Vector fields and admissible embeddings for quiver moduli

TL;DR

This paper introduces a new construction for quiver moduli spaces that simplifies sheaf cohomology calculations and reveals a deep connection between vector fields and Hochschild cohomology, also establishing an admissible embedding of derived categories.

Contribution

It presents a double framing construction that reduces complex cohomology computations to line bundle methods and links vector fields to Hochschild cohomology, with implications for derived category embeddings.

Findings

Vector fields are isomorphic to Hochschild cohomology in many cases.

The universal representation can be viewed as a Fourier-Mukai kernel.

The construction simplifies cohomology calculations for quiver moduli.

Abstract

We introduce a double framing construction for moduli spaces of quiver representations. It allows us to reduce certain sheaf cohomology computations involving the universal representation, to computations involving line bundles, making them amenable to methods from geometric invariant theory. We will use this to show that in many good situations the vector fields on the moduli space are isomorphic as a vector space to the first Hochschild cohomology of the path algebra. We also show that considering the universal representation as a Fourier-Mukai kernel in the appropriate sense gives an admissible embedding of derived categories.