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DT INVARIANTS FROM VERTEX ALGEBRAS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 24 / Issue 1 / January 2025
- Published online by Cambridge University Press:
- 20 September 2024, pp. 291-339
- Print publication:
- January 2025
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- Article
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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.
