{1}^{−2})^{L(0)}w_{a′_{3}}, w_{a_{2}}⟩  (4.5)

By the locality between vertex operators on V -modules and intertwining operators and the definition of σ_{23}, there exists a positive integer N such that

x_{2}^{−N}e^{πi(h_{a_{1}}+h_{a_{3}})}

·⟨e^{x_{1}^{−1}L(−1)}σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(e^{x_{2}L(1)}(e^{−πi}x_{2}^{−2})^{L(0)}w_{a_{1}}, x_{2}^{−1} − x_{1}^{−1}) ·

·Y_{W^{a′_{3}}}(u, −x_{1}^{−1})e^{x_{1}L(1)}(e^{−πi}x_{1}^{−2})^{L(0)}w_{a′_{3}}, w_{a_{2}}⟩

= ((x_{2}^{−1} − x_{1}^{−1}) − (−x_{1}^{−1})^{N}e^{πi(h_{a_{1}}+h_{a_{3}})} ·

·⟨e^{x_{1}^{−1}L(−1)}σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(e^{x_{2}L(1)}(e^{−πi}x_{2}^{−2})^{L(0)}w_{a_{1}}, x_{2}^{−1} − x_{1}^{−1}) ·

·Y_{W^{a′_{3}}}(u, −x_{1}^{−1})e^{x_{1}L(1)}(e^{−πi}x_{1}^{−2})^{L(0)}w_{a′_{3}}, w_{a_{2}}⟩

= ((x_{2}^{−1} − x_{1}^{−1}) − (−x_{1}^{−1})^{N}e^{πi(h_{a_{1}}+h_{a_{3}})} ·

·⟨e^{x_{1}^{−1}L(−1)}Y_{W_{a′_{2}}}(u, −x_{1}^{−1}) ·

·σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(e^{x_{2}L(1)}(e^{−πi}x_{2}^{−2})^{L(0)}w_{a_{1}}, z_{2}^{−1} − z_{1}^{−1}) ·

·e^{z_{1}L(1)}(e^{−πi}z_{1}^{−2})^{L(0)}w_{a′_{3}}, w_{a_{2}}⟩

= x_{2}^{−N}e^{πi(h_{a_{1}}+h_{a_{3}})} ·

·⟨σ_{12}(Y_{W_{a′_{2}}})(σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(e^{x_{2}L(1)}(e^{−πi}x_{2}^{−2})^{L(0)}w_{a_{1}}, x_{2}^{−1} − x_{1}^{−1}) ·

·e^{x_{1}L(1)}(e^{−πi}x_{1}^{−2})^{L(0)}w_{a′_{3}}, x_{1}^{−1})u, w_{a_{2}}⟩

= x_{2}^{−N}e^{πi(h_{a_{1}}+h_{a_{3}}−h_{a_{2}})}⟨e^{πih_{a_{1}}}Y_{a′_{2}e;1}^{a′_{2}}(e^{x_{1}L(1)}(e^{−πi}x_{1}^{−2})^{L(0)}(e^{πi}x_{1}^{2})^{L(0)}e^{−x_{1}L(1)} ·

·σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(e^{x_{2}L(1)}(e^{−πi}x_{2}^{−2})^{L(0)}w_{a_{1}}, x_{2}^{−1} − x_{1}^{−1}) ·

·e^{x_{1}L(1)}(e^{−πi}x_{1}^{−2})^{L(0)}w_{a′_{3}}, x_{1}^{−1})u, w_{a_{2}}⟩

= x_{2}^{−N}e^{πi(h_{a_{1}}+h_{a_{3}}−h_{a_{2}})}⟨u, σ_{23}(Y_{a′_{2}e;1}^{a′_{2}})((e^{πi}x_{1}^{2})^{L(0)}e^{−x_{1}L(1)} ·

·σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(e^{x_{2}L(1)}(e^{−πi}x_{2}^{−2})^{L(0)}w_{a_{1}}, x_{2}^{−1} − x_{1}^{−1}) ·

·e^{x_{1}L(1)}(e^{−πi}x_{1}^{−2})^{L(0)}w_{a′_{3}}, x_{1})w_{a_{2}}⟩

= x_{2}^{−N}e^{πi(h_{a_{1}}+h_{a_{3}}−h_{a_{2}})}⟨u, Y_{a′_{2}a_{2};1}^{e}((e^{−πi}x_{1}^{2})^{L(0)}e^{−x_{1}L(1)} ·

·σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(e^{x_{2}L(1)}(e^{−πi}x_{2}^{−2})^{L(0)}w_{a_{1}}, x_{2}^{−1} − x_{1}^{−1}) ·

·e^{x_{1}L(1)}(e^{−πi}x_{1}^{−2})^{L(0)}w_{a′_{3}}, x_{1})w_{a_{2}}⟩

= x_{2}^{−N}e^{πi(h_{a_{1}}+h_{a_{3}}−h_{a_{2}})} ·

·⟨u, Y_{a′_{2}a_{2};1}^{e}(σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(w_{a_{1}}, e^{πi}(x_{1} − x_{2}))w_{a′_{3}}, x_{1})w_{a_{2}}⟩

= x_{2}^{−N}⟨u, Y_{a′_{2}a_{2};1}^{e}(e^{(x_{2}−x_{1})L(−1)}e^{−πiΔ(σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}})))}e^{(x_{1}−x_{2})L(−1)} ·

·σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))(w_{a_{1}}, e^{πi}(x_{1} − x_{2}))w_{a′_{3}}, z_{1})w_{a_{2}}⟩

= x_{2}^{−N}⟨u, Y_{a′_{2}a_{2};1}^{e}(e^{(x_{2}−x_{1})L(−1)} · ·σ_{12}(σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}})))(w_{a_{1}}, x_{1} − x_{2})w_{a′_{3}}, x_{1})w_{a_{2}}⟩.

(4.6)

Using (4.5), (4.6), the L(−1)-derivative property for σ_{12}(σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}))) and the equality σ_{12}σ_{23} = σ_{123}, we obtain

E(⟨u, Y_{a′_{3}a_{3};1}^{e}(w_{a′_{3}}, z_{1})σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}})(w_{a_{1}}, z_{2})w_{a_{2}}⟩)

= E(⟨u, Y_{a′_{2}a_{2};1}^{e}(e^{(z_{2}−z_{1})L(−1)} ·
·σ_{12}(σ_{23}(σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}})))(w_{a_{1}}, z_{1} − z_{2})w_{a′_{3}}, z_{1})w_{a_{2}}⟩)
= E(⟨u, Y_{a′_{2}a_{2};1}^{e}(σ_{123}^{2}(Y_{a_{2}a′_{3};i}^{a′_{1}})(w_{a_{1}}, z_{1} − z_{2})w_{a′_{3}}, z_{2})w_{a_{2}}⟩)

Thus we obtain

F (Y_{a′_{3}a_{3};1}^{e} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{a′_{2}a_{2};1}^{e} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}})) = δ_{in}.  (4.7)

Similarly, we can prove

F (Y_{a_{2}a′_{2};1}^{e} ⊗ σ_{123}^{2}(Y_{a_{2}a′_{3};n}^{a′_{1}}); Y_{a′_{1}a_{1};1}^{e} ⊗ Y_{a_{2}a′_{3};j}^{a′_{1}}) = δ_{nj}  (4.8)

Using (4.7) and (4.8), we see that (4.4) becomes

∑_{k=1}^{N_{a′_{1}a_{3}}^{a_{2}}} F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{3}a_{3};1}^{e}; Y_{a′_{1}a_{3};k}^{a_{2}} ⊗ Y_{a_{2}a′_{3};j}^{a′_{1}})·

·F (Y_{a_{4}a_{3};k}^{a_{2}} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{ea_{2};m}^{a_{2}} ⊗ Y_{a′_{1}a_{1};l}^{e})
= δ_{ij}F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a_{2}a′_{2};1}^{e}).  (4.9)

Summing over i = 1, …, ℕ_{a_{1}a_{3}}^{a_{2}} on both sides in the special case j = i of (4.9), we obtain (4.1).

We now prepare to prove the second formula. Recall from Section 2 the maps Ψ_{a_{1},a_{2},e}^{1,1} : ∐_{a∈A} W^{a} ⊗ W^{a′} → 𝔾_{1;2} for a_{1}, a_{2} ∈ A and the projection π : F_{1;2} → F_{1;2}^{e}. For any f ∈ F_{1;2}, we shall, for convenience, denote (π(f))(w_{a} ⊗ w_{a′}) by π(f(w_{a} ⊗ w_{a′})).

We need the following lemma:

Lemma 4.2 For a_{1}, a_{2} ∈ A, w_{a_{2}} ∈ W^{a_{2}} and w_{a′_{2}} ∈ W^{a′_{2}}, we have Ψ_{a_{1},a_{2},e}^{1,1}(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2} − 1; τ)

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} ∑_{a_{4}∈A} ∑_{k=1}^{N_{a_{4}a_{1}}^{a_{1}}} ∑_{l=1}^{N_{a_{2}a′_{2}}^{a_{4}}} e^{−2πi(h_{a_{3}}−h_{a_{1}})} ·

·F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; Y_{a_{2}a_{3};i}^{a_{1}} ⊗ Y_{a′_{2}a_{1};j}^{a_{3}}) ·
·F (Y_{a_{2}a_{3};i}^{a_{1}} ⊗ Y_{a′_{2}a_{1};j}^{a_{3}}; Y_{a_{4}a_{1};k}^{a_{1}} ⊗ Y_{a_{2}a′_{2};l}^{a_{4}}) ·
·E(Tr_{W^{a_{1}}}Y_{a_{4}a_{1};k}^{a_{1}}(U(e^{2πiz_{2}}) ·
·Y_{a_{2}a′_{2};l}^{a_{4}}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

(4.10)

and Ψ_{a_{1},a_{2},e}^{1,1}(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2} + τ; τ)

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} ∑_{a_{4}∈A} ∑_{k=1}^{N_{a_{4}a_{3}}^{a_{3}}} ∑_{l=1}^{N_{a_{2}a′_{2}}^{a_{4}}} e^{πi(−2h_{a_{2}}+h_{a_{4}})}

F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}}) ⊗ σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})) ·
·F (Y_{a_{2}a′_{1};i}^{a′_{3}} ⊗ σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}}); Y_{a_{4}a′_{3};k}^{a′_{3}} ⊗ Y_{a_{2}a′_{2};l}^{a_{4}}) ·

·E(Tr_{W^{a_{3}}}Y_{a_{4}a_{3};k}^{a_{3}}(U(e^{2πiz_{2}})Y_{a_{2}a′_{2};l}^{a_{4}}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ).

(4.11)

In particular, for any a_{1}, a_{2} ∈ A, the maps from ∐_{a∈A} W^{a} ⊗W^{a′} to the space of single-valued analytic functions on M̃_{1}^{2} given by

w_{a} ⊗ w_{a′} ↦ Ψ_{a_{1},a_{2},e}^{1,1}(w_{a}, w_{a′}; z_{1}, z_{2} − 1; τ),
w_{a} ⊗ w_{a′} ↦ Ψ_{a_{1},a_{2},e}^{1,1}(w_{a}, w_{a′}; z_{1}, z_{2} + τ; τ)

for w_{a} ∈ W^{a} and w_{a′} ∈ W^{a′} are in F_{1;2}. Proof. Using the definition of Ψ_{a_{1},a_{2},e}^{1,1}, the associativity properties (1.5), (1.6) and (1.1), we have Ψ_{a_{1},a_{2},e}^{1,1}(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2} − 1; τ)

= E(Tr_{W^{a_{1}}}Y_{ea_{1};1}^{a_{1}}(U(e^{2πi(z_{2}−1)}) ·

·Y_{a_{2}a′_{2};1}^{e}(w_{a_{2}}, z_{1} − (z_{2} − 1))w_{a′_{2}}, e^{2πi(z_{2}−1)})q_{τ}^{L(0)− \frac{c}{24}} )

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; Y_{a_{2}a_{3};i}^{a_{1}} ⊗ Y_{a′_{2}a_{1};j}^{a_{3}}) ·

·E(Tr_{W^{a_{1}}}Y_{a_{2}a_{3};i}^{a_{1}}(U(e^{2πiz_{1}})w_{a_{2}}, e^{2πiz_{1}}) ·

·Y_{a′_{2}a_{1};j}^{a_{3}}(U(e^{2πi(z_{2}−1)})w_{a′_{2}}, e^{2πi(z_{2}−1)})q_{τ}^{L(0)− \frac{c}{24}} )

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; Y_{a_{2}a_{3};i}^{a_{1}} ⊗ Y_{a′_{2}a_{1};j}^{a_{3}}) ·

·E(Tr_{W^{a_{1}}}Y_{a_{2}a_{3};i}^{a_{1}}(U(e^{2πiz_{1}})w_{a_{2}}, e^{2πiz_{1}}) ·

·Y_{a′_{2}a_{1};j}^{a_{3}}(U(e^{−2πi}e^{2πiz_{2}})w_{a′_{2}}, e^{−2πi}e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} e^{−2πi(h_{a_{3}}−h_{a_{1}})}F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; Y_{a_{2}a_{3};i}^{a_{1}} ⊗ Y_{a′_{2}a_{1};j}^{a_{3}}) ·

·E(Tr_{W^{a_{1}}}Y_{a_{2}a_{3};i}^{a_{1}}(U(e^{2πiz_{1}})w_{a_{2}}, e^{2πiz_{1}}) ·

·Y_{a′_{2}a_{1};j}^{a_{3}}(U(e^{2πiz_{2}})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} e^{−2πi(h_{a_{3}}−h_{a_{1}})}F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; Y_{a_{2}a_{3};i}^{a_{1}} ⊗ Y_{a′_{2}a_{1};j}^{a_{3}}) ·

· ∑_{a_{4}∈A} ∑_{k=1}^{N_{a_{4}a_{1}}^{a_{1}}} ∑_{l=1}^{N_{a_{2}a′_{2}}^{a_{4}}} F (Y_{a_{2}a_{3};i}^{a_{1}} ⊗ Y_{a′_{2}a_{1};j}^{a_{3}}; Y_{a_{4}a_{1};k}^{a_{1}} ⊗ Y_{a_{2}a′_{2};l}^{a_{4}}) ·

·E(Tr_{W^{a_{1}}}Y_{a_{4}a_{1};k}^{a_{1}}(U(e^{2πiz_{2}}) ·

·Y_{a_{2}a′_{2};l}^{a_{4}}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ),

proving (4.10). The proof of (4.11) is more complicated. Using the definition of Ψ_{a_{1},a_{2},e}^{1,1}, the associativity properties (1.5), (1.6), the L(0)-conjugation property, the property of traces, and (1.1),

Ψ_{a_{1},a_{2},e}^{1,1}(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2} + τ; τ)

= E(Tr_{W^{a_{1}}}Y_{ea_{1};1}^{a_{1}}(U(e^{2πi(z_{2}+τ)}) ·

·Y_{a_{2}a′_{2};1}^{e}(w_{a_{2}}, z_{1} − (z_{2} + τ))w_{a′_{2}}, e^{2πi(z_{2}+τ)})q_{τ}^{L(0)− \frac{c}{24}} )

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}}) ⊗ σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})) ·

·E(Tr_{W^{a_{1}}}σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}})(U(e^{2πiz_{1}})w_{a_{2}}, e^{2πiz_{1}}) ·

·σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})(U(e^{2πi(z_{2}+τ)})w_{a′_{2}}, e^{2πi(z_{2}+τ)})q_{τ}^{L(0)− \frac{c}{24}} )

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}}) ⊗ σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})) ·

·E(Tr_{W^{a_{1}}}σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}})(U(e^{2πiz_{1}})w_{a_{2}}, e^{2πiz_{1}}) ·

·σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})(U(q_{τ}e^{2πiz_{2}})w_{a′_{2}}, q_{τ}e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}}) ⊗ σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})) ·

·E(Tr_{W^{a_{1}}}σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}})(U(e^{2πiz_{1}})w_{a_{2}}, e^{2πiz_{1}}) ·

·q_{τ}^{L(0)− \frac{c}{24}} σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})(U(e^{2πiz_{2}})w_{a′_{2}}, e^{2πiz_{2}}))

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}}) ⊗ σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})) ·

·E(Tr_{W^{a_{3}}}σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})(U(e^{2πiz_{2}})w_{a′_{2}}, e^{2πiz_{2}}) ·

·σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}})(U(e^{2πiz_{1}})w_{a_{2}}, e^{2πiz_{1}})q_{τ}^{L(0)− \frac{c}{24}} ). (4.12)

Using (2.21), the relations σ_{23}^{2} = 1, σ_{23}σ_{12} = σ_{123} and the genus-one associativity, we have E(Tr_{W^{a_{3}}}σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})(U(e^{2πiz_{2}})w_{a′_{2}}, e^{2πiz_{2}})·

·σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}})(U(e^{2πiz_{1}})w_{a_{2}}, e^{2πiz_{1}})q_{τ}^{L(0)− \frac{c}{24}} )

= E(Tr_{W^{a_{3}}}σ_{23}^{2}(σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}}))(U(e^{2πiz_{2}})w_{a′_{2}}, e^{2πiz_{2}}) ·
·σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}})(U(e^{2πiz_{1}})w_{a_{2}}, e^{2πiz_{1}})q_{τ}^{L(0)− \frac{c}{24}} )
= e^{−2πih_{a_{2}}}E(Tr_{W^{a′_{3}}}Y_{a_{2}a′_{1};i}^{a′_{3}}(U(e^{2πiz_{1}})e^{πiL(0)}w_{a_{2}}, e^{−2πiz_{1}}) ·

·σ_{23}(σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}}))(U(e^{2πiz_{2}})e^{πiL(0)}w_{a′_{2}}, e^{−2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

= e^{−2πih_{a_{2}}}E(Tr_{W^{a′_{3}}}Y_{a_{2}a′_{1};i}^{a′_{3}}(U(e^{−2πiz_{1}})e^{πiL(0)}w_{a_{2}}, e^{−2πiz_{1}}) ·
·σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}})(U(e^{−2πiz_{2}})e^{πiL(0)}w_{a′_{2}}, e^{−2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

(4.13)

We now prove E(Tr_{W^{a′_{3}}}Y_{a_{2}a′_{1};i}^{a′_{3}}(U(e^{−2πiz_{1}})e^{πiL(0)}w_{a_{2}}, e^{−2πiz_{1}})·

·σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}})(U(e^{−2πiz_{2}})e^{πiL(0)}w_{a′_{2}}, e^{−2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

= ∑_{a_{4}∈A} ∑_{k=1}^{N_{a_{4}a_{3}}^{a_{3}}} ∑_{l=1}^{N_{a_{2}a′_{2}}^{a_{4}}} F (Y_{a_{2}a′_{1};i}^{a′_{3}} ⊗ σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}}); Y_{a_{4}a′_{3};k}^{a′_{3}} ⊗ Y_{a_{2}a′_{2};l}^{a_{4}}) ·

·E(Tr_{W^{a′_{3}}}Y_{a_{4}a′_{3};k}^{a′_{3}}(U(e^{−2πiz_{2}})Y_{a_{2}a′_{2};l}^{a_{4}}(e^{πiL(0)}w_{a_{2}}, e^{πi}(z_{1} − z_{2})) ·

·e^{πiL(0)}w_{a′_{2}}, e^{−2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ).

(4.14)

To prove (4.14), we need only prove their restrictions to a subregion of M̃_{1}^{2} are equal. So we need only prove that Tr_{W^{a′_{3}}}Y_{a_{2}a′_{1};i}^{a′_{3}}(U(e^{−2πiz_{1}})e^{πiL(0)}w_{a_{2}}, e^{−2πiz_{1}})·

·σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}})(U(e^{−2πiz_{2}})e^{πiL(0)}w_{a′_{2}}, e^{−2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}}

= ∑_{a_{4}∈A} ∑_{k=1}^{N_{a_{4}a_{3}}^{a_{3}}} ∑_{l=1}^{N_{a_{2}a′_{2}}^{a_{4}}} F (Y_{a_{2}a′_{1};i}^{a′_{3}} ⊗ σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}}); Y_{a_{4}a′_{3};k}^{a′_{3}} ⊗ Y_{a_{2}a′_{2};l}^{a_{4}}) ·

·Tr_{W^{a′_{3}}}Y_{a_{4}a′_{3};k}^{a′_{3}}(U(e^{−2πiz_{2}})Y_{a_{2}a′_{2};l}^{a_{4}}(e^{πiL(0)}w_{a_{2}}, e^{πi}(z_{1} − z_{2})) ·

·e^{πiL(0)}w_{a′_{2}}, e^{−2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

(4.15)

holds when |q_{τ}| < |e^{−2πiz_{2}}| < |e^{−2πiz_{1}}| < 1 and 0 < |e^{2πi(−z_{1}+z_{2})} − 1| < 1. From (2.9), we see that in this region the left-hand side of (4.15) is equal to

∑_{a_{4}∈A} ∑_{k=1}^{N_{a_{4}a_{3}}^{a_{3}}} ∑_{l=1}^{N_{a_{2}a′_{2}}^{a_{4}}} F (Y_{a_{2}a′_{1};i}^{a′_{3}} ⊗ σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}}); Y_{a_{4}a′_{3};k}^{a′_{3}} ⊗ Y_{a_{2}a′_{2};l}^{a_{4}})·

·Tr_{W^{a′_{3}}}Y_{a_{4}a′_{3};k}^{a′_{3}}(U(e^{−2πiz_{2}})Y_{a_{2}a′_{2};l}^{a_{4}}(e^{πiL(0)}w_{a_{2}}, (−z_{1} + z_{2})) ·

·e^{πiL(0)}w_{a′_{2}}, e^{−2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

(4.16)

Now in this region, because |e^{−2πiz_{2}}| < |e^{−2πiz_{1}}|, the imaginary part of z_{1} must be bigger than the imaginary part of z_{2}. Thus z_{1} − z_{2} is in the upper half plane. This means that arg(z_{1} − z_{2}) < π and arg(z_{1} − z_{2}) + π < 2π. So we have arg(−(z_{1} − z_{2})) = arg(z_{1} − z_{2}) + π. Now for any n ∈ ℂ, by our convention,

(−z_{1} + z_{2})^{n} = e^{n log(−z_{1}+z_{2})}
= e^{n log(−(z_{1}−z_{2}))}
= e^{n(log |−(z_{1}−z_{2})|+i arg(−(z_{1}−z_{2})))}
= e^{n(log |(z_{1}−z_{2})|+i arg(z_{1}−z_{2})+πi)}
= e^{n(log(z_{1}−z_{2})+πi)}
= (e^{πi}(z_{1} − z_{2}))^{n}.

This shows that indeed when |q_{τ}| < |e^{−2πiz_{2}}| < |e^{−2πiz_{1}}| < 1 and 0 < |e^{2πi(−z_{1}+z_{2})} − 1| < 1, (4.16) is equal to the right-hand side of (4.15) and (4.15) holds. Consequently, we obtain (4.14).

Using (2.22) and the L(0)-conjugation formula, we have E(Tr_{W^{a′_{3}}}Y_{a_{4}a′_{3};k}^{a′_{3}}(U(e^{−2πiz_{2}})Y_{a_{2}a′_{2};l}^{a_{4}}(e^{πiL(0)}w_{a_{2}}, e^{πi}(z_{1} − z_{2}))·

·e^{πiL(0)}w_{a′_{2}}, e^{−2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

= e^{πh_{a_{4}}}E(Tr_{W^{a_{3}}}Y_{a_{4}a_{3};k}^{a_{3}}(U(e^{2πiz_{2}})e^{−πiL(0)} ·

·Y_{a_{2}a′_{2};l}^{a_{4}}(e^{πiL(0)}w_{a_{2}}, e^{πi}(z_{1} − z_{2}))e^{πiL(0)}w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

= e^{πh_{a_{4}}}E(Tr_{W^{a_{3}}}Y_{a_{4}a_{3};k}^{a_{3}}(U(e^{2πiz_{2}})Y_{a_{2}a′_{2};l}^{a_{4}}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ).

(4.17)

Combining (4.12), (4.13), (4.14) and (4.17), we obtain (4.11).

For a_{1}, a_{2} ∈ A, we define α(Ψ_{a_{1},a_{2},e}^{1,1}) and β(Ψ_{a_{1},a_{2},e}^{1,1}) by

(α(Ψ_{a_{1},a_{2},e}^{1,1}))(w_{a}, w_{a′}; z_{1}, z_{2}; τ) = π(Ψ_{a_{1},a_{2},e}^{1,1}(w_{a}, w_{a′}; z_{1}, z_{2} − 1; τ)),
(β(Ψ_{a_{1},a_{2},e}^{1,1}))(w_{a}, w_{a′}; z_{1}, z_{2}; τ) = π(Ψ_{a_{1},a_{2},e}^{1,1}(w_{a}, w_{a′}; z_{1}, z_{2} + τ; τ))

for a ∈ A, w_{a} ∈ W^{a} and w_{a′} ∈ W^{a′}. Proposition 4.3 For a_{1}, a_{2} ∈ A, we have

α(Ψ_{a_{1},a_{2},e}^{1,1}) = e^{−2πih_{a_{2}}}(B^{(−1)})^{2}(Y_{a_{1}e;1}^{a_{1}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{1}e;1}^{a_{1}} ⊗ Y_{a′_{2}a_{2};1}^{e})Ψ_{a_{1},a_{2},e}^{1,1} (4.18)

and

β(Ψ_{a_{1},a_{2},e}^{1,1}) = e^{−2πih_{a_{2}}} ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{1}a_{1};1}^{e}; Y_{a′_{3}a_{1};j}^{a_{2}} ⊗ Y_{a_{2}a′_{1};i}^{a′_{3}}) ·
·F (Y_{a′_{3}a_{1};j}^{a_{2}} ⊗ σ_{123}(Y_{a_{2}a′_{1};i}^{a′_{3}}); Y_{ea_{2};l}^{a_{2}} ⊗ Y_{a′_{3}a_{3};k}^{e})Ψ_{a_{3};a_{2}}.

(4.19)

Proof. Using the definitions of α, π and Ψ_{a_{1},a_{2},e}^{1,1}(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2}; τ), (4.10) and (3.1), we have

(α(Ψ_{a_{1},a_{2},e}^{1,1}))(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2}; τ) = π(Ψ_{a_{1},a_{2},e}^{1,1}(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2} − 1; τ))

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} ∑_{a_{4}∈A} ∑_{k=1}^{N_{a_{4}a_{1}}^{a_{1}}} ∑_{l=1}^{N_{a_{2}a′_{2}}^{a_{4}}} e^{−2πi(h_{a_{3}}−h_{a_{1}})} ·

·F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; Y_{a_{2}a_{3};i}^{a_{1}} ⊗ Y_{a′_{2}a_{1};j}^{a_{3}}) ·

·F (Y_{a_{2}a_{3};i}^{a_{1}} ⊗ Y_{a′_{2}a_{1};j}^{a_{3}}; Y_{a_{4}a_{1};k}^{a_{1}} ⊗ Y_{a_{2}a′_{2};l}^{a_{4}}) ·

·π(E(Tr_{W^{a_{1}}}Y_{a_{4}a_{1};k}^{a_{1}}(U(e^{2πiz_{2}}) ·

·Y_{a_{2}a′_{2};l}^{a_{4}}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ))

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} e^{−2πi(h_{a_{3}}−h_{a_{1}})}F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; Y_{a_{2}a_{3};i}^{a_{1}} ⊗ Y_{a′_{2}a_{1};j}^{a_{3}}) ·

·F (Y_{a_{2}a_{3};i}^{a_{1}} ⊗ Y_{a′_{2}a_{1};j}^{a_{3}}; Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}) ·

·E(Tr_{W^{a_{1}}}Y_{ea_{1};1}^{a_{1}}(U(e^{2πiz_{2}}) ·

·Y_{a_{2}a′_{2};1}^{e}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

= e^{−2πih_{a_{2}}} ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} F (Y_{a_{1}e;1}^{a_{1}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{3}a_{2};i}^{a_{1}} ⊗ Y_{a_{1}a′_{2};j}^{a_{3}}) ·

·e^{−2πi(h_{a_{3}}−h_{a_{1}}−h_{a_{2}})}F^{−1}(Y_{a_{3}a_{2};i}^{a_{1}} ⊗ Y_{a_{1}a′_{2};j}^{a_{3}}; Y_{a_{1}e;1}^{a_{1}} ⊗ Y_{a′_{2}a_{2};1}^{e}) ·

·E(Tr_{W^{a_{1}}}Y_{ea_{1};1}^{a_{1}}(U(e^{2πiz_{2}}) ·

·Y_{a_{2}a′_{2};1}^{e}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

= e^{−2πih_{a_{2}}}(B^{(−1)})^{2}(Y_{a_{1}e;1}^{a_{1}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{1}e;1}^{a_{1}} ⊗ Y_{a′_{2}a_{2};1}^{e}) ·

·Ψ_{a_{1},a_{2},e}^{1,1}(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2}; τ),

proving (4.18).

Using the definitions of β, π and Ψ_{a_{1},a_{2},e}^{1,1}(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2}; τ), (4.11), we have

(β(Ψ_{a_{1},a_{2},e}^{1,1}))(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2}; τ) = π(Ψ_{a_{1},a_{2},e}^{1,1}(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2} + τ; τ))

= ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} ∑_{a_{4}∈A} ∑_{k=1}^{N_{a_{4}a_{3}}^{a_{3}}} ∑_{l=1}^{N_{a_{2}a′_{2}}^{a_{4}}} e^{πi(−2h_{a_{2}}+h_{a_{4}})} ·

·F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}}) ⊗ σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})) · ·F (Y_{a_{2}a′_{1};i}^{a′_{3}} ⊗ σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}}); Y_{a_{4}a′_{3};k}^{a′_{3}} ⊗ Y_{a_{2}a′_{2};l}^{a_{4}}) · ·π(E(Tr_{W^{a_{3}}}Y_{a_{4}a_{3};k}^{a_{3}}(U(e^{2πiz_{2}})Y_{a_{2}a′_{2};l}^{a_{4}}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ))

= e^{−2πih_{a_{2}}} ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}}) ⊗ σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})) ·

·F (Y_{a_{2}a′_{1};i}^{a′_{3}} ⊗ σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}}); Y_{ea′_{3};1}^{a′_{3}} ⊗ Y_{a_{2}a′_{2};1}^{e}) ·

·E(Tr_{W^{a_{3}}}Y_{ea_{3};1}^{a_{3}}(U(e^{2πiz_{2}})Y_{a_{2}a′_{2};1}^{e}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} )

= e^{−2πih_{a_{2}}} ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}}) ⊗ σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})) ·

·F (Y_{a_{2}a′_{1};i}^{a′_{3}} ⊗ σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}}); Y_{ea′_{3};k}^{a′_{3}} ⊗ Y_{a_{2}a′_{2};l}^{e}) ·

·Ψ_{a_{3};a_{2}}(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2}; τ).

Thus we obtain β(Ψ_{a_{1},a_{2},e}^{1,1})

= e^{−2πih_{a_{2}}} ∑_{a_{3}∈A} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}}) ⊗ σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})) ·

·F (Y_{a_{2}a′_{1};i}^{a′_{3}} ⊗ σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}}); Y_{ea′_{3};k}^{a′_{3}} ⊗ Y_{a_{2}a′_{2};l}^{e})Ψ_{a_{3};a_{2}}.  (4.20)

Now using (3.5), (3.12) and the relations σ_{12}σ_{23} = σ_{123}, σ_{12}σ_{13} = σ_{132} and σ_{123}σ_{132} = σ_{132}σ_{123} = 1, we have

F^{−1}(Y_{ea_{1};1}^{a_{1}} ⊗ Y_{a_{2}a′_{2};1}^{e}; σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}}) ⊗ σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})) = F (σ_{12}(Y_{ea_{1};1}^{a_{1}}) ⊗ σ_{12}(Y_{a_{2}a′_{2};1}^{e}); σ_{12}(σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}})) ⊗ σ_{12}(σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}}))) = F (Y_{a_{1}e;1}^{a_{1}} ⊗ Y_{a′_{2}a_{2};1}^{e}; σ_{12}(σ_{23}(Y_{a_{2}a′_{1};i}^{a′_{3}})) ⊗ σ_{12}(σ_{13}(Y_{a′_{3}a_{1};j}^{a_{2}})))

= F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{1}a_{1};1}^{e}; σ_{123}(σ_{132}(Y_{a′_{3}a_{1};j}^{a_{2}})) ⊗ σ_{132}(σ_{123}(Y_{a_{2}a′_{1};i}^{a′_{3}})))
= F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{1}a_{1};1}^{e}; Y_{a′_{3}a_{1};j}^{a_{2}} ⊗ Y_{a_{2}a′_{1};i}^{a′_{3}})  (4.21)

and

F (Y_{a_{2}a′_{1};i}^{a′_{3}} ⊗ σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}}); Y_{ea′_{3};k}^{a′_{3}} ⊗ Y_{a_{2}a′_{2};l}^{e})

= F (σ_{132}(σ_{123}(Y_{a′_{3}a_{1};j}^{a_{2}})) ⊗ σ_{123}(Y_{a_{2}a′_{1};i}^{a′_{3}}); σ_{123}(Y_{a_{2}a′_{2};l}^{e}) ⊗ σ_{132}(Y_{ea′_{3};k}^{a′_{3}}))

= F (Y_{a′_{3}a_{1};j}^{a_{2}} ⊗ σ_{123}(Y_{a_{2}a′_{1};i}^{a′_{3}}); Y_{ea_{2};l}^{a_{2}} ⊗ Y_{a′_{3}a_{3};k}^{e}).  (4.22)

From (4.20)–(4.22), we obtain (4.19).

By Proposition 2.11, Ψ_{a_{1},a_{2},e}^{1,1}, a_{1} ∈ A form a basis of F_{1;2}^{e}. For fixed a_{2} ∈ A, we use α_{a_{1}}^{a_{3}}(a_{2}) and β_{a_{1}}^{a_{3}}(a_{2}), a_{1}, a_{3} ∈ A, to denote the matrix elements of α and β, respectively, under the basis Ψ_{a_{1},a_{2},e}^{1,1}, a_{1} ∈ A.

Corollary 4.4 The matrix elements α_{a_{1}}^{a_{3}}(a_{2}) and β_{a_{1}}^{a_{3}}(a_{2}), a_{1}, a_{3} ∈ A, are given by

α_{a_{1}}^{a_{3}}(a_{2}) = δ_{a_{1}a_{3}}(B^{(−1)})^{2}(Y_{a_{1}e;1}^{a_{1}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{1}e;1}^{a_{1}} ⊗ Y_{a′_{2}a_{2};1}^{e})  (4.23)

and

β_{a_{1}}^{a_{3}}(a_{2}) = e^{−2πih_{a_{2}}} ∑_{i=1}^{N_{a_{2}a_{3}}^{a_{1}}} ∑_{j=1}^{N_{a′_{2}a_{1}}^{a_{3}}} F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{1}a_{1};1}^{e}; Y_{a′_{3}a_{1};j}^{a_{2}} ⊗ Y_{a_{2}a′_{1};i}^{a′_{3}}) ·
·F (Y_{a′_{3}a_{1};j}^{a_{2}} ⊗ σ_{123}(Y_{a_{2}a′_{1};i}^{a′_{3}}); Y_{ea_{2};l}^{a_{2}} ⊗ Y_{a′_{3}a_{3};k}^{e}).

(4.24)

Proof. This corollary follows directly from the definition of α_{a_{1}}^{a_{3}}(a_{2}) and β_{a_{1}}^{a_{3}}(a_{2}), a_{1}, a_{3} ∈ A, (4.18) and (4.19).

It is also easy to establish the relationship between α and β:

Proposition 4.5 We have the following formula:

SαS^{−1} = β.  (4.25)

Proof. We have (β(S(Ψ_{a_{1},a_{2},e}^{1,1})))(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2}; τ)

= π(S(Ψ_{a_{1},a_{2},e}^{1,1})(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2} + τ; τ))

= π(E(Tr_{W^{a_{1}}}Y_{ea_{1};1}^{a_{1}}(U(e^{−2πi \frac{z_{2}+τ}{τ}}) (−\frac{1}{τ})^{L(0)} ·

·Y_{a_{2}a′_{2};1}^{e}(w_{a_{2}}, z_{1} − z_{2} + τ)w_{a′_{2}}, e^{−2πi \frac{z_{2}+τ}{τ}})q_{− \frac{1}{τ}}^{L(0)− \frac{c}{24}} ))

= π(E(Tr_{W^{a_{1}}}Y_{ea_{1};1}^{a_{1}}(U(e^{2πi(− \frac{z_{2}}{τ} −1)}) ·

·Y_{a_{2}a′_{2};1}^{e}( (−\frac{1}{τ})^{L(0)} w_{a_{2}}, −\frac{1}{τ}z_{1} − (−\frac{1}{τ}z_{2} − 1)) ·

· (−\frac{1}{τ})^{L(0)} w_{a′_{2}}, e^{2πi(− \frac{z_{2}}{τ} −1)})q_{− \frac{1}{τ}}^{L(0)− \frac{c}{24}} ))

= π (Ψ_{a_{1},a_{2},e}^{1,1} ((−\frac{1}{τ})^{L(0)} w_{a_{2}}, (−\frac{1}{τ})^{L(0)} w_{a′_{2}}; −\frac{1}{τ}z_{1}, −\frac{1}{τ}z_{2} − 1; −\frac{1}{τ}))

= (α(Ψ_{a_{1},a_{2},e}^{1,1})) ((−\frac{1}{τ})^{L(0)} w_{a_{2}}, (−\frac{1}{τ})^{L(0)} w_{a′_{2}}; −\frac{1}{τ}z_{1}, −\frac{1}{τ}z_{2}; −\frac{1}{τ})

= (S(α(Ψ_{a_{1},a_{2},e}^{1,1})))(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2}; τ). Thus we obtain

βS = Sα

or equivalently (4.25).

With the basis of the spaces of intertwining operators we choose in the beginning of this section, we have

Ψ_{a_{1}}(u; τ) = Tr_{W^{a_{1}}}Y_{ea_{1};1}^{a_{1}}(U(e^{2πiz})u, e^{2πiz})q_{τ}^{L(0)− \frac{c}{24}}
= Tr_{W^{a_{1}}}Y_{W^{a_{1}}}(U(e^{2πiz})u, e^{2πiz})q_{τ}^{L(0)− \frac{c}{24}}

for a_{1} ∈ A, u ∈ V and

Ψ_{a_{1},a_{2},e}^{1,1}(w_{a_{2}}, w_{a′_{2}}; z_{1}, z_{2}; τ) = E(Tr_{W^{a_{1}}}Y_{ea_{1};1}^{a_{1}}(U(e^{2πiz_{2}})Y_{a_{2}a′_{2};1}^{e}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ) = E(Tr_{W^{a_{1}}}Y_{W^{a_{1}}}(U(e^{2πiz_{2}})Y_{a_{2}a′_{2};1}^{e}(w_{a_{2}}, z_{1} − z_{2})w_{a′_{2}}, e^{2πiz_{2}})q_{τ}^{L(0)− \frac{c}{24}} ) for a_{1}, a_{2} ∈ A, w_{a_{2}} ∈ W^{a_{2}} and w_{a′_{2}} ∈ W^{a′_{2}}.

Since Ψ_{a_{1}} for a_{1} ∈ A are linear independent, they form a basis of F_{1;1}. Thus we know that there exist unique S_{a_{1}}^{a_{3}} ∈ ℂ for a_{1}, a_{3} ∈ A such that

(S(Ψ_{a_{1}}))(u; τ) = ∑_{a_{2}∈A} S_{a_{1}}^{a_{2}}Ψ_{a_{1}}(u; τ),  (4.26)
S(Ψ_{a_{1}}) = ∑_{a_{3}∈A} S_{a_{1}}^{a_{3}}Ψ_{a_{3}}.  (4.27)

Clearly we also have

S(Ψ_{a_{1},a_{2},e}^{1,1}) = ∑_{a_{3}∈A} S_{a_{1}}^{a_{3}}Ψ_{a_{3}a_{2},e}^{1,1}

for a_{1}, a_{2} ∈ A.

Let u = 1 in (4.26). Then we see that S_{a_{1}}^{a_{2}}, a_{1}, a_{2} ∈ A, are the entries of the matrix representing the action of the modular transformation τ ↦ −1/τ on the space spanned by shifted graded dimensions (vacuum characters) of irreducible V -modules. Since the set of shifted graded dimensions (vacuum characters) of irreducible modules is linearly independent and is in bijection with the set A, we can view S as a linear operator on the vector space spanned by A.

The following theorem gives the second Moore-Seiberg formula in [MS1]:

Theorem 4.6 For a_{1}, a_{2}, a_{3} ∈ A, we have

∑_{a_{4}∈A} S_{a_{1}}^{a_{4}}(B^{(−1)})^{2}(Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e})(S^{−1})_{a_{4}}^{a_{3}}

= ∑_{i=1}^{N_{a_{1}a_{2}}^{a_{3}}} ∑_{k=1}^{N_{a′_{1}a_{3}}^{a_{2}}} F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{3}a_{3};1}^{e}; Y_{a′_{1}a_{3};k}^{a_{2}} ⊗ Y_{a_{2}a′_{3};i}^{a′_{1}}) ·

·F (Y_{a′_{1}a_{3};k}^{a_{2}} ⊗ σ_{123}(Y_{a_{2}a′_{3};i}^{a′_{1}}); Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a′_{1}a_{1};1}^{e}) (4.28)

Proof. This follows from (4.25), (4.23) and (4.24) immediately.

Corollary 4.7 For a_{1}, a_{2}, a_{3} ∈ A, we have

∑_{a_{4}∈A} S_{a_{1}}^{a_{4}}(B^{(−1)})^{2}(Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e})(S^{−1})_{a_{4}}^{a_{3}}

= N_{a_{1}a_{2}}^{a_{3}}F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a_{2}a′_{2};1}^{e}).  (4.29)

Proof. This follows immediately from (4.9) and (4.28).

5 The main theorem and the Verlinde formula

In this section, using the results obtained in the preceding section, we prove the main theorem, the Verlinde conjecture, of the present paper, derive the Verlinde formula for fusion rules and prove that (S_{a_{1}}^{a_{2}}) is symmetric.

First, we have the following:

Proposition 5.1 For a_{2} ∈ A,

F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a_{2}a′_{2};1}^{e}) ≠ 0.

Proof. If

F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a_{2}a′_{2};1}^{e}) = 0,

then by (4.29),

(B^{(−1)})^{2}(Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e}) = 0

for a_{4} ∈ A. But we know that

(B^{(−1)})^{2}(Y_{ee;1}^{e} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{ee;1}^{e} ⊗ Y_{a′_{2}a_{2};1}^{e}) = 1.

Contradiction.

For a_{2} ∈ A, let ℕ(a_{2}) be the matrices whose entries are N_{a_{1}a_{2}}^{a_{3}} = N_{a_{2}a_{1}}^{a_{3}} for a_{1}, a_{3} ∈ A, that is,

N (a_{2}) = (N_{a_{1}a_{2}}^{a_{3}}) = (N_{a_{2}a_{1}}^{a_{3}}).

Then we have the main result of the present paper:

Theorem 5.2 Let V be a simple vertex operator algebra satisfying the conditions in Section 1. Then we have

∑_{a_{1},a_{3}∈A}(S^{−1})_{a_{4}}^{a_{1}}N_{a_{1}a_{2}}^{a_{3}}S_{a_{3}}^{a_{5}} = δ_{a_{4}}^{a_{5}} \frac{(B^{(−1)})^{2}(Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e})}{F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a_{2}a′_{2};1}^{e})}.

(5.1)

In particular, the matrix S diagonalizes the matrices N (a_{2}) for all a_{2} ∈ A.

Proof. By Proposition 5.1, we can rewrite (4.29) as (5.1). Since the righthand side of (5.1) are entries of diagonal matrices, S diagonalizes the N (a_{2}) for all a_{2} ∈ A.

We now prove the Verlinde formula for fusion rules. We first need the following:

Proposition 5.3 The square S^{2} viewed as a linear operator on the vector space spanned by A is equal to the linear operator obtained from the map ′ : A → A.

Proof. By definition, we have

(S^{2}(Ψ_{a}))(u; τ) = (S(Ψ_{a}) ((−\frac{1}{τ})^{L(0)} u; −\frac{1}{τ})

= Ψ_{a} ((−\frac{1}{−\frac{1}{τ}} )^{L(0)} (−\frac{1}{τ})^{L(0)} u; −\frac{1}{−\frac{1}{τ}} )

= Ψ_{a} (τ^{L(0)} (−\frac{1}{τ})^{L(0)} u; τ)
= Ψ_{a} (e^{(log τ+log(− \frac{1}{τ}))L(0)}u; τ) .  (5.2)

Note that both τ and −\frac{1}{τ} are in the upper half plane. So 0 < arg τ, arg(−\frac{1}{τ}) < π. Thus by our convention,

arg τ + arg(−\frac{1}{τ}) = arg(−1) = π.

So we have

log τ + log (−\frac{1}{τ}) = log |τ| + i arg τ log |−\frac{1}{τ}| + i arg (−\frac{1}{τ})
= πi.  (5.3)

Using (5.3) and (2.20), we see that the right-hand side of (5.2) is equal to

Ψ_{a}(e^{πiL(0)}u; τ) = Tr_{W^{a}}Y_{W^{a}}(U(e^{−2πiz})e^{πiL(0)}u, e^{−2πiz})q_{τ}^{L(0)− \frac{c}{24}}
= Tr_{W^{a′}}Y_{W^{a′}}(U(e^{2πiz_{2}})u, e^{2πiz})q_{τ}^{L(0)− \frac{c}{24}} )
= Ψ_{a′} (u; τ)  (5.4)

Combining (5.2) and (5.4), we obtain

S^{2}(Ψ_{a}) = Ψ_{a′},

proving the conclusion.

An immediate consequence of the proposition above is the following: Corollary 5.4 The inverse S^{−1} of S is equal to S ∘ ′ =′ ∘S. In particular, we have

(S^{−1})_{a_{1}}^{a_{2}} = S_{a_{1}}^{a′_{2}}
= S_{a′_{1}}^{a_{2}}  (5.5)

for a_{1}, a_{2} ∈ A.

Now we have the following Verlinde formula for fusion rules: Theorem 5.5 Let V be a vertex operator algebra satisfying the conditions in Section 1. Then we have

N_{a_{1}a_{2}}^{a_{3}} = ∑_{a_{4}∈A} \frac{S_{a_{1}}^{a_{4}}S_{a_{2}}^{a_{4}}S_{a_{4}}^{a′_{3}}}{S_{0}^{a_{4}}} .  (5.6)

Proof. Let

λ_{a_{2}}^{a_{4}} = \frac{(B^{(−1)})^{2}(Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e})}{F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a_{2}a′_{2};1}^{e})}  (5.7)

for a_{2}, a_{4} ∈ A. Then by (5.1), we have

∑_{a_{1},a_{3}∈A}(S^{−1})_{a_{4}}^{a_{1}}N_{a_{1}a_{2}}^{a_{3}}S_{a_{3}}^{a_{5}} = δ_{a_{4}}^{a_{5}}λ_{a_{2}}^{a_{4}},

or equivalently

N_{a_{1}a_{2}}^{a_{3}} = ∑_{a_{4}∈A} S_{a_{1}}^{a_{4}}λ_{a_{2}}^{a_{4}}(S^{−1})_{a_{4}}^{a_{3}}.  (5.8)

Using (5.5), we see that (5.8) becomes

N_{a_{1}a_{2}}^{a_{3}} = ∑_{a_{4}∈A} S_{a_{1}}^{a_{4}}λ_{a_{2}}^{a_{4}}S_{a_{4}}^{a′_{3}}.  (5.9)

We know that N_{ea_{2}}^{a_{3}} = δ_{a_{2}}^{a_{3}}. Combining this fact with (5.8), we obtain

δ_{a_{2}}^{a_{3}} = ∑_{a_{4}∈A} S_{e}^{a_{4}}λ_{a_{2}}^{a_{4}}(S^{−1})_{a_{4}}^{a_{3}}.

Thus we have

λ_{a_{2}}^{a_{4}} = \frac{S_{a_{2}}^{a_{4}}}{S_{e}^{a_{4}}} .  (5.10)

Substituting (5.10) into (5.9), we obtain (5.6).

Finally we have: Theorem 5.6 The matrix (S_{a_{1}}^{a_{2}}) is symmetric. Proof. Rewriting (5.8) as

∑_{a_{1}}(S^{−1})_{a_{4}}^{a_{1}}N_{a_{1}a_{2}}^{a_{3}} = λ_{a_{2}}^{a_{4}}(S^{−1})_{a_{4}}^{a_{3}},  (5.11)

and then letting a_{3} = a_{4} = e in (5.11) and using N_{a_{1}a_{2}}^{e} = δ_{a_{1}}^{a′_{2}}, we obtain

(S^{−1})_{e}^{a′_{2}} = λ_{a_{2}}^{e}(S^{−1})_{e}^{e}.  (5.12)

Using (5.5), (5.12) can be written as

S_{e}^{a_{2}} = λ_{a_{2}}^{e}S_{e}^{e}.  (5.13)

By (5.10), (5.13) and (5.7), S_{a_{2}}^{a_{4}} = λ_{a′_{4}}^{e}S_{e}^{e}λ_{a_{2}}^{a_{4}}

= \frac{S_{e}^{e}((B^{(−1)})^{2}(Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e}))}{F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a_{2}a′_{2};1}^{e})F (Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{4}a_{4};1}^{e}; Y_{ea_{4};1}^{a_{4}} ⊗ Y_{a_{4}a′_{4};1}^{e})}.

(5.14)

By (5.14), we obtain

S_{a_{2}}^{a′_{4}} = \frac{S_{e}^{e}((B^{(−1)})^{2}(Y_{a′_{4}e;1}^{a′_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a′_{4}e;1}^{a′_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e}))}{F (Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{ea_{2};1}^{a_{2}} ⊗ Y_{a_{2}a′_{2};1}^{e})F (Y_{a′_{4}e;1}^{a′_{4}} ⊗ Y_{a_{4}a′_{4};1}^{e}; Y_{ea′_{4};1}^{a′_{4}} ⊗ Y_{a′_{4}a_{4};1}^{e})}.

(5.15)

From (3.1), (3.12), (3.5) and the choice of the basis of the spaces of intertwining operators when some of the modules involved are V, we have

(B^{(−1)})^{2}(Y_{a′_{4}e;1}^{a′_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e}; Y_{a′_{4}e;1}^{a′_{4}} ⊗ Y_{a′_{2}a_{2};1}^{e})
= (B^{(−1)})^{2}(Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a_{4}a′_{4};1}^{e}; Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a_{4}a′_{4};1}^{e}).  (5.16)

Using (5.16) and (5.14), we see that the right-hand side of (5.15) is equal to

\frac{S_{e}^{e}((B^{(−1)})^{2}(Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a_{4}a′_{4};1}^{e}; Y_{a_{2}e;1}^{a_{2}} ⊗ Y_{a_{4}a′_{4};1}^{e}))}{F (Y_{a′_{2}e;1}^{a′_{2}} ⊗ Y_{a_{2}a′_{2};1}^{e}; Y_{ea′_{2};1}^{a′_{2}} ⊗ Y_{a′_{2}a_{2};1}^{e})F (Y_{a_{4}e;1}^{a_{4}} ⊗ Y_{a′_{4}a_{4};1}^{e}; Y_{ea′_{4};1}^{a′_{4}} ⊗ Y_{a_{4}a′_{4};1}^{e})} = S_{a′_{4}}^{a_{2}}.

(5.17)

The formulas (5.15) and (5.17) gives

S_{a_{2}}^{a′_{4}} = S_{a′_{4}}^{a_{2}},

proving that (S_{a_{1}}^{a_{2}}) is symmetric.

References

[AN] T. Abe and K. Nagatomo, Finiteness of conformal blocks over the projective line, in: Vertex operator algebras in mathematics and physics (Toronto, 2000), ed. S. Berman, Y. Billig, Y.-Z. Huang and J. Lepowsky, Fields Inst. Commun., Vol. 39, Amer. Math. Soc., Providence, 2003, 1–12.

[AM] G. Anderson and G. Moore, Rationality in conformal field theory, Comm. Math. Phys.117 (1988), 441–450.

[BL] A. Beauville and Y. Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), 385–419.

[B] R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA 83 (1986), 3068–3071.

[DLM] C. Dong, H. Li and G. Mason, Modular-invariance of trace functions in orbifold theory and generalized Moonshine, Comm. Math. Phys. 214 (2000), 1–56.

[F] G. Faltings, A proof for the Verlinde formula, J. Alg. Geom. 3 (1994), 347–374.

[FHL] I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, preprint, 1989; Memoirs Amer. Math. Soc. 104, 1993.

[FLM] I. B. Frenkel, J. Lepowsky, and A. Meurman, Vertex operator algebras and the Monster, Pure and Appl. Math., Vol. 134, Academic Press, New York, 1988.

[GN] M. R. Gaberdiel and A. Neitzke, Rationality, quasirationality and finite W-algebras, Comm. Math. Phys. 238 (2003), 305–331.

[H1] Y.-Z. Huang, A theory of tensor products for module categories for a vertex operator algebra, IV, J. Pure Appl. Alg. 100 (1995), 173-216.

[H2] Y.-Z. Huang, Virasoro vertex operator algebras, (nonmeromorphic) operator product expansion and the tensor product theory, J. Alg. 182 (1996), 201–234.

[H3] Y.-Z. Huang, Two-dimensional conformal geometry and vertex operator algebras, Progress in Mathematics, Vol. 148, 1997, Birkhäuser, Boston.

[H4] Y.-Z. Huang, Generalized rationality and a “Jacobi identity” for intertwining operator algebras, Selecta Math. (N. S.), 6 (2000), 225–267.

[H5] Y.-Z. Huang, Differential equations and intertwining operators, Duke Math. J., to appear.

[H6] Y.-Z. Huang, Differential equations, duality and modular invariance, to appear; math.QA/0303049.

[HL1] Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, I, Selecta Math. (N.S.) 1 (1995), 699-756.

[HL2] Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, II, Selecta Math. (N.S.) 1 (1995), 757–786.

[HL3] Y.-Z. Huang and J. Lepowsky, Tensor products of modules for a vertex operator algebra and vertex tensor categories, in: Lie Theory and Geometry, in honor of Bertram Kostant, ed. R. Brylinski, J.-L. Brylinski, V. Guillemin, V. Kac, Birkhäuser, Boston, 1994, 349– 383.

[HL4] Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, III, J. Pure Appl. Alg. 100 (1995), 141-171.

[KNR] S. Kumar, M. S. Narasimhan and A. Ramanathan, Infinite Grassmannians and moduli spaces of G-bundles, Math. Ann. 300 (1994), 41–75.

[LL] J. Lepowsky and H. Li, Introduction to vertex operator algebras and their representations, Progress in Mathematics, Vol. 227, 2004 Birkhüser, Boston.

[L] H. Li, Some finiteness properties of regular vertex operator algebras, J. Alg. 212 (1999), 495–514.

[M] M. Miyamoto, Intertwining operators and modular invariance, to

appear; math.QA/0010180.

[MS1] G. Moore and N. Seiberg, Polynomial equations for rational conformal field theories, Phys. Lett. B 212 (1988), 451–460.

[MS2] G. Moore and N. Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), 177–254.

[S1] G. B. Segal, The definition of conformal field theory, preprint, 1988.

[S2] G. B. Segal, Two-dimensional conformal field theories and modular functors, in: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, 1988, Hilger, Bristol, 1989, 22– 37.

[TUY] A. Tsuchiya, K. Ueno and Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, in: Advanced Studies in Pure Math., Vol. 19, Kinokuniya Company Ltd., Tokyo, 1989, 459–565.

[V] E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B300 (1988), 360–376.

[Z] Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237–307.

Max Planck Institute of Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany

Institute of Mathematics, Fudan University, Shanghai, China

Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019 (permanent address)

E-mail address: yzhuang@math.rutgers.edu