Donaldson-Thomas theory for categories of homological dimension one with potential

TL;DR

This paper provides an axiomatic framework for Donaldson-Thomas theory in categories of homological dimension one with potential, establishing key results and connecting different approaches and realizations.

Contribution

It introduces an axiomatic presentation of Donaldson-Thomas theory for these categories, proving standard results and relating functions with arbitrary potential to zero potential cases.

Findings

Established rigorous proofs of integration map, wall-crossing, and PT-DT correspondence.

Proved equivalence of Kontsevich-Soibelman and Joyce-Song approaches.

Provided geometric interpretation of Donaldson-Thomas functions across various realizations.

Abstract

The aim of the paper is twofold. Firstly, we give an axiomatic presentation of Donaldson-Thomas theory for categories of homological dimension at most one with potential. In particular, we provide rigorous proofs of all standard results concerning the integration map, wall-crossing, PT-DT correspondence, etc. following Kontsevich and Soibelman. We also show the equivalence of their approach and the one given by Joyce and Song. Secondly, we relate Donaldson-Thomas functions for such a category with arbitrary potential to those with zero potential under some mild conditions. As a result of this, we obtain a geometric interpretation of Donaldson-Thomas functions in all known realizations, i.e. mixed Hodge modules, perverse sheaves and constructible functions.