# The generic Markov cohomological Hall algebra is not spherically generated

## TL;DR

This paper challenges a conjecture about cohomological Hall algebras by showing they depend on the choice of potential and are not always spherically generated.

## Contribution

The paper provides a counterexample to a conjecture and proposes a modification by excluding non-spherical parts.

## Key findings

- Low-degree BPS invariants show the cohomological Hall algebra is not spherically generated.
- The algebra depends on the choice of infinitely mutable potential.
- A modification to the conjecture is suggested using character decomposition.

## 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 HQ,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 HQ,W according to the characters of a discrete symmetry group.