### Abstract

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V_{(n)} = 0 for n < 0, V_{0} = 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) |
---|---|

Pages (from-to) | 5352-5356 |

Number of pages | 5 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 102 |

Issue number | 15 |

DOIs | |

State | Published - Apr 12 2005 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- General

### Cite this

}

**Vertex operator algebras, the Verlinde conjecture, and modular tensor categories.** / Huang, Yi-Zhi.

Research output: Contribution to journal › Article

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 -