Reduced Donaldson-Thomas invariants and the ring of dual numbers

TL;DR

This paper develops a framework using equivariant Grothendieck rings and Hall algebras to compute reduced Donaldson-Thomas invariants, confirming conjectured multiple cover formulas in specific cases.

Contribution

It introduces A-equivariant structures and integration maps to analyze reduced DT invariants systematically, providing explicit calculations for certain threefolds.

Findings

Computed reduced DT invariants for K3×E and abelian threefolds

Verified multiple cover formulas in special cases

Established a new algebraic framework for DT invariants

Abstract

Let $A$ be an abelian variety. We introduce $A$-equivariant Grothendieck rings and $A$-equivariant motivic Hall algebras, and endow them with natural integration maps to the ring of dual numbers. The construction allows a systematic treatment of reduced Donaldson-Thomas invariants by Hall algebra techniques. We calculate reduced Donaldson-Thomas invariants for $\mathrm{K3} \times E$ and abelian threefolds for several imprimitive curve classes. This verifies (in special cases) multiple cover formulas conjectured by Oberdieck-Pandharipande and Bryan-Oberdieck-Pandharipande-Yin.