Purity of critical cohomology and Kac's conjecture

Abstract

We provide a new proof of the Kac positivity conjecture for an arbitrary quiver $Q$. The ingredients are the cohomological integrality theorem in Donaldson-Thomas theory, dimensional reduction, and an easy purity result. These facts imply the purity of the cohomological Donaldson-Thomas invariants for a quiver with potential $(\tilde{Q},W)$ associated to $Q$, which in turn implies positivity of the Kac polynomials for $Q$.

15 pages. v6: further minor improvements, Lemma 4.4 is cleared up. v7: further minor changes thanks to referee's suggestions

References in corpus (5)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
• Motivic Donaldson-Thomas invariants and McKay correspondence
• The integrality conjecture and the cohomology of preprojective stacks
• Donaldson-Thomas theory for categories of homological dimension one with potential

Cited by in corpus (2)

• Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras
• The critical CoHA of a quiver with potential