VERTEX OPERATOR ALGEBRAS AND THE VERLINDE CONJECTURE

By Yi-Zhi Huang. 2004

209 citations recorded by the source corpus.

We prove the Verlinde conjecture in the following general form: Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0, V(0) = ℂ1 and V′ is isomorphic to V as a V-module. (ii) Every ℕ-gradable weak V-module is completely reducible. (iii) V is C2-cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V-module is completely reducible.) Then the matrices formed by the fusion rules among the 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. Using this result, we obtain the Verlinde formula for the fusion rules. We also prove that the matrix associated to the modular transformation τ ↦ -1/τ is symmetric.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0b728f5983755dc54b0793b5d3d3f3389af00d6a

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Citations in the corpus

Regularity of fixed-point vertex operator subalgebras (2016, methodology): “We also use the assumption sT,T 6= 0, which follows from the Verlinde formula, whose proof in [Huang 2004] assumes T is of “CFT type”.”
Fusion categories for affine vertex algebras at admissible levels (2018, methodology): “By now one understands this well in the instance of strongly rational vertex operator algebras [1, 2].”
A remark on the C2-cofiniteness condition on vertex algebras (2010, background): “This condition has been assumed in many significant theories of vertex operator algebras, such as [6,16,17,25,26,29].”
Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I: Introduction and Strongly Graded Algebras and their Generalized Modules (2010, background): “e Verlinde conjecture without the assumption of the axioms for a rational conformal ﬁeld theory. The Verlinde conjecture for suitable vertex operator algebras was proved in 2004 by the ﬁrst author in [H12] (without the assumption of the axioms for a rational conformal ﬁeld theory), and its proof in ”