BPS cohomology for 2-Calabi—Yau categories

Abstract

The integrality conjecture predicts how the refined Donaldson—Thomas invariants of Abelian categories are determined by smaller BPS invari- ants. In this thesis, we prove the cohomological integrality conjecture for 2-Calabi—Yau categories, that is, we prove that the underlying mixed Hodge structure of the Borel—Moore homology of the moduli stacks of objects is isomorphic to a symmetric algebra generated by BPS cohomology and the C*-equivariant cohomology of a point.

2-Calabi—Yau (2CY) categories are homological dimension 2 categories together with bifunctorial pairing on the shifted extension groups. This pairing can roughly be thought of as a symplectic form on the category. The ubiquity of 2CY categories is illustrated by the following list of examples: categories of coherent sheaves on symplec- tic surfaces, local systems on Riemann surfaces, categories of Higgs bundles, and representations of preprojective algebras of quivers.

The cohomological Hall algebra (CoHA) A of a 2CY category plays a prominent role. Using the CoHA we identify the BPS algebra as the universal enveloping algebra of a generalised Kac—Moody (GKM) Lie algebra which we define to be the BPS Lie algebra of the 2CY category. Thus we realise the BPS cohomology as the underlying mixed Hodge structure of the GKM Lie algebra; we prove a Poincare ́—Birkhoff—Witt- type isomorphism for the CoHA in terms of the BPS Lie algebra and the C*-equivariant-equivariant cohomology of a point.

The explicit description of the BPS Lie algebra as a GKM algebra reveals a core aspect of our result: the BPS Lie algebra is generated in terms of the intersection cohomology of good moduli spaces of objects in the 2CY category. In the case of totally negative 2CY categories (pairs of nonzero objects of the category have negative Euler pairing) we find the BPS Lie algebra to be the free Lie algebra generated by the intersection cohomology. The main results of the thesis are based on joint work with Ben Davison and Lucien Hennecart.