The threefold way to quantum periods: WKB, TBA equations and q-Painlevé
We show that TBA equations defined by the BPS spectrum of $5d$ $\mathcal{N}=1$ $SU(2)$ Yang-Mills on $S^1\times \mathbb{R}^4$ encode the q-Painlevé III$_3$ equation. We find a fine-tuned stratum in the physical moduli space of the theory where solutions to TBA equations can be obtained exactly, and...
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| Format: | Article |
| Language: | English |
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SciPost
2023-09-01
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| Series: | SciPost Physics |
| Online Access: | https://scipost.org/SciPostPhys.15.3.112 |
| _version_ | 1827807625848291328 |
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| author | Fabrizio Del Monte, Pietro Longhi |
| author_facet | Fabrizio Del Monte, Pietro Longhi |
| author_sort | Fabrizio Del Monte, Pietro Longhi |
| collection | DOAJ |
| description | We show that TBA equations defined by the BPS spectrum of $5d$ $\mathcal{N}=1$ $SU(2)$ Yang-Mills on $S^1\times \mathbb{R}^4$ encode the q-Painlevé III$_3$ equation. We find a fine-tuned stratum in the physical moduli space of the theory where solutions to TBA equations can be obtained exactly, and verify that they agree with the algebraic solutions to q-Painlevé. Switching from the physical moduli space to that of stability conditions, we identify two one-parameter deformations of the fine-tuned stratum, where the general solution of the q-Painlevé equation in terms of dual instanton partition functions continues to provide explicit TBA solutions. Motivated by these observations, we propose a further extensions of the range of validity of this correspondence, under a suitable identification of moduli. As further checks of our proposal, we study the behavior of exact WKB quantum periods for the quantum curve of local $\mathbb{P}^1\times\mathbb{P}^1$. |
| first_indexed | 2024-03-11T22:00:03Z |
| format | Article |
| id | doaj.art-d35eb7ecb491477a9ba5cf8cbd76809b |
| institution | Directory Open Access Journal |
| issn | 2542-4653 |
| language | English |
| last_indexed | 2024-03-11T22:00:03Z |
| publishDate | 2023-09-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics |
| spelling | doaj.art-d35eb7ecb491477a9ba5cf8cbd76809b2023-09-25T11:29:49ZengSciPostSciPost Physics2542-46532023-09-0115311210.21468/SciPostPhys.15.3.112The threefold way to quantum periods: WKB, TBA equations and q-PainlevéFabrizio Del Monte, Pietro LonghiWe show that TBA equations defined by the BPS spectrum of $5d$ $\mathcal{N}=1$ $SU(2)$ Yang-Mills on $S^1\times \mathbb{R}^4$ encode the q-Painlevé III$_3$ equation. We find a fine-tuned stratum in the physical moduli space of the theory where solutions to TBA equations can be obtained exactly, and verify that they agree with the algebraic solutions to q-Painlevé. Switching from the physical moduli space to that of stability conditions, we identify two one-parameter deformations of the fine-tuned stratum, where the general solution of the q-Painlevé equation in terms of dual instanton partition functions continues to provide explicit TBA solutions. Motivated by these observations, we propose a further extensions of the range of validity of this correspondence, under a suitable identification of moduli. As further checks of our proposal, we study the behavior of exact WKB quantum periods for the quantum curve of local $\mathbb{P}^1\times\mathbb{P}^1$.https://scipost.org/SciPostPhys.15.3.112 |
| spellingShingle | Fabrizio Del Monte, Pietro Longhi The threefold way to quantum periods: WKB, TBA equations and q-Painlevé SciPost Physics |
| title | The threefold way to quantum periods: WKB, TBA equations and q-Painlevé |
| title_full | The threefold way to quantum periods: WKB, TBA equations and q-Painlevé |
| title_fullStr | The threefold way to quantum periods: WKB, TBA equations and q-Painlevé |
| title_full_unstemmed | The threefold way to quantum periods: WKB, TBA equations and q-Painlevé |
| title_short | The threefold way to quantum periods: WKB, TBA equations and q-Painlevé |
| title_sort | threefold way to quantum periods wkb tba equations and q painleve |
| url | https://scipost.org/SciPostPhys.15.3.112 |
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