From Painlevé equations to ${\cal N}=2$ susy gauge theories: prolegomena
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arXiv
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| Natura: | Preprint |
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2024
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| author | Fioravanti, Davide Rossi, Marco |
| author_facet | Fioravanti, Davide Rossi, Marco |
| contents | We study the linear problems in $z,t$ (time) associated to the Painlevé III$_3$ and III$_1$ equations when the Painlevé solution $q(t)$ approaches a pole or a zero. In this limit the problem in $z$ for the Painlevé III$_3$ reduces to the modified Mathieu equation, while that for the Painlevé III$_1$ to the Doubly Confluent Heun Equation. These equations appear as Nekrasov-Shatashvili quantisations/deformations of Seiberg-Witten differentials for $SU(2)$ ${\cal N}=2$ super Yang-Mills gauge theory with number of flavours $N_f=0$ and $N_f=2$, respectively. These results allow us to conjecture that this link holds for any Painlevé equation relating each of them to a different matter theory, which is actually the same as in the well-established Painlevé gauge correspondence, but {\it with another deformation ($Ω$-background)}. An explicit expression for the dual gauge period (and then prepotential) is also found. As a by-product, a new solution to the connexion problem is illustrated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_21148 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From Painlevé equations to ${\cal N}=2$ susy gauge theories: prolegomena Fioravanti, Davide Rossi, Marco High Energy Physics - Theory Mathematical Physics Exactly Solvable and Integrable Systems We study the linear problems in $z,t$ (time) associated to the Painlevé III$_3$ and III$_1$ equations when the Painlevé solution $q(t)$ approaches a pole or a zero. In this limit the problem in $z$ for the Painlevé III$_3$ reduces to the modified Mathieu equation, while that for the Painlevé III$_1$ to the Doubly Confluent Heun Equation. These equations appear as Nekrasov-Shatashvili quantisations/deformations of Seiberg-Witten differentials for $SU(2)$ ${\cal N}=2$ super Yang-Mills gauge theory with number of flavours $N_f=0$ and $N_f=2$, respectively. These results allow us to conjecture that this link holds for any Painlevé equation relating each of them to a different matter theory, which is actually the same as in the well-established Painlevé gauge correspondence, but {\it with another deformation ($Ω$-background)}. An explicit expression for the dual gauge period (and then prepotential) is also found. As a by-product, a new solution to the connexion problem is illustrated. |
| title | From Painlevé equations to ${\cal N}=2$ susy gauge theories: prolegomena |
| topic | High Energy Physics - Theory Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2412.21148 |