Poisson Vertex Algebra of Seiberg-Witten Theory
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866913004535152640 |
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| author | Khan, Ahsan Z. |
| author_facet | Khan, Ahsan Z. |
| contents | The space of local operators in the $Q$-cohomology of the holomorphic-topological supercharge in a four-dimensional $\mathcal{N}=2$ theory carries the structure of a Poisson vertex algebra. This note studies the Poisson vertex algebra associated to the pure $\mathcal{N}=2$ gauge theory with gauge group $SU(2)$. We propose an explicit Poisson vertex algebra $A$, claimed to be isomorphic to the algebra of holomorphic-topological observables to all orders in perturbation theory. We compute the Hilbert-Poincaré series of $A$ and show that it refines the Schur index of the pure $SU(2)$ theory. We show that $A$ admits a further differential $Q_{\text{inst}}$ which we hypothesize captures non-perturbative corrections, and compute the cohomology of this differential. We thus present an explicit candidate for the space of non-perturbative holomorphic-topological observables of Seiberg-Witten theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03500 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Poisson Vertex Algebra of Seiberg-Witten Theory Khan, Ahsan Z. High Energy Physics - Theory Mathematical Physics Quantum Algebra The space of local operators in the $Q$-cohomology of the holomorphic-topological supercharge in a four-dimensional $\mathcal{N}=2$ theory carries the structure of a Poisson vertex algebra. This note studies the Poisson vertex algebra associated to the pure $\mathcal{N}=2$ gauge theory with gauge group $SU(2)$. We propose an explicit Poisson vertex algebra $A$, claimed to be isomorphic to the algebra of holomorphic-topological observables to all orders in perturbation theory. We compute the Hilbert-Poincaré series of $A$ and show that it refines the Schur index of the pure $SU(2)$ theory. We show that $A$ admits a further differential $Q_{\text{inst}}$ which we hypothesize captures non-perturbative corrections, and compute the cohomology of this differential. We thus present an explicit candidate for the space of non-perturbative holomorphic-topological observables of Seiberg-Witten theory. |
| title | Poisson Vertex Algebra of Seiberg-Witten Theory |
| topic | High Energy Physics - Theory Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2604.03500 |