Abstract

Let A be a C-linear additive category coming from geometry, e.g. an abelian category or derived category of coherent sheaves on a smooth projective scheme, or of representations of a quiver with relations. Then we can form two moduli stacks of objects in A: the usual moduli stack M, and the “projective linear” moduli stack M^{, in which we rigidify M by quotienting out isotropy groups by multiples of the identity. There is a morphism \pi : M —> M}{pl} which is a [/G_m]-fibration.   We form the homology H_(M,Q), H_, by which we mean the ordinary homology of the topological classifying space of the stacks.  I will explain how under very weak assumptions (basically we need total Ext groups Ext(E,F) to be finite-dimensional for all E,F in A) we can give H_ the structure of a graded vertex algebra, and H_(M{pl},Q) the structure of a graded Lie algebra (which is that constructed from the vertex algebra H_ by Borcherds).  These vertex and Lie algebras appear naturally in the study of enumerative invariants (e.g. of semistable coherent sheaves on projective surfaces, or Fano 4-folds), and this was how I found them.   The vertex algebras can be computed explicitly in a lot of cases, and usually turn out to be some form of super-lattice vertex algebra. In particular, by work of my student Jacob Gross, if A = D^{b coh(X) for X a projective curve, surface, or smooth toric variety, then we can write down H_}(M,Q), H_*(M{pl},Q) with their vertex and Lie algebra structures completely explicitly (the formulae are complicated).  The vertex algebra construction admits a lot of interesting generalizations. For example, there is a version for equivariant homology if a group G acts on A, M, M^{pl}. For complex-oriented generalized homology theories, such as K-theory, one can define a variant of vertex algebra defined using the formal group law attached to the generalized homology theory. Work by my student Chenjing Bu defines “twisted” representations of a vertex algebra on homology of moduli spaces of self-dual objects in a self-dual abelian category  (e.g. vector bundles with orthogonal or symplectic structures). For “odd Calabi-Yau” categories A there is a variant which produces a vertex Lie algebra.