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.

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U2 - 10.1073/pnas.0409901102

DO - 10.1073/pnas.0409901102

M3 - Article

C2 - 15809423

AN - SCOPUS:17244380729

SN - 0027-8424

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

IS - 15

ER -