# Seminars & Colloquia Calendar

Lie Group Quantum Mathematics Seminar

## Towards vector bundles on the moduli space of curves from strongly finite VOAs

#### Angela Gibney, University of Pennsylvania

Location:  Zoom
Date & time: Friday, 25 March 2022 at 12:10PM - 1:10PM

Abstract  Given any vertex operator algebra V, Zhu defined an associative algebra A(V), and showed that to any A(V)-module, one can associate an admissible V-module. This ultimately gives rise to a functor taking n-tuples of finite dimensional A(V)-modules to a sheaf of coinvariants (and its dual sheaf of conformal blocks) on the moduli space of stable n-pointed curves of genus g.  If V is rational and C_2-cofinite, so A(V) is finite and semi-simple, much is known about these sheaves, including that they are coherent (fibers
are finite dimensional) and satisfy a factorization property. Factorization ultimately allows one to show they are vector bundles.  In this talk I will describe a program in which we are aiming for analogous results after removing the assumption of rationality, and weakening C_2-cofiniteness.   As a first step, we replace the standard factorization formula with an inductive one that holds for sheaves defined by modules over any VOA of CFT-type.  As an application, we show that if V is of CFT-type and A(V) is finite, then sheaves of coinvariants and conformal blocks are coherent.  This is a preliminary description of new and ongoing joint work with Daniel Krashen and Chiara Damiolini.