REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

Jethro van Ekeren

ABSTRACT: In my previous talk I explained Zhu's theorem on modular invariance of vertex algebras. In his proof Zhu assumed that certain numbers, called the conformal weights of the vertex algebra, were integers. There are many interesting examples of vertex algebras for which this condition fails (especially when we pass to the supersymmetric world), and although Zhu's theorem generally also fails for these vertex algebras, one expects some analogue of it to hold.

In this talk I will explain how to modify Zhu's constructions to prove that if we include certain so-called twisted modules alongside the ordinary ones, then we recover modular invariance. I will conclude with examples (a superconformal vertex algebra in particular) demonstrating that vertex algebras with non-integer conformal weights can give rise to nontrivial modular forms of non-integer weight.