Special polynomials related to the supersymmetric eight-vertex model. III. Painlev\'e VI equation

Abstract

We prove that certain polynomials previously introduced by the author can be identified with tau functions of Painlev\'e VI, obtained from one of Picard's algebraic solutions by acting with a four-dimensional lattice of B\"acklund transformations. For particular lines in the lattice, this proves conjectures of Bazhanov and Mangazeev. As applications, we describe the behaviour of the corresponding solutions near the singular points of Painlev\'e VI, and obtain several new properties of our polynomials.