Cohomological Hall algebras for 3-Calabi-Yau categories

TL;DR

This paper constructs cohomological Hall algebras for 3-Calabi-Yau categories, providing a mathematical framework for BPS states and proving a conjecture on Donaldson-Thomas invariants, with applications to 3-manifold theories.

Contribution

It introduces a new construction of cohomological Hall algebras for 3-Calabi-Yau categories with strong orientation data, confirming a conjecture by Joyce and extending Donaldson-Thomas theory.

Findings

Construction of cohomological Hall algebras for 3-Calabi-Yau categories.

Proof of Joyce's conjecture on Donaldson-Thomas perverse sheaves.

Development of a parabolic induction map for 3-manifold invariants.

Abstract

The aim of this paper is to construct the cohomological Hall algebras for $3$-Calabi--Yau categories admitting a strong orientation data. This can be regarded as a mathematical definition of the algebra of BPS states, whose existence was first mathematically conjectured by Kontsevich and Soibelman. Along the way, we prove Joyce's conjecture on the functorial behaviour of the Donaldson--Thomas perverse sheaves for the attractor Lagrangian correspondence of $(-1)$-shifted symplectic stacks. This result allows us to construct a parabolic induction map for cohomological Donaldson--Thomas invariants of $G$-local systems on $3$-manifolds for a reductive group $G$, which can be regarded as a $3$-manifold analogue of the Eisenstein series functor in the geometric Langlands program.