Extended Heun Hierarchy in Quantum Seiberg-Witten Geometry
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908754097733632 |
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| author | Yang, Peng Wang, Yi-Rong Zhang, Kilar |
| author_facet | Yang, Peng Wang, Yi-Rong Zhang, Kilar |
| contents | We investigate the quantum geometry of the Seiberg-Witten curve for $\mathcal{N}=2$, $\mathrm{SU(2)}^n$ linear quiver gauge theories. By applying the Weyl quantization prescription to the algebraic curve, we derive the corresponding second-order differential equation and demonstrate that it is isomorphic to the Extended Heun Equation with $n+3$ regular singular points. The physical parameters of the gauge theory are linked to the canonical coefficients of the Heun equation via a polynomial representation of the Seiberg-Witten curve. This framework provides the necessary mathematical foundation to apply non-perturbative gauge-theoretic techniques, such as instanton counting, to spectral problems in gravitational physics, most notably for higher-dimensional black holes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_05204 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extended Heun Hierarchy in Quantum Seiberg-Witten Geometry Yang, Peng Wang, Yi-Rong Zhang, Kilar High Energy Physics - Theory High Energy Astrophysical Phenomena General Relativity and Quantum Cosmology High Energy Physics - Phenomenology Mathematical Physics We investigate the quantum geometry of the Seiberg-Witten curve for $\mathcal{N}=2$, $\mathrm{SU(2)}^n$ linear quiver gauge theories. By applying the Weyl quantization prescription to the algebraic curve, we derive the corresponding second-order differential equation and demonstrate that it is isomorphic to the Extended Heun Equation with $n+3$ regular singular points. The physical parameters of the gauge theory are linked to the canonical coefficients of the Heun equation via a polynomial representation of the Seiberg-Witten curve. This framework provides the necessary mathematical foundation to apply non-perturbative gauge-theoretic techniques, such as instanton counting, to spectral problems in gravitational physics, most notably for higher-dimensional black holes. |
| title | Extended Heun Hierarchy in Quantum Seiberg-Witten Geometry |
| topic | High Energy Physics - Theory High Energy Astrophysical Phenomena General Relativity and Quantum Cosmology High Energy Physics - Phenomenology Mathematical Physics |
| url | https://arxiv.org/abs/2601.05204 |