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Lie Group Quantum Mathematics Seminar

Conformal blocks for C_2-cofinite vertex operator algebras

Hao Zhang, Tsinghua University

Location:  Hill 705
Date & time: Friday, 19 April 2024 at 12:10PM - 1:10PM

Vertex operator algebras (VOAs) mathematically characterize 2d conformal field theories. For C_2-cofinite and rational VOAs (plus some minor assumptions), Yi-Zhi Huang proved that their representation categories are rigid modular tensor categories. For C_2-cofinite (not necessarily rational) VOAs, Huang-Lepowsky-Zhang proved that their representation categories are braided tensor categories, which mean that the rigidity and modularity are still open problems. Geometrically, one of the main reasons is that, for C_2-cofinite VOAs, their conformal blocks associated to compact Riemann surfaces of genera 1 were not clear at that time. In 2023, Yi-Zhi Huang proved that such conformal blocks can be expressed by pseudotraces of genus 0 conformal blocks (intertwining operators). This result is expected to be used to prove rigidity (and hence modularity).

In this talk, I will introduce a systematic approach towards higher genus conformal blocks for C_2-cofinite VOAs and explain the relationship between our work and Yi-Zhi Huang’s results above. This is based on an ongoing project (arXiv: 2305.10180) joint with Bin Gui.