The generic Markov cohomological Hall algebra is not spherically generated

Area of Science:

• Algebraic Geometry
• Mathematical Physics
• Representation Theory

Background:

• The study of cohomological Hall algebras (CHAs) is crucial for understanding BPS invariants.
• Gaiotto et al. (2024) proposed a conjecture regarding spherical generation and potential independence of CHAs.
• Markov quivers and infinitely mutable potentials are key structures in this field.

Purpose of the Study:

• To calculate refined Bogomol'nyi-Prasad-Sommerfield (BPS) invariants for Jacobi algebras associated with Markov quivers and potentials.
• To investigate the properties of the resulting critical cohomological Hall algebra (CHA).
• To test and potentially modify a conjecture by Gaiotto et al. concerning CHAs.

Main Methods:

• Calculation of low-degree refined BPS invariants.
• Analysis of the Jacobi algebra derived from a Markov quiver and an infinitely mutable potential.
• Examination of the critical cohomological Hall algebra's generation properties.

Main Results:

• The critical CHA is not necessarily spherically generated.
• The critical CHA is dependent on the choice of the infinitely mutable potential.
• A counterexample to the Gaiotto et al. conjecture is provided.

Conclusions:

• The original conjecture by Gaiotto et al. requires modification.
• A method for modifying the conjecture is proposed, involving decomposition based on discrete symmetry groups for cubic potentials.
• This research advances the understanding of CHAs and their relation to BPS invariants.