A boson-fermion correspondence in cohomological Donaldson-Thomas theory

TL;DR

This paper develops a fermionization method for the cohomological Hall algebra of preprojective algebra representations, revealing new relationships between Donaldson-Thomas invariants and homology of representation stacks.

Contribution

It introduces a fermionization procedure that switches cohomological parity in the BPS Lie algebra and computes invariants for central extensions of preprojective algebras using deformed dimensional reduction.

Findings

Determined cohomological Donaldson-Thomas invariants of central extensions.

Computed Borel-Moore homology of representation stacks for all dimension vectors.

Unified results on cohomology of deformed and undeformed preprojective algebra representations.

Abstract

We introduce and study a fermionization procedure for the cohomological Hall algebra $\mathcal{H}_{\Pi_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson--Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel-Moore homology of the stack of representations of the $\mu$-deformed preprojective algebra introduced by Crawley-Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras, and my earlier results on the Borel-Moore homology of the stack of representations of the undeformed preprojective algebra.