DT invariants from vertex algebras
Résumé
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 $\mathcal{H}_Q$ is naturally identified with the graded dual vector space of the principal free vertex algebra associated to the Euler form of $Q$; the product of $\mathcal{H}_Q$ arises from an isomorphism of the latter with the universal enveloping vertex algebra of a certain vertex Lie algebra. This leads to a new interpretation of Donaldson--Thomas invariants of $Q$ (and, in particular, re-proves their positivity), and to a new interpretation of CoHA modules made of cohomologies of non-commutative Hilbert schemes.