Group actions on quiver varieties and applications

Victoria Hoskins, Florent Schaffhauser

Published 2016-12-20Version 1

We study two types of actions on moduli spaces of quiver representations over a field $k$ and we decompose their fixed loci using group cohomology. First, for a perfect field $k$, we study the action of the absolute Galois group of $k$ on the $\overline{k}$-valued points of this quiver moduli space; the fixed locus is the set of $k$-rational points and we obtain a decomposition of this fixed locus indexed by elements in the Brauer group of $k$. Second, we study algebraic actions of finite groups of quiver automorphisms on these moduli spaces; the fixed locus is decomposed using group cohomology and we give a modular interpretation of each component. As an application, we construct branes in hyperk\"ahler quiver varieties, as fixed loci of such actions.