The generic Markov cohomological Hall algebra is not spherically generated

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> be the Markov quiver, and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>W</mml:mi></mml:math> be an infinitely mutable potential for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>H</mml:mi> <mml:mrow><mml:mi>Q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>W</mml:mi></mml:mrow> </mml:msub> </mml:math> is not necessarily spherically generated and is not independent of the choice of infinitely mutable potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>W</mml:mi></mml:math> . This leads to a counterexample to a conjecture of Gaiotto <i>et al</i>. (Gaiotto <i>et al</i>. 2024 Categories of line defects and cohomological Hall algebras. <i>arXiv</i>. §2.1), but also suggestions for how to modify it. In the case of generic cubic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>W</mml:mi></mml:math> , we discuss a way to modify the conjecture by excluding the non-spherical part via the decomposition of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>H</mml:mi> <mml:mrow><mml:mi>Q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>W</mml:mi></mml:mrow> </mml:msub> </mml:math> according to the characters of a discrete symmetry group.