M-theory, enumerative geometry, and representation theory of affine Lie algebras

 I will review some well-established relationships between four manifolds and vertex algebras that can be deduced from studying the M5-brane worldvolume theory, and outline some of the corresponding results in mathematics which have been understood so far. I will then describe a proposal of Gaiotto-Rapcak to generalize these ideas to the setting of multiple M5 branes wrapping divisors in toric Calabi-Yau threefolds, and explain work in progress on understanding the mathematical implications of this proposal as a complex network of relationships between the enumerative geometry of sheaves on threefolds and the representation theory of affine Lie algebras.