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DT INVARIANTS FROM VERTEX ALGEBRAS

Part of: Projective and enumerative geometry Representation theory of rings and algebras Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  20 September 2024

Vladimir Dotsenko Open the ORCID record for Vladimir Dotsenko [Opens in a new window]  and
Sergey Mozgovoy
Vladimir Dotsenko*
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes, 67000 Strasbourg, France
Sergey Mozgovoy
Affiliation:
School of Mathematics, Trinity College Dublin, Dublin 2, Ireland and Hamilton Mathematics Institute, Dublin 2, Ireland (mozgovoy@maths.tcd.ie)
*
vdotsenko@unistra.fr
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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.

Keywords

cohomological Hall algebrasvertex Lie algebrasymmetric quivernon-commutative Hilbert scheme

MSC classification

Primary: 17B69: Vertex operators; vertex operator algebras and related structures
Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 16G20: Representations of quivers and partially ordered sets

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 24 , Issue 1 , January 2025 , pp. 291 - 339
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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