Representations of cohomological Hall algebras and Donaldson-Thomas theory with classical structure groups

TL;DR

This paper introduces cohomological Hall modules for quivers with involution, extending Donaldson-Thomas theory to include classical structure groups, and proves key conjectures about their properties and invariants.

Contribution

It develops a new class of representations called CoHM for cohomological Hall algebras, generalizes Donaldson-Thomas invariants to orientifold settings, and proves several conjectures about their structure and properties.

Findings

Proves integrality of orientifold Donaldson-Thomas invariants.

Establishes freeness of CoHM for specific quivers.

Constructs explicit bases for CoHM of finite type quivers.

Abstract

We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution $\sigma$ and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to define a generalization of the cohomological Donaldson-Thomas theory of quivers which allows the quiver representations to have orthogonal and symplectic structure groups. The associated invariants are called orientifold Donaldson-Thomas invariants. We prove the integrality conjecture for orientifold Donaldson-Thomas invariants of $\sigma$-symmetric quivers. We also formulate precise conjectures regarding the geometric meaning of these invariants and the freeness of the CoHM of a $\sigma$-symmetric quiver. We prove the freeness conjecture for disjoint union quivers, loop quivers and the affine Dynkin quiver of type $\widetilde{A}_1$. We also verify the geometric conjecture in a number of examples. Finally, we describe the CoHM of finite type quivers by constructing explicit Poincar\'{e}-Birkhoff-Witt type bases of these representations.