Beijing Institute of Mathematical Sciences and Applications

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.