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The critical CoHA of a self dual quiver with potential

Ben Davison

Number 33
Author Ben Davison
Year 2013

In this paper we provide an explanation for the many beautiful infinite product formulas for generating functions of refined DT invariants for symmetric quivers with potential: they are characteristic functions of free supercommutative algebras. We prove this by first showing that, for a quiver with potential that satisfies a notion of self duality, introduced below, the critical cohomological Hall algebra is supercommutative, and admits a kind of localised coproduct, which is enough to guarantee freeness. The primitive graded pieces of the space of generators of the cohomological Hall algebra are a putative definition of the space of BPS states in String Theory, and we discuss the issue of their finite-dimensionality. We consider the conjecture that in the presence of the above-mentioned self duality these spaces are always finite-dimensional, and finish with some examples, in which, amongst other things, this conjecture can be seen to hold. Some of these examples use a cohomological dimensional reduction of the sort used by Behrend, Bryan and Szendr\H{o}i to calculate the motivic DT invariants of C3. We prove the required isomorphism holds in the appendix. The final example pulls together all of the technology in this paper to describe the link between critical CoHAs and character varieties: we build a free supercommutative critical CoHA out of the cohomology of untwisted character varieties, and present some evidence that the spaces of primitive generators in this case are given by the cohomology of their twisted counterparts. In a separate paper we discuss our other main example, which yields a new proof of the Kac conjecture.

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