Painleve III Equation: Tau Functions, Representation Theory and q-deformations

Student: Shchechkin Anton

Supervisor: Mikhail Bershtein

Educational Programme: Mathematics and Mathematical Physics (Master)

Year of Graduation: 2016

We propose $q$-deformation of the Gamayun-Iorgov-Lisovyy formula for Painlev\'e $\tau$ function. Namely we propose formula for $\tau$ function for $q$-difference Painlev\'e equation corresponding to $A_7^{(1)}{}'$ surface (and $A_1^{(1)}$ symmetry) in Sakai classification. In this formula $\tau$ function equals the series of $q$-Virasoro Whittaker conformal blocks (equivalently Nekrasov partition functions for pure $SU(2)$ 5d theory). $\tau$ functions of $q$-difference Painlev\'e equation should be consistent with the action of symmetry group of the equation. For our formula this reduces to (conjectural) bilinear relations on Nekrasov partition functions and fiber-base duality of topological vertex.

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