On Solutions of the Fuji-Suzuki-Tsuda System

Abstract

We derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlevé VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also initiate the study of algebraic braid group dynamics of semi-degenerate monodromy, and obtain as a byproduct a direct isomonodromic proof of the AGT-W relation for c = N − 1.

Crossing invariant correlation functions at c = 1 from isomonodromic τ functions

“…Theory of the first type already appeared, for instance, in [46]. Some steps towards the study of the orbits of the braid group were already done in [36], where the action of three involutions on the monodromy data is written.…”

“…In this section we give self-contained description of the moduli space of SL(2, C) and SU(2) flat connections on sphere with four punctures. Almost all content of this section can be found as well in [23], [29], [30] and [36].…”

Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations

“…and can be tested, both numerically and exactly for some degenerate values of the Wcharges θ of the fields [20,32]. In (5.2) the normalization of conformal block B w (•; q) is chosen to be B w (•; q) = 1 + O(q) and C (•) w (•) as usually denote the corresponding 3point structure constants (all these quantities in the case of W (sl N ) = W N blocks with N > 2 depend on extra parameters {µ, ν}, being the coordinates on the moduli space of flat connections on 3-punctured sphere, and for their generic values the conformal blocks B w (•; q) are not defined algebraically, see [20] for more details).…”