Two-point correlation functions of scaling fields in the Dirac theory on the Poincare disk

TL;DR

This paper verifies and completes the description of two-point functions of scaling fields in a Dirac theory on the Poincare disk using Painleve VI transcendents, including fixing constants and analyzing asymptotics.

Contribution

It fixes integration constants in Painleve VI solutions and calculates long-distance expansions and normalizations of two-point functions in curved space.

Findings

Confirmed Painleve VI description of two-point functions

Derived long-distance asymptotic expansion

Calculated normalization of scaling fields

Abstract

A result from Palmer, Beatty and Tracy suggests that the two-point function of certain spinless scaling fields in a free Dirac theory on the Poincare disk can be described in terms of Painleve VI transcendents. We complete and verify this description by fixing the integration constants in the Painleve VI transcendent describing the two-point function, and by calculating directly in a Dirac theory on the Poincare disk the long distance expansion of this two-point function and the relative normalization of its long and short distance asymptotics. The long distance expansion is obtained by developing the curved-space analogue of a form factor expansion, and the relative normalization is obtained by calculating the one-point function of the scaling fields in question. The long distance expansion in fact provides part of the solution to the connection problem associated with the Painleve VI equation involved. Calculations are done using the formalism of angular quantization.