Jethro van Ekeren
IMPA
Conformal blocks for vertex superalgebras.
Abstract: In this talk I will introduce, and generalise, Zhu's theory connecting vertex algebras and modular forms. Let $V$ be a vertex algebra, subject to certain conditions. Zhu's main theorem provides an explicit characterisation of geometric objects associated to $V$, called torus conformal blocks, in terms of characters of $V$-modules. It follows that certain normalised characters of $V$-modules are elliptic modular forms, a result which in turn generalises many well-known appearances of these functions in representation theory. I will give a similar characterisation of torus conformal blocks for a vertex superalgebra in terms of supercharacters and certain `odd' characters of its modules. As a simple application I will determine a character of a module over the Neveu-Schwartz Lie superalgebra whose explicit computation seems to otherwise require detailed knowledge of singular vectors.