Solving Heun's equation using conformal blocks

TL;DR

This paper explores the semi-classical analysis of conformal blocks to solve Heun's equation, providing a new computational approach and insights relevant to spectral problems, CFT, and black hole physics.

Contribution

It introduces a heavy-light factorization mechanism for 5-point degenerate conformal blocks, enabling practical computation of Heun's solutions and broadening applications in physics.

Findings

Decoupling of heavy and light contributions in conformal blocks.

A practical method for computing Floquet type Heun solutions.

Applications to spectral problems, CFT, and black hole physics.

Abstract

It is known that the classical limit of the second order BPZ null vector decoupling equation for the simplest two 5-point degenerate spherical conformal blocks yields: (i) the normal form of the Heun equation with the complex accessory parameter determined by the 4-point classical block on the sphere, and (ii) a pair of the Floquet type linearly independent solutions. A key point in a derivation of the above result is the classical asymptotic of the 5-point degenerate blocks in which the so-called heavy and light contributions decouple. In the present work the semi-classical heavy-light factorization of the 5-point degenerate conformal blocks is studied. In particular, a mechanism responsible for the decoupling of the heavy and light contributions is identified. Moreover, it is shown that the factorization property yields a practical method of computation of the Floquet type Heun's solutions. Finally, it should be stressed that tools analyzed in this work have a broad spectrum of applications, in particular, in the studies of spectral problems with the Heun class of potentials, sphere-torus correspondence in 2d CFT, the KdV theory, the connection problem for the Heun equation and black hole physics. These applications are main motivations for the present work.