Chern-Simons functions on toric Calabi-Yau threefolds and Donaldson-Thomas theory

TL;DR

This paper constructs global Chern-Simons functions on toric Calabi-Yau threefolds, linking them to moduli spaces of sheaves and applying these to prove conjectures and derive recursion formulas for Donaldson-Thomas invariants.

Contribution

It introduces a method to construct Chern-Simons functions on toric Calabi-Yau stacks and applies this to prove Joyce's integrality conjecture and derive recursion formulas.

Findings

Proved Joyce's integrality conjecture for generalized DT invariants.

Established a dimension reduction formula for virtual motives.

Derived recursion formulas for motivic Donaldson-Thomas invariants.

Abstract

In this paper, we give a construction of the global Chern-Simons functions for toric Calabi-Yau stacks of dimension three using strong exceptional collections. The moduli spaces of sheaves on such stacks can be identified with critical loci of these functions. We give two applications of these functions. First, we prove Joyce's integrality conjecture of generalized DT invariants on local surfaces. Second, we prove a dimension reduction formula for virtual motives, which leads to two recursion formulas for motivic Donaldson-Thomas invariants.