On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk

Abstract

Dirac Hamiltonian on the Poincaré disk in the presence of an Aharonov–Bohm flux and a uniform magnetic field admits a one-parameter family of self-adjoint extensions. We determine the spectrum and calculate the resolvent for each element of this family. Explicit expressions for Green’s functions are then used to find Fredholm determinant representations for the tau function of the Dirac operator with two branch points on the Poincaré disk. Isomonodromic deformation theory for the Dirac equation relates this tau function to a one-parameter class of solutions of the Painlevé VI equation with γ=0. We analyze long-distance behavior of the tau function, as well as the asymptotics of the corresponding Painlevé VI transcendents as s→1. Considering the limit of flat space, we also obtain a class of solutions of the Painlevé V equation with β=0.