Isomonodromic deformation and tau function II

Introduction

Last semester, we talk about the basic theory of isomonodromic deformations of Fuchsian systems. In this semester, we will continue this topic and present the Isomonodromy/CFT correspondence. In the first part, we will discuss the rank 2 case, the Painleve/CFT correpondence, where the generic Painleve VI tau function can be interpreted as 4-point correlator of primary fields of arbitrary dimensions in 2d CFT with central charge c=1. On the other hand, the AGT combinatorial representation of conformal blocks helps us to obtain completely explicit expansions of tau(t) near the singular points. In particular, we will discuss examples of conformal blocks arising from Riccati, Picard, Chazy and algebraic soultions of Painleve VI. In the second part, we will discuss the higher rank case: the correspondence between isomonodromic deformations of higher-rank Fuchsian systems and conformal field theory with higher-spin (or W-)symmetry. I will talk about the construction of monodromy fields and W-primary fields in the free-fermionic framework and use it to give the Fredholm-determinant representation of the corresponding isomonodromic tau function.

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 research fellow. 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, quantum groups and KZ equations, W algebra and W symmetry, Augmentation representation.