DT invariants from vertex algebras

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.

arXiv comment43 pages

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Cited by in corpus (1)

• Koszul algebras and Donaldson-Thomas invariants