The generic Markov cohomological Hall algebra is not spherically generated.

Abstract

Let $Q$ be the Markov quiver, and let $W$ be an infinitely mutable potential for $Q$ . We calculate some low-degree refined Bogomol'nyi-Prasad-Sommerfield (BPS) invariants for the resulting Jacobi algebra and use them to show that the critical cohomological Hall algebra $H_{Q,W}$ 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 et al. (Gaiotto et al. 2024 Categories of line defects and cohomological Hall algebras. arXiv. §2.1), 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 $H_{Q,W}$ according to the characters of a discrete symmetry group.