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.