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DT invariants from vertex algebras

Vladimir Dotsenko, Sergey Mozgovoy

TL;DR

The paper develops a unified framework linking cohomological Hall algebras for symmetric quivers with vertex (and conformal) algebras by showing that the graded dual $\\mathcal H_Q^{\vee}$ carries a natural cocommutative vertex bialgebra structure and is isomorphic to the universal enveloping vertex algebra of a vertex Lie algebra $C$. This identification explains the CoHA product as a shuffle product arising from the principal free vertex algebra associated to the Euler form $\\chi$ of the quiver, and it yields a new, conceptually clear proof of the positivity of refined Donaldson–Thomas invariants via $C/\\partial C$. The work also provides new descriptions and spanning sets for CoHA-modules (non-commutative Hilbert schemes) and develops a framework to relate CoHA-modules to free vertex algebras through modified coproducts and coactions. Collectively, these results illuminate deep structural connections between CoHAs, vertex algebras, and DT theory, offering a path toward Koszul duality perspectives and potential generalizations to quivers with potentials."

Abstract

We obtain a new interpretation of the cohomological Hall algebra $\mathcal{H}_Q$ of a symmetric quiver $Q$ in the context of the theory of vertex algebras. Namely, we show that the graded dual of $\mathcal{H}_Q$ is naturally identified with the underlying vector space of the principal free vertex algebra associated to the Euler form of $Q$. Properties of that vertex algebra are shown to account for the key results about $\mathcal{H}_Q$. In particular, it has a natural structure of a vertex bialgebra, leading to a new interpretation of the product of $\mathcal{H}_Q$. Moreover, it is isomorphic to the universal enveloping vertex algebra of a certain vertex Lie algebra, which leads to a new interpretation of Donaldson--Thomas invariants of $Q$ (and, in particular, re-proves their positivity). Finally, it is possible to use that vertex algebra to give a new interpretation of CoHA modules made of cohomologies of non-commutative Hilbert schemes.

DT invariants from vertex algebras

TL;DR

The paper develops a unified framework linking cohomological Hall algebras for symmetric quivers with vertex (and conformal) algebras by showing that the graded dual carries a natural cocommutative vertex bialgebra structure and is isomorphic to the universal enveloping vertex algebra of a vertex Lie algebra . This identification explains the CoHA product as a shuffle product arising from the principal free vertex algebra associated to the Euler form of the quiver, and it yields a new, conceptually clear proof of the positivity of refined Donaldson–Thomas invariants via . The work also provides new descriptions and spanning sets for CoHA-modules (non-commutative Hilbert schemes) and develops a framework to relate CoHA-modules to free vertex algebras through modified coproducts and coactions. Collectively, these results illuminate deep structural connections between CoHAs, vertex algebras, and DT theory, offering a path toward Koszul duality perspectives and potential generalizations to quivers with potentials."

Abstract

We obtain a new interpretation of the cohomological Hall algebra of a symmetric quiver in the context of the theory of vertex algebras. Namely, we show that the graded dual of is naturally identified with the underlying vector space of the principal free vertex algebra associated to the Euler form of . Properties of that vertex algebra are shown to account for the key results about . In particular, it has a natural structure of a vertex bialgebra, leading to a new interpretation of the product of . Moreover, it is isomorphic to the universal enveloping vertex algebra of a certain vertex Lie algebra, which leads to a new interpretation of Donaldson--Thomas invariants of (and, in particular, re-proves their positivity). Finally, it is possible to use that vertex algebra to give a new interpretation of CoHA modules made of cohomologies of non-commutative Hilbert schemes.

Paper Structure

This paper contains 41 sections, 30 theorems, 254 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Borel--Moore homology
  4. Graded vector spaces
  5. Laurent series
  6. Characters and Poincaré series
  7. Plethystic exponential
  8. CoHA and CoHA-modules
  9. Definition of CoHA
  10. Shuffle algebra description
  11. Modules over CoHA
  12. Characters of CoHA and CoHA modules
  13. Vertex algebras and conformal algebras
  14. Graded vertex algebras
  15. Classical vertex (super)algebras
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $Q$ be a symmetric quiver, $L=\bZ^{Q_0}$ and $\chi\xspace$ be the Euler form of $Q$. Then the coalgebra $\cH_Q^{\vee}$ has a canonical structure of a cocommutative connected vertex bialgebra. The space of primitive elements is a vertex Lie algebra (also having a structure of a Lie algebra) such that $\cH_Q^{\vee}$ is isomorphic to the universal enveloping vertex algebra of $C$ (as vertex bial

Theorems & Definitions (63)

  • Theorem 1.1: Theorem \ref{['th:DT']}
  • Theorem 2.1
  • proof
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Remark 4.1
  • ...and 53 more