The generic Markov cohomological Hall algebra is not spherically generated

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.