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

Research output: Contribution to journalArticle

45 Citations (Scopus)

### Abstract

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0, V0 = ℂ1, and the contragredient module 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 announce a proof of the Verlinde conjecture for V, that is, of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation τ → -1/τ on the space of characters of irreducible V-modules. We discuss some consequences of the Verlinde conjecture, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of τ → -1/τ, and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of V-modules when V satisfies in addition the condition that irreducible V-modules not equivalent to V have no nonzero elements of weight 0. In particular, the category of V-modules has a natural structure of modular tensor category.

Original language English (US) 5352-5356 5 Proceedings of the National Academy of Sciences of the United States of America 102 15 https://doi.org/10.1073/pnas.0409901102 Published - Apr 12 2005

### Fingerprint

Weights and Measures

• General

### Cite this

@article{6039f4e865ed453faf28cc6f25f57737,
title = "Vertex operator algebras, the Verlinde conjecture, and modular tensor categories",
abstract = "Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0, V0 = ℂ1, and the contragredient module 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 announce a proof of the Verlinde conjecture for V, that is, of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation τ → -1/τ on the space of characters of irreducible V-modules. We discuss some consequences of the Verlinde conjecture, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of τ → -1/τ, and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of V-modules when V satisfies in addition the condition that irreducible V-modules not equivalent to V have no nonzero elements of weight 0. In particular, the category of V-modules has a natural structure of modular tensor category.",
author = "Yi-Zhi Huang",
year = "2005",
month = "4",
day = "12",
doi = "10.1073/pnas.0409901102",
language = "English (US)",
volume = "102",
pages = "5352--5356",
journal = "Proceedings of the National Academy of Sciences of the United States of America",
issn = "0027-8424",
number = "15",

}

In: Proceedings of the National Academy of Sciences of the United States of America, Vol. 102, No. 15, 12.04.2005, p. 5352-5356.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Vertex operator algebras, the Verlinde conjecture, and modular tensor categories

AU - Huang, Yi-Zhi

PY - 2005/4/12

Y1 - 2005/4/12

N2 - Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0, V0 = ℂ1, and the contragredient module 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 announce a proof of the Verlinde conjecture for V, that is, of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation τ → -1/τ on the space of characters of irreducible V-modules. We discuss some consequences of the Verlinde conjecture, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of τ → -1/τ, and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of V-modules when V satisfies in addition the condition that irreducible V-modules not equivalent to V have no nonzero elements of weight 0. In particular, the category of V-modules has a natural structure of modular tensor category.

AB - Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0, V0 = ℂ1, and the contragredient module 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 announce a proof of the Verlinde conjecture for V, that is, of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation τ → -1/τ on the space of characters of irreducible V-modules. We discuss some consequences of the Verlinde conjecture, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of τ → -1/τ, and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of V-modules when V satisfies in addition the condition that irreducible V-modules not equivalent to V have no nonzero elements of weight 0. In particular, the category of V-modules has a natural structure of modular tensor category.

UR - http://www.scopus.com/inward/record.url?scp=17244380729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17244380729&partnerID=8YFLogxK

U2 - 10.1073/pnas.0409901102

DO - 10.1073/pnas.0409901102

M3 - Article

C2 - 15809423

AN - SCOPUS:17244380729

VL - 102

SP - 5352

EP - 5356

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

SN - 0027-8424

IS - 15

ER -