Perturbative connection formulas for Heun equations

TL;DR

This paper develops a systematic method to compute connection formulas for Heun equations using large order asymptotics, enabling verification of a conjecture linking Heun connection matrices to Virasoro conformal blocks.

Contribution

It introduces a systematic approach to compute asymptotic amplitudes for Heun equations, facilitating the verification of a conjecture relating connection formulas to conformal blocks.

Findings

Computed asymptotic amplitudes to arbitrary order for Heun equations.

Confirmed the conjecture relating Heun connection matrices to Virasoro conformal blocks.

Provided a new tool for analyzing linear ODEs with Fuchsian singularities.

Abstract

Connection formulas relating Frobenius solutions of linear ODEs at different Fuchsian singular points can be expressed in terms of the large order asymptotics of the corresponding power series. We demonstrate that for the usual, confluent and reduced confluent Heun equation, the series expansion of the relevant asymptotic amplitude in a suitable parameter can be systematically computed to arbitrary order. This allows to check a recent conjecture of Bonelli-Iossa-Panea Lichtig-Tanzini expressing the Heun connection matrix in terms of quasiclassical Virasoro conformal blocks.