Chiral homology, the Zhu algebra, and Rogers-Ramanujan

Abstract: Graded dimensions of rational vertex algebras are modular functions. The proof of this celebrated theorem by Y. Zhu centres on geometric objects attached to elliptic curves known as conformal blocks, and their behaviour in the limit as the underlying curve becomes singular. In this limit, roughly speaking, conformal blocks pass to the degree zero Hochschild homology of Zhu's associative algebra. On the other hand, conformal blocks have been interpreted by Beilinson and Drinfeld as the degree zero component of a theory of chiral homology. It is therefore natural to wonder if the relationship extends to higher homological degrees. We are indeed able to extend this story to homological degree 1 for classically free vertex algebras, and in the process we discover relations with objects of number theory such as the Rogers-Ramanujan identity and its generalisations. This is joint work with R. Heluani and G. Andrews.

algebraic geometryrepresentation theory

Audience: researchers in the topic

Algebra and Geometry Seminar @ HKUST

Series comments: Algebra and Geometry seminar at the Hong Kong University of Science and Technology (HKUST).

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