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BPS cohomology in geometry and representation theory

Ben Davison

TL;DR

The work develops a comprehensive framework for BPS invariants of stacks of objects, defining refined BPS cohomology via vanishing cycles, mixed Hodge structures, and plethystic exponentials. It proves a cohomological integrality (PBW) theorem that expresses the total cohomology as a symmetric algebra on BPS data, and constructs BPS sheaves on good moduli spaces to organize the invariants categorically. The survey then connects these invariants to geometric representation theory through CoHAs, PBW-type structures, and Nakajima quiver varieties, and extends the theory to 2-Calabi–Yau categories, Maulik–Okounkov Yangians, and nonabelian Hodge theory, with dimensional-reduction and wall-crossing as unifying principles. Beyond moduli of objects, it discusses BPS cohomology for more general shifted-stacks, including local systems on 3-manifolds, and highlights conjectural links to Betti Langlands duality and broader topological applications.

Abstract

We motivate and survey the theory of BPS invariants of categories and BPS cohomology of stacks, indicating applications in enumerative geometry and representation theory, as well as recent advances.

BPS cohomology in geometry and representation theory

TL;DR

The work develops a comprehensive framework for BPS invariants of stacks of objects, defining refined BPS cohomology via vanishing cycles, mixed Hodge structures, and plethystic exponentials. It proves a cohomological integrality (PBW) theorem that expresses the total cohomology as a symmetric algebra on BPS data, and constructs BPS sheaves on good moduli spaces to organize the invariants categorically. The survey then connects these invariants to geometric representation theory through CoHAs, PBW-type structures, and Nakajima quiver varieties, and extends the theory to 2-Calabi–Yau categories, Maulik–Okounkov Yangians, and nonabelian Hodge theory, with dimensional-reduction and wall-crossing as unifying principles. Beyond moduli of objects, it discusses BPS cohomology for more general shifted-stacks, including local systems on 3-manifolds, and highlights conjectural links to Betti Langlands duality and broader topological applications.

Abstract

We motivate and survey the theory of BPS invariants of categories and BPS cohomology of stacks, indicating applications in enumerative geometry and representation theory, as well as recent advances.
Paper Structure (26 sections, 10 theorems, 44 equations)

This paper contains 26 sections, 10 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. BPS cohomology of moduli stacks of objects
  3. Cohomology of stacks of modules
  4. BPS invariants from BPS cohomology
  5. BPS sheaves and BPS cohomology
  6. Relative integrality
  7. Perverse filtration
  8. BPS cohomology of stacks of objects in 3-Calabi--Yau categories
  9. Dimensional reduction
  10. Calculating refined BPS invariants
  11. Examples from flopping curves
  12. Kac polynomials as refined BPS invariants
  13. Cohomological wall-crossing and quantum dilogarithm identities
  14. Why "integrality"?
  15. BPS cohomology and geometric representation theory
  16. ...and 11 more sections

Key Result

Theorem 2.2

Let $A$ be presented as in one of the cases (1), (2) or (3) above, and define ${\mathcal{H}}_{A}\coloneqq \bigoplus_{\mathbf{d}\in{\mathbb N}^{Q_0}} {\mathcal{H}}_{A,\mathbf{d}}$, one of the three types of ${\mathbb Z}_{\mathop{\mathrm{coh}}\nolimits}\oplus {\mathbb N}^{Q_0}$-graded mixed Hodge stru

Theorems & Definitions (25)

  • Remark 2.1: Mixed Hodge structures
  • Theorem 2.2
  • Remark 2.3
  • Proposition 2.4: MR4132957
  • Theorem 2.5: MR4132957 Theorem A
  • Theorem 3.1: MR4000572
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • ...and 15 more