Beijing Institute of Mathematical Sciences and Applications

Zheng Wei Liu

(Tsinghua) Hao Zhang

Friday, April 14, 2023 9:30 AM - 10:30 AM

JCY-1

Tencent 494 8360 9451 (2023)

Vertex operator algebras (VOAs) are a mathematically rigorous formulation of 2d conformal field theory (CFT). In 2008, Yi-Zhi Huang proved a significant result that the representation category of a "rational" and "C_2 cofinite" VOA V (plus some minor assumptions) is a modular tensor category. Huang's proof relies on a deep understanding of the geometric properties of conformal blocks (a crucial concept in VOA) associated to compact Rieman surfaces of genera 0 and 1. In recent years, these geometric properties have also been proved for higher genus conformal blocks associated to C_2 cofinite and rational VOAs. However, much less was known for C_2 cofinite but non-rational VOAs. Indeed, only the genus-0 story was clear, which led to Huang-Lepowsky-Zhang's construction of braided tensor categories for the representations of C_2 cofinite VOAs. The categories are not necessarily semisimple since V is not assumed to be rational. In this talk, we present an ongoing project that gives a systematic approach to the geometric properties of conformal blocks of C_2 cofinite but non-rational VOAs, associated to compact Riemann surfaces of all genera. Our approach generalizes simultaneously Huang-Lepowsky-Zhang's tensor category theory and higher level Zhu's algebras.