BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories

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Abstract: We determine the structure of the BPS algebra of 2-Calabi-Yau Abelian categories whose stack of objects admits a good moduli space. We prove that this algebra is isomorphic to the positive part of the enveloping algebra of a generalised Kac-Moody Lie algebra generated by the intersection cohomology of certain connected components (corresponding to roots) of the good moduli space. Some major examples include the BPS algebras of (1) the category of semistable coherent sheaves of given slope on a K3 or more generally quasiprojective symplectic surface (2) preprojective algebras of quivers, (3) multiplicative preprojective algebras and (4) fundamental groups of (quiver) Riemann surfaces. We define the BPS Lie algebras of 2-Calabi-Yau categories and prove that they coincide with the ones obtained by dimensional reduction and the critical cohomological Hall algebra. Consequences include (1) A proof in full generality of the positivity conjecture for absolutely cuspidal polynomials of Bozec-Schiffmann, a strengthening of the Kac positivity conjecture (2) A proof of the cohomological integrality conjecture for local K3 surfaces (3) A lowest weight vector description for the cohomology (in all degrees) of Nakajima quiver varieties.

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