A boson-fermion correspondence in cohomological Donaldson-Thomas theory

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.

arXiv commentv3: accepted version, minor typos fixed v2: 22 pages, added results on BM homology of stacks of reps of deformed preprojective algebras, added references, corrected typos. v1: 20 pages, prepared for the 2020 British Mathematical Colloquium in Glasgow