2004/12/13 by Yi-Zhi Huang
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebra representation #Algebraic structures and combinatorial models #Combinatorics #Conjecture #Fusion #Fusion rules #Graph #Mathematics #Nonlinear Waves and Solitons #Operator (biology) #Operator algebra #Pure mathematics #Tensor (intrinsic definition) #Vertex (graph theory) #Vertex operator algebra #hep-th #math.AG #math.QA #msc:17B69 #msc:81T40
paper · pdf · doi:10.1073/pnas.0409901102
published as Proc.Nat.Acad.Sci. 102 (2005) 5352-5356 · 18 pages. To appear in the Proc. Natl. Acad. Sci. USA
arxiv created 2004/12/13 · openalex publication_date 2005/04/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let V be a simple vertex operator algebra satisfying the following conditions: ( i ) V ( n ) = 0 for n < 0, \documentclass[12pt]minimal \usepackageamsmath \usepackagewasysym \usepackageamsfonts \usepackageamssymb \usepackageamsbsy \usepackagemathrsfs \setlength\oddsidemargin-69pt \begindocument V0=ℂ1\enddocument , and the contragredient module V ' is isomorphic to V as a V -module; ( ii ) every \documentclass[12pt]minimal \usepackageamsmath \usepackagewasysym \usepackageamsfonts \usepackageamssymb \usepackageamsbsy \usepackagemathrsfs \setlength\oddsidemargin-69pt \begindocument ℕ-gradable\enddocument 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.