Heun and Mathieu functions as solutions of the Dirac equation

TL;DR

This paper explores how Heun and Mathieu functions serve as solutions to the Dirac equation in specific curved spacetimes, highlighting the transition from Mathieu to Heun functions in higher dimensions and the application of spectral boundary conditions.

Contribution

It demonstrates the emergence of Heun functions as solutions to the Dirac equation in five-dimensional Nutku helicoid metrics, extending previous four-dimensional results and analyzing boundary conditions.

Findings

Heun functions appear as solutions in five-dimensional cases.

Mathieu functions are solutions in four-dimensional backgrounds.

Spectral boundary conditions are necessary due to metric singularities.

Abstract

We give examples of where the Heun function exists as solutions of wave equations encountered in general relativity. While the Dirac equation written in the background of Nutku helicoid metric yields Mathieu functions as its solutions in four spacetime dimensions, the trivial generalization to five dimensions results in the double confluent Heun function. We reduce this solution to the Mathieu function with some transformations. We must apply Atiyah-Patodi-Singer spectral boundary conditions to this system since the metric has a singularity at the origin.