bibkey: huang2008verlinde authors: Yi-Zhi Huang year: 2008 title: Vertex operator algebras and the Verlinde conjecture doi: 10.1142/S0219199708002727 claim: Verlinde diagonalization and formula for the canonical genus-one S action on trace functions with insertions, under the stated VOA hypotheses. strata_touched: [] license: citation-only triage: anchor

Verlinde theorem for vertex operator algebras

Huang’s arXiv:math/0406291v3 abstract and introduction state the complete hypotheses: a simple complex VOA V; V_n = 0 for n < 0; V_0 = C times the vacuum; V is isomorphic to its contragredient as a V-module; every nonnegatively integer-gradable weak V-module is completely reducible; and C2-cofiniteness, meaning dim(V/C2(V)) is finite. Theorem 5.2 diagonalizes fusion-rule matrices and Theorem 5.5 gives the Verlinde formula.

The introduction defines the canonical S matrix through the genus-one trace functions with arbitrary state insertions and their coordinate transformations. Setting the insertion to the vacuum gives ordinary characters, but those observations need not identify that matrix: inequivalent contragredient modules can have identical graded dimensions and hence identical characters. Recovering fusion rules from observations therefore requires this canonical S action, or observations rich enough to identify it. An arbitrary matrix compatible with ordinary character transformations is insufficient. The theorem does not apply to a bare rational Fock module or to an unspecified CFT; scalar extension to C alone does not supply the VOA structure or the listed hypotheses.

Verified locator

• Versioned manuscript, introduction, Theorems 5.2 and 5.5: https://arxiv.org/abs/math/0406291v3
• Published DOI: https://doi.org/10.1142/S0219199708002727