# RIGIDITY AND MODULARITY OF VERTEX TENSOR CATEGORIES

```@article{Huang2005RIGIDITYAM,
title={RIGIDITY AND MODULARITY OF VERTEX TENSOR CATEGORIES},
author={Yi-Zhi Huang},
journal={Communications in Contemporary Mathematics},
year={2005},
volume={10},
pages={871-911}
}```
• Yi-Zhi Huang
• Published 25 February 2005
• Mathematics
• Communications in Contemporary Mathematics
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 C2-cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V-module is completely reducible.) Using the results obtained by the author in the formulation and proof of the general version of the…
181 Citations
Vertex operator algebras, the Verlinde conjecture, and modular tensor categories.
• Yi-Zhi Huang
• Mathematics
Proceedings of the National Academy of Sciences of the United States of America
• 2005
A proof of the Verlinde conjecture for V is announced of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation tau |--> -1/tau on the space of characters of irreducing V- modules.
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## References

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Vertex operator algebras, the Verlinde conjecture, and modular tensor categories.
• Yi-Zhi Huang
• Mathematics
Proceedings of the National Academy of Sciences of the United States of America
• 2005
A proof of the Verlinde conjecture for V is announced of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation tau |--> -1/tau on the space of characters of irreducing V- modules.
VERTEX OPERATOR ALGEBRAS AND THE VERLINDE CONJECTURE
We prove the Verlinde conjecture in the following general form: 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
A theory of tensor products for module categories for a vertex operator algebra, I
• Mathematics
• 1993
This is the first part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a “vertex tensor category” structure
Differential equations and intertwining operators
We show that if every module W for a vertex operator algebra V satisfies the condition that the dimension of W/C_1(W) is less than infinity, where C_1(W) is the subspace of W spanned by elements of
Logarithmic tensor product theory for generalized modules for a conformal vertex algebra. Part I
• Mathematics
• 2006
This work generalizes the tensor product theory for modules for a vertex operator algebra to suitable module categories for a ''conformal vertex algebra'' or even more generally, for a "M\"obius vertex algebra.
Differential equations, duality and modular invariance
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Intertwining operator algebras and vertex tensor categories for affine Lie algebras
• Mathematics
• 1997
We apply the general theory of tensor products of modules for a vertex operator algebra developed in our papers hep-th/9309076, hep-th/9309159, hep-th/9401119, q-alg/9505018, q-alg/9505019 and
Virasoro vertex operator algebras, the (nonmeromorphic) operator product expansion and the tensor product theory
Abstract A theory of tensor products of modules for a vertex operator algebra is being developed by Lepowsky and the author. To use this theory, one first has to verify that the vertex operator
Tensor structures arising from affine Lie algebras. III
• Mathematics
• 1993
This paper is a part of the series [KL]; however, it can be read independently of the first two parts. In [D3], Drinfeld proved the existence of an equivalence between a tensor category of