Isomonodromic deformation and tau function III

In this semester, we will continue discussing the isomonodromic deformations. We will focus on the two topics:

1. Riemann-Hilbert problems with quasi-permutation monodromies and isomonodromic tau-function

In general, the RH problem cannot be solved in terms of special functions. But, for an arbitrary quasi-permutation monodromy group, the RH problem was solved outside of the so-called Malgrange divisor. This divisor is characterized in terms of the theta-divisor on the Jacobi manifold of an auxiliary compact Riemann surface realized as an appropriate branched covering of CP^1. The solution is given in terms of a generalization of Szego kernel on the Riemann surface. Correspondingly, the solutions of Schlesinger equation and tau function can be computed. We will explain the details as well as discussing the relation between this tau function and exact conformal block.

2. Isospectral deformation and Widom constant

Tau functions play a central role in the theory of integrable equations, both in fields of isospectral and isomonodromic deformations. On the side of isospectral deformations, Sato defined the tau function starting from his interpretation of the KP hierarchy in terms of the geometry of Grassmannian manifolds. Even through the definitions of tau functions (isomonodromic tau functions and Sato-Segal-Wilson tau functions) look so different in both the worlds of isomonodromic and isospectral deformations, tau functions coincide with a simple object: Widom constant.

1. Riemann-Hilbert problems with quasi-permutation monodromies and isomonodromic tau-function

In general, the RH problem cannot be solved in terms of special functions. But, for an arbitrary quasi-permutation monodromy group, the RH problem was solved outside of the so-called Malgrange divisor. This divisor is characterized in terms of the theta-divisor on the Jacobi manifold of an auxiliary compact Riemann surface realized as an appropriate branched covering of CP^1. The solution is given in terms of a generalization of Szego kernel on the Riemann surface. Correspondingly, the solutions of Schlesinger equation and tau function can be computed. We will explain the details as well as discussing the relation between this tau function and exact conformal block.

2. Isospectral deformation and Widom constant

Tau functions play a central role in the theory of integrable equations, both in fields of isospectral and isomonodromic deformations. On the side of isospectral deformations, Sato defined the tau function starting from his interpretation of the KP hierarchy in terms of the geometry of Grassmannian manifolds. Even through the definitions of tau functions (isomonodromic tau functions and Sato-Segal-Wilson tau functions) look so different in both the worlds of isomonodromic and isospectral deformations, tau functions coincide with a simple object: Widom constant.

Lecturer

Date

12th March ~ 27th May, 2024

Location

Weekday | Time | Venue | Online | ID | Password |
---|---|---|---|---|---|

Monday | 13:30 - 16:55 | A3-2a-201 | ZOOM 02 | 518 868 7656 | BIMSA |

Prerequisite

Basic theory of compact Riemann surface, linear algebra

Reference

1. D. Korotkin, Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices, Mathematische Annalen 329 (2004), no. 2, 335–364.

2. P. Gavrylenko and A. Marshakov, Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations. JHEP 02 (2016) 181.

3. M. Cafasso, P. Gavrylenko, O. Lisovyy, Tau functions as Widom constants. Commun. Math. Phys. 365, 741 (2019).

2. P. Gavrylenko and A. Marshakov, Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations. JHEP 02 (2016) 181.

3. M. Cafasso, P. Gavrylenko, O. Lisovyy, Tau functions as Widom constants. Commun. Math. Phys. 365, 741 (2019).

Audience

Advanced Undergraduate

Video Public

No

Notes Public

Yes

Language

Chinese

Lecturer Intro

Xinxing Tang, received a bachelor's degree in basic mathematics from the School of Mathematics, Sichuan University in 2013, and received a doctorate from Beijing International Center for Mathematical Research, Peking University in 2018. From 2018 to 2021, he worked as a postdoctoral fellow at the Yau Mathematical Sciences Center, Tsinghua University, and joined Yanqi Lake Beijing Institute of Mathematical Sciences and Applications in 2021 as assistant professor. Research interests include: integrable systems, especially infinite-dimensional integrable systems that appear in GW theory and LG theory, and are interested in understanding the algebraic structure of infinite symmetries and related calculations. Other interests include: mixed Hodge structures, isomonodromic deformation theory, KZ equations.