BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories
Ben Davison, Lucien Hennecart, Sebastian Schlegel Mejia
TL;DR
This work establishes a unified, geometric framework linking BPS algebras from 2-Calabi–Yau categories to generalised Kac–Moody algebras. By constructing the BPS algebra as the degree-zero piece of a cohomological Hall algebra and identifying it with the positive half of a GK Lorentz-type algebra, the authors encode root data via intersection cohomology of good moduli spaces. They prove a relative PBW isomorphism and deduce cohomological integrality results, providing a broad toolkit to study DT theory, Nakajima quiver varieties, and quiver cuspidal polynomials; they also connect to 3d theories via dimensional reduction. The theory yields positivity for absolutely cuspidal polynomials, a complete decomposition of Nakajima cohomology into lowest-weight modules for BPS Lie algebras, and a clear geometric interpretation of cuspidal data through intersection cohomology, with far-reaching implications for representation theory and enumerative geometry.
Abstract
We determine the structure of the BPS algebra of 2-Calabi-Yau Abelian categories for which the 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 surface or, more generally, quasiprojective symplectic surface, (2) semistable Higgs bundles on a smooth projective curve, (3) preprojective algebras of quivers, (4) multiplicative preprojective algebras and (5) 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 from the critical cohomological Hall algebra in the case in which the 2-Calabi-Yau category is the category of representations of a preprojective algebra. Consequences include (1) A proof in full generality of the Bozec-Schiffmann positivity conjecture for absolutely cuspidal polynomials, a strengthening of the Kac positivity conjecture (2) A proof of the cohomological integrality conjecture for the category of semistable coherent sheaves on local K3 surfaces (3) A description of the cohomology (in all degrees) of Nakajima quiver varieties as direct sums of irreducible lowest weight representations over the BPS Lie algebra.