Poles Distribution of PVI Transcendents close to a Critical Point (summer 2011)

TL;DR

This paper analyzes the distribution of poles of Painleve VI transcendents near a critical point, revealing they accumulate along two rays, with applications to quantum cohomology of complex projective space.

Contribution

It determines the pole distribution of Painleve VI transcendents near critical points, including specific examples from quantum cohomology.

Findings

Poles accumulate at the critical point along two rays

Distribution pattern is asymptotic near the critical point

Application to quantum cohomology of 2P^1

Abstract

The distribution of the poles of branches of the Painleve' VI transcendents associated to semi-simple Frobenius manifolds is determined close to a critical point. It is shown that the poles accumulate at the critical point, asymptotically along two rays. The example of the Frobenius manifold given by the quantum cohomology of the two-dimensional complex projective space is also considered.