Convolution for quiver varieties via cup product on a Morse complex

Abstract

Convolution in Borel-Moore homology plays an important role in Nakajima's construction of representations of the Heisenberg algebra and of modified enveloping algebras of Kac-Moody algebras. In its most basic form, convolution between two quiver varieties is given by pullback and then pushforward via the Hecke correspondence for quivers. In previous work we showed that the Hecke correspondence has a Morse-theoretic interpretation in terms of spaces of flow lines. The goal of this paper is to show that the topological information that defines generators for Nakajima's representations can be encoded in the cup product for a Morse complex defined on the smooth space of representations of a quiver without relations, and then pulling back to the subvariety of representations that do satisfy a given set of relations. The results are valid for the main motivating example of Nakajima quivers, as well as other quivers with relations derived from these (for example handsaw quivers). For the norm square of a moment map on the space of representations of a quiver, the usual Morse-Bott-Smale transversality condition on the space of flow lines fails, however a weaker version of transversality is still satisfied. A major part of the paper is spent developing a general theory in this setting of weak transversality from which one can recover the usual construction of the differentials and cup product on the Morse complex by adding an intermediate step of taking cup product with a certain Euler class, which is explicitly computable for the space of representations of a quiver.