Painlevé/CFT correspondence on a torus. (English) Zbl 1509.34100
Summary: This Review details the relationship between the isomonodromic tau-function and conformal blocks on a torus with one simple pole. It is based on the author’s talk at ICMP 2021.
©2022 American Institute of Physics
©2022 American Institute of Physics
MSC:
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
| 34M56 | Isomonodromic deformations for ordinary differential equations in the complex domain |
Software:
DLMFReferences:
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| [31] | The reason behind this is computational and is detailed in Refs. 5 and 7. |
| [32] | For the ease of notation, we the keep the notation of the kernels \(\mathsf{a}, \mathsf{b}, \mathsf{c}, \mathsf{d} \). |
| [33] | The explicit form of the Fourier coefficients \(\mathsf{a}_{s; \beta}^{- r; \alpha}, \mathsf{b}_{- s; \beta}^{- r; \alpha}, \mathsf{c}_{s; \beta}^{r; \alpha}, \mathsf{d}_{- s; \beta}^{r; \alpha}\) is not relevant for the purposes of this Review but is computed in Ref. 5. |
| [34] | Since s ∈ I, the hole positions in the corresponding Maya diagram m are \(\mathsf{h}(\mathsf{m}) = \left\{- s_1, \ldots, - s_k\right\} \), and since r ∈ J, the particle positions are \(\mathsf{p}(\mathsf{m}) = \left\{r_1, \ldots, r_l\right\} \). |
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| [36] | Refer to Eq. (2.45) in Ref. 7 for the full computation. |
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| [39] | Bershtein, M.; Gavrylenko, P.; Grassi, A., Quantum spectral problems and isomonodromic deformations · Zbl 1507.81089 |
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