A boson-fermion correspondence in cohomological Donaldson-Thomas theory

We introduce and study a fermionisation procedure for the cohomological Hall algebra 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.