In a few cases, for a quasiprojective suface S it is known that there are vertex algebras which provide formulas for enumerative invariants of moduli spaces of sheaves on S. I will explain a number of examples where this is known or expected to hold and focus on extending this structure to the case of elliptic surfaces. I will describe a close relationship between the geometry of moduli spaces of sheaves on elliptic surfaces and the construction of vertex representations of toroidal lie algebras and their supersymmetric and logarithmic extensions.