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Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants

Maxim Kontsevich, Yan Soibelman

TL;DR

This work defines a Cohomological Hall algebra (COHA) as an algebra structure on the cohomology of stacks of representations, replacing constructible functions with (equivariant) cohomology and extending it to smooth algebras and potentials via exponential mixed Hodge structures. It develops two parallel strands: an off-shell (rapid-decay) COHA using EMHS and a on-shell (critical) COHA built from vanishing cycles, then proves associativity and explicit product formulas, including torus localization and Heisenberg-type gradings. A central theme is the motivic Donaldson-Thomas theory: generating series (DT-series) factorize by stability slopes, satisfy wall-crossing formulas, and admit integrality properties of exponents via admissible series and factorization systems. The paper also extends COHA to equivariant settings, complex cobordism, and motives, and discusses the relationship to categorical DT theory and mutations of quivers with potentials, offering a versatile bridge between representation theory, algebraic geometry, and string-theoretic BPS-state frameworks.

Abstract

We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves or functions. In order to take into account the potential we introduce a generalization of theory of mixed Hodge structures, related to exponential integrals. Generating series of our Cohomological Hall algebra is a generalization of the motivic Donaldson-Thomas invariants introduced in arXiv:0811.2435. Also we prove a new integrality property of motivic Donaldson-Thomas invariants.

Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants

TL;DR

This work defines a Cohomological Hall algebra (COHA) as an algebra structure on the cohomology of stacks of representations, replacing constructible functions with (equivariant) cohomology and extending it to smooth algebras and potentials via exponential mixed Hodge structures. It develops two parallel strands: an off-shell (rapid-decay) COHA using EMHS and a on-shell (critical) COHA built from vanishing cycles, then proves associativity and explicit product formulas, including torus localization and Heisenberg-type gradings. A central theme is the motivic Donaldson-Thomas theory: generating series (DT-series) factorize by stability slopes, satisfy wall-crossing formulas, and admit integrality properties of exponents via admissible series and factorization systems. The paper also extends COHA to equivariant settings, complex cobordism, and motives, and discusses the relationship to categorical DT theory and mutations of quivers with potentials, offering a versatile bridge between representation theory, algebraic geometry, and string-theoretic BPS-state frameworks.

Abstract

We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves or functions. In order to take into account the potential we introduce a generalization of theory of mixed Hodge structures, related to exponential integrals. Generating series of our Cohomological Hall algebra is a generalization of the motivic Donaldson-Thomas invariants introduced in arXiv:0811.2435. Also we prove a new integrality property of motivic Donaldson-Thomas invariants.

Paper Structure

This paper contains 60 sections, 45 theorems, 418 equations.

Table of Contents

  1. Introduction
  2. Cohomological Hall algebra of a quiver without potential
  3. Stacks of representations and their cohomology
  4. Multiplication
  5. Proof of associativity
  6. Explicit formula for the product
  7. Example: quivers with one vertex
  8. Symmetric case
  9. Grading by the Heisenberg group and twisted graded algebras
  10. Non-symmetric example: quiver $A_2$
  11. Questions about other algebraic structures on $\mathcal{H}$
  12. Generalization of COHA for smooth algebras
  13. Cohomology theories
  14. Smooth algebras
  15. Cohomological Hall algebra for an $I$-bigraded smooth algebra
  16. ...and 45 more sections

Key Result

Theorem 1

The product $\mathsf{m}$ on $\mathcal{H}$ is associative.

Theorems & Definitions (77)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Conjecture 1
  • Proposition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • ...and 67 more