Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants

TL;DR

This paper introduces a new type of Cohomological Hall algebra based on the cohomology of representation stacks, generalizes motivic Donaldson-Thomas invariants, and proves their integrality property.

Contribution

It defines a novel Cohomological Hall algebra using cohomology of stacks, extending the theory of motivic Donaldson-Thomas invariants with exponential Hodge structures.

Findings

Defined a new Cohomological Hall algebra for quivers with potentials.

Generalized motivic Donaldson-Thomas invariants via cohomology and exponential Hodge structures.

Proved a new integrality property of motivic Donaldson-Thomas invariants.

Abstract

We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves or functions. In order to take into account the potential we introduce a generalization of theory of mixed Hodge structures, related to exponential integrals. Generating series of our Cohomological Hall algebra is a generalization of the motivic Donaldson-Thomas invariants introduced in arXiv:0811.2435. Also we prove a new integrality property of motivic Donaldson-Thomas invariants.