The Geometry and Topology of Twisted Quiver Varieties

Abstract

Quivers have a rich history of being used to construct algebraic varieties via their representations in the category of vector spaces. It is also natural to consider quiver representations in a larger category, namely that of vector bundles on some complex variety equipped with a fixed locally free sheaf that twists the morphisms.

For A-type quivers, such representations can be identified with the critical points of a Morse-Bott function on the moduli space of twisted Higgs bundles. Hence these ``twisted quiver varieties'' can be used to extract topological information about the Higgs bundle moduli space. We find a formula for the dimension of the moduli space of twisted representations of A-type quivers and geometric descriptions when each node of the quiver is represented by a line bundle. We then specialize to the so-called ``argyle quivers'', studied using Bradlow-Daskaloploulous stability parameters and pullback diagrams. Next we focus on the Riemann sphere P1 and obtain explicit expressions for the twisted quiver varieties as well as a stratification of these spaces via collisions of invariant zeroes of polynomials. We apply these results to some low-rank Higgs bundle moduli spaces.

We then study representations of cyclic quivers, which can be viewed as corresponding to certain deformations of the Hitchin representations in non-abelian Hodge theory. When all of the ranks are 1, we describe the moduli spaces as subvarieties of the Hitchin system. We also draw out descriptions of the twisted quiver varieties for when the underlying curve is P1 and extend this to some other labellings of the quiver.

We close with a discussion of possible applications of these ideas to hyperpolygon spaces as well as possible directions that use the motivic approach to moduli theory.