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.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2021
- DOI:
- 10.48550/arXiv.2109.09788
- arXiv:
- arXiv:2109.09788
- Bibcode:
- 2021arXiv210909788D
- Keywords:
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- Mathematics - Representation Theory;
- High Energy Physics - Theory;
- Mathematics - Algebraic Geometry;
- 17B37 (Primary) 14N35 (Secondary)
- E-Print:
- v3: 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