Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 123, 27 pages      arXiv:1806.08650      https://doi.org/10.3842/SIGMA.2018.123
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

On Solutions of the Fuji-Suzuki-Tsuda System

Pavlo Gavrylenko abc, Nikolai Iorgov ad and Oleg Lisovyy e
a) Bogolyubov Institute for Theoretical Physics, 03143 Kyiv, Ukraine
b) Center for Advanced Studies, Skolkovo Institute of Science and Technology, 143026 Moscow, Russia
c) National Research University Higher School of Economics, International Laboratory of Representation Theory and Mathematical Physics, Moscow, Russia
d) Kyiv Academic University, 36 Vernadsky Ave., 03142 Kyiv, Ukraine
e) Institut Denis-Poisson, Université de Tours, Parc de Grandmont, 37200 Tours, France

Received June 22, 2018, in final form October 30, 2018; Published online November 11, 2018

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 $c=N-1$.

Key words: isomonodromic deformations; Painlevé equations; Fredholm determinants.

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