Betti structures of hypergeometric equations

TL;DR

This paper investigates Betti structures in hypergeometric equations using advanced sheaf theory and Riemann-Hilbert correspondence, providing criteria for solutions to be defined over specific subfields of complex numbers.

Contribution

It introduces a group theoretic criterion for the algebraic definability of solutions of hypergeometric systems within the framework of enhanced ind-sheaves.

Findings

Established a criterion for solutions to be defined over subfields of complex numbers.

Connected hypergeometric systems to exponentially twisted Gauss-Manin systems.

Applied the irregular Riemann-Hilbert correspondence to analyze Betti structures.

Abstract

We study Betti structures in the solution complexes of confluent hypergeometric equations. We use the framework of enhanced ind-sheaves and the irregular Riemann-Hilbert correspondence of D'Agnolo-Kashiwara. The main result is a group theoretic criterion that ensures that enhanced solutions of such systems are defined over certain subfields of the complex numbers. The proof uses a description of the hypergeometric systems as exponentially twisted Gauss-Manin systems of certain Laurent polynomials.