[Paper Review] BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories

Ben Davison, Lucien Hennecart+1 more|arXiv (Cornell University)|Mar 22, 2023

Algebraic structures and combinatorial models4 citations

TL;DR

This paper establishes a deep connection between 2-Calabi–Yau categories and generalized Kac–Moody Lie algebras by showing that the BPS algebra of such categories with a good moduli space is isomorphic to the universal enveloping algebra of the positive part of a generalized Kac–Moody algebra. The Cartan data of this algebra are encoded in the intersection cohomology of connected components of the moduli space, unifying diverse geometric and representation-theoretic structures and proving major conjectures in combinatorics and algebraic 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.

Motivation & Objective

• To establish a structural link between BPS algebras in 2-Calabi–Yau categories and generalized Kac–Moody Lie algebras.
• To prove that the BPS algebra arises as the universal enveloping algebra of the positive part of a generalized Kac–Moody algebra.
• To show that the Cartan datum of this Lie algebra is determined by intersection cohomology of connected components of the good moduli space of objects.
• To apply the framework to prove long-standing conjectures in representation theory and enumerative geometry, including the Bozec–Schiffmann positivity conjecture and cohomological integrality for Nakajima quiver varieties.

Proposed method

• The authors use the BPS algebra construction from [DHS22], which is defined via the Borel–Moore homology of the moduli stack of objects in a 2-Calabi–Yau category.
• They analyze the geometry of the good moduli space of objects, focusing on connected components corresponding to roots, and compute their intersection cohomology to define the Cartan matrix of the generalized Kac–Moody algebra.
• The BPS Lie algebra is constructed via dimensional reduction from the critical cohomological Hall algebra, particularly in the case of preprojective algebras.
• The structure of the BPS algebra is shown to be isomorphic to the universal enveloping algebra of the positive part of a generalized Kac–Moody algebra, using injective maps from the BPS sheaf into the cohomological Hall algebra.
• The framework is applied to key examples: semistable sheaves on K3 surfaces, Higgs bundles on curves, preprojective algebras, and multiplicative preprojective algebras.
• The method relies on mixed Hodge modules and perverse sheaves, with equivariant and nilpotent variants introduced to refine the structure and prove further positivity results.

Experimental results

Research questions

• RQ1 How is the BPS algebra of a 2-Calabi–Yau category with a good moduli space of objects related to generalized Kac–Moody Lie algebras?
• RQ2 What is the precise geometric and cohomological data that determines the Cartan matrix of the generalized Kac–Moody algebra arising from such a category?
• RQ3 Can the BPS Lie algebra be constructed via dimensional reduction from the critical cohomological Hall algebra in the case of preprojective algebras?
• RQ4 Does the BPS algebra structure unify known results on Heisenberg and Kac–Moody algebras acting on cohomology of quiver varieties?
• RQ5 Can this framework prove the Bozec–Schiffmann positivity conjecture for absolutely cuspidal polynomials and cohomological integrality for Nakajima quiver varieties?

Key findings

• The BPS algebra of a 2-Calabi–Yau category with a good moduli space is isomorphic to the universal enveloping algebra of the positive part of a generalized Kac–Moody Lie algebra.
• The Cartan datum of this Lie algebra is explicitly given by the intersection cohomology of connected components of the good moduli space of objects.
• The BPS Lie algebra coincides with the dimensional reduction of the critical cohomological Hall algebra in the case of preprojective algebras.
• The framework proves the Bozec–Schiffmann positivity conjecture for absolutely cuspidal polynomials in full generality, strengthening the Kac positivity conjecture.
• The cohomology of Nakajima quiver varieties decomposes as a direct sum of irreducible lowest weight modules over the BPS Lie algebra, resolving a long-standing question of Nakajima.
• The cohomological integrality conjecture for semistable coherent sheaves on local K3 surfaces is proven using this framework.