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The generic Markov CoHA is not spherically generated

Ben Davison

TL;DR

This paper investigates the Kontsevich–Soibelman cohomological Hall algebra $\mathcal{H}_{Q,W}$ for the Markov quiver with infinitely mutable potentials $W$ and probes two central conjectures: spherical generation by the spherical subalgebra $\mathcal{S}_Q$ and independence of $\mathcal{H}_{Q,W}$ from the choice of $W$. By computing low-degree refined BPS invariants and analyzing stability-filtered partition functions, the authors demonstrate that for generic $W$ the degree $(1,1,1)$ piece of $\mathcal{H}_{Q,W}$ has larger dimension than the corresponding spherical part, showing that $\mathcal{H}_{Q,W}$ is not spherically generated and, in general, depends on $W$. They further isolate a non-spherical summand that can be traced to a $G$-symmetry action, suggesting a refined picture in which $\mathcal{H}_{Q,W}$ may decompose as a product of a spherical subalgebra with a non-spherical exterior- or symmetry-driven factor. The results motivate a modified conjecture: for certain choices (e.g., excluding non-spherical generators or restricting to nilpotent/quasi-homogeneous $W$), a stable spherical core persists, and dimensional reduction alongside explicit BPS computations provides a practical framework to test these refinements.

Abstract

Let $Q$ be the Markov quiver, and let $W$ be an infinitely mutable potential for $Q$. We calculate some low degree refined BPS invariants for the resulting Jacobi algebra, and use them to show that the critical cohomological Hall algebra $\mathcal{H}_{Q,W}$ 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, Grygoryev and Li \cite[§2.1]{GGL}, 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 $\mathcal{H}_{Q,W}$ according to the characters of a discrete symmetry group.

The generic Markov CoHA is not spherically generated

TL;DR

This paper investigates the Kontsevich–Soibelman cohomological Hall algebra for the Markov quiver with infinitely mutable potentials and probes two central conjectures: spherical generation by the spherical subalgebra and independence of from the choice of . By computing low-degree refined BPS invariants and analyzing stability-filtered partition functions, the authors demonstrate that for generic the degree piece of has larger dimension than the corresponding spherical part, showing that is not spherically generated and, in general, depends on . They further isolate a non-spherical summand that can be traced to a -symmetry action, suggesting a refined picture in which may decompose as a product of a spherical subalgebra with a non-spherical exterior- or symmetry-driven factor. The results motivate a modified conjecture: for certain choices (e.g., excluding non-spherical generators or restricting to nilpotent/quasi-homogeneous ), a stable spherical core persists, and dimensional reduction alongside explicit BPS computations provides a practical framework to test these refinements.

Abstract

Let be the Markov quiver, and let be an infinitely mutable potential for . We calculate some low degree refined BPS invariants for the resulting Jacobi algebra, and use them to show that the critical cohomological Hall algebra is not necessarily spherically generated, and is not independent of the choice of infinitely mutable potential . This leads to a counterexample to a conjecture of Gaiotto, Grygoryev and Li \cite[§2.1]{GGL}, but also suggestions for how to modify it. In the case of generic cubic , we discuss a way to modify the conjecture, by excluding the non-spherical part via the decomposition of according to the characters of a discrete symmetry group.

Paper Structure

This paper contains 11 sections, 8 theorems, 45 equations.

Table of Contents

  1. Preliminaries
  2. Setup
  3. The conjectures
  4. The Markov quiver
  5. Potentials for the Markov quiver
  6. BPS invariants for generic $W$ and small dimension vectors
  7. Counterexamples
  8. Spherical (non) generation
  9. Excluding non-spherical generators
  10. Dependence on $W$
  11. Acknowledgements

Key Result

Lemma 2.1

MR1264417 After applying a graded linear isomorphism $\Phi\colon \mathbb{C} Q\rightarrow \mathbb{C} Q$, i.e. an isomorphism taking arrows to linear combinations of arrows, we may write $W$ in one of the following five forms where $W_{\geq 6}$ is the sum of all of the homogeneous pieces of $W$ of degree at least $6$, i.e. a linear combination of cyclic paths of length at least $6$. Moreover, case

Theorems & Definitions (15)

  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3
  • ...and 5 more