Gamma functions, monodromy and Frobenius constants

TL;DR

This paper explores the relationship between Frobenius constants, gamma functions, and periods in the context of differential operators related to mirror symmetry, extending previous results to more general cases and providing explicit formulas.

Contribution

It establishes a link between Frobenius constants and gamma function coefficients, showing these constants are periods, and extends the theory to general regular singularities with explicit formulas for hypergeometric cases.

Findings

Frobenius constants are shown to be periods.

Relation between Frobenius constants and gamma function coefficients is established.

Explicit generating functions for Frobenius constants in hypergeometric cases are provided.

Abstract

In their paper on the gamma conjecture in mirror symmetry, Golyshev and Zagier introduce what we refer to as Frobenius constants associated to an ordinary linear differential operator L with a reflection type singularity. These numbers describe the variation around the reflection point of Frobenius solutions to L defined near other singular points. Golyshev and Zagier show that in certain geometric cases Frobenius constants are periods, and they raise the question quite generally how to describe these numbers motivically. In this paper we give a relation between Frobenius constants and Taylor coefficients of generalized gamma functions, from which it follows that Frobenius constants of Picard--Fuchs differential operators are periods. We also study the relation between these constants and periods of limiting Hodge structures. This is a major revision of the previous version of the manuscript. The notion of Frobenius constants and our main result are extended to the general case of regular singularities with any sets of local exponents. In addition, the generating function of Frobenius constants is given explicitly for all hypergeometric connections.