The generic Markov CoHA is not spherically generated
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915142252363776 |
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| author | Davison, Ben |
| author_facet | Davison, Ben |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05009 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The generic Markov CoHA is not spherically generated Davison, Ben Representation Theory High Energy Physics - Theory 16G20 (primary), 14A22 (secondary) 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. |
| title | The generic Markov CoHA is not spherically generated |
| topic | Representation Theory High Energy Physics - Theory 16G20 (primary), 14A22 (secondary) |
| url | https://arxiv.org/abs/2502.05009 |