Abstract

Vertex algebras” are complicated algebraic structures coming from Physics, which also play an important role in Mathematics in areas such as monstrous moonshine and geometric Langlands. I will explain a new geometric construction of vertex algebras, which seems to be unknown. The construction applies in many situations in algebraic geometry, differential geometry, and representation theory, and produces vast numbers of new examples. It is also easy to generalize the construction in several ways to produce different types of vertex algebra, quantum vertex algebras, representations of vertex algebras, … It seems to be related to work by Grojnowski, Nakajima and others, which produces representations of interesting infinite-dimensional Lie algebras on the homology of moduli schemes such as Hilbert schemes. Suppose A is a nice abelian category (such as coherent sheaves coh(X) on a smooth complex projective variety X, or representations mod-CQ of a quiver Q) or T is a nice triangulated category (such as D^{b coh(X) or D}bmod-CQ) over C. Let M be the moduli stack of objects in A or T, as an Artin stack or higher stack. Consider the homology H_ over some ring R. Given a little extra data on M, for which there are natural choices in our examples, I will explain how to define the structure of a graded vertex algebra on H_(M). By a standard construction, one can then define a graded Lie algebra from the vertex algebra; roughly speaking, this is a Lie algebra structure on the homology H_ of a “projective linear” version M ^{of the moduli stack M. For example, if we take T = D}bmod-CQ, the vertex algebra H_(M) is the lattice vertex algebra attached to the dimension vector lattice Z^{Q_0} of Q with the symmetrized intersection form. The degree zero part of the graded Lie algebra contains the associated Kac-Moody algebra. There is also a differential-geometric version, involving putting a vertex algebra structure on homology of moduli stacks of connections on a compact manifold X equipped with an elliptic complex E.