The generic Markov CoHA is not spherically generated

TL;DR

This paper demonstrates that the critical cohomological Hall algebra associated with the Markov quiver and a mutable potential is not always spherically generated and depends on the potential, providing a counterexample to a prior conjecture.

Contribution

It provides the first explicit counterexample to the conjecture that the generic Markov CoHA is spherically generated, and suggests modifications to the conjecture based on symmetry considerations.

Findings

Counterexample to the sphericity conjecture for Markov CoHA

Dependence of CoHA structure on the choice of potential W

Proposal to modify the conjecture by excluding non-spherical parts

Abstract

Let $Q$ be the Markov quiver, and let $W$ be an infinitely mutable potential for $Q$. We calculate some low degree refined BPS invariants for the resulting Jacobi algebra, and use them to show that the critical cohomological Hall algebra $\mathcal{H}_{Q,W}$ is not necessarily spherically generated, and is not independent of the choice of infinitely mutable potential $W$. This leads to a counterexample to a conjecture of Gaiotto, Grygoryev and Li \cite[\S 2.1]{GGL}, but also suggestions for how to modify it. In the case of generic cubic $W$, we discuss a way to modify the conjecture, by excluding the non-spherical part via the decomposition of $\mathcal{H}_{Q,W}$ according to the characters of a discrete symmetry group.