Sixth Painlev\'e Equation, Universal Elliptic Curve, and Mirror of $\bold{P}^2$

TL;DR

This paper introduces an algebro-geometric framework for Painlevé VI using universal elliptic curves, connecting elliptic functions and quantum cohomology of the projective plane.

Contribution

It presents a novel geometric realization of Painlevé VI on a twisted cotangent bundle of the universal elliptic curve with marked points.

Findings

Hamiltonian form realized on elliptic curve bundle

Links between Painlevé VI, elliptic functions, and quantum cohomology

Provides a geometric setting for Painlevé VI analysis

Abstract

An algebro-geometric setting for the study of the Painlev\'e VI equation is introduced. Hamiltonian form of the equation is realized on a twisted relative cotangent bundle to the universal elliptic curve with labelled points of order two. Relations with the theory of elliptic functions and the quantum cohomology of projective plane are discussed.