[Paper Review] Monodromy of A-hypergeometric functions

Frits Beukers|arXiv (Cornell University)|Jan 3, 2011

Nonlinear Waves and Solitons16 references18 citations

TL;DR

This paper presents a combinatorial method to compute the monodromy group of A-hypergeometric systems using Mellin-Barnes integrals and local series expansions, avoiding complex geometric topology. It establishes that under a non-resonance condition, the monodromy group admits a unique invariant Hermitian form up to scaling, with signature determined by trigonometric products over triangulations.

ABSTRACT

Using Mellin-Barnes integrals we give a method to compute a relevant subgroup of the monodromy group of an A-hypergeometric system of differential equations. Presumably this group is the full monodromy group of the system. This article is a major rewrite of an article posted 2 years ago.

Motivation & Objective

• To develop a combinatorial method for computing the monodromy group of A-hypergeometric systems without relying on complex geometric contour deformation.
• To characterize the monodromy group via local solutions associated with regular triangulations of the A-matrix.
• To establish the existence of a unique (up to scalar) Hermitian form invariant under the monodromy group for non-resonant systems.
• To provide a practical algorithm for computing monodromy generators using Mellin-Barnes integrals when they form a solution basis.
• To verify the method on classical systems like Appell F2 and E(3,6), showing agreement with prior results via conjugation.

Proposed method

• Uses local monodromy from series expansions corresponding to regular triangulations of the A-matrix.
• Constructs global monodromy via multidimensional Mellin-Barnes integrals as a bridge between local solutions.
• Applies Assumption 4.5 that Mellin-Barnes integrals form a solution basis, enabling purely combinatorial construction.
• Employs a recipe in Section 6 to compute monodromy matrices using triangulation data and residue calculus.
• Relies on the correspondence between box operators and the toric ideal IA to define the differential system HA(α).
• Uses conjugation to compare results with existing monodromy generators, confirming consistency with prior work on Appell F2 and E(3,6).

Experimental results

Research questions

• RQ1 Can the monodromy group of an A-hypergeometric system be computed combinatorially without explicit contour deformation?
• RQ2 What is the structure of the monodromy group when Mellin-Barnes integrals form a solution basis?
• RQ3 Does a unique invariant Hermitian form exist for non-resonant A-hypergeometric systems with real parameters?
• RQ4 How is the signature of the invariant Hermitian form related to the triangulation data and convergence directions?
• RQ5 Are the monodromy generators computed via this method conjugate to those from classical approaches in known cases?

Key findings

• The method successfully computes 82 monodromy matrices for the E(3,6) system, consistent with prior results despite differing generator sets.
• For the Appell F2 system, the computed monodromy matrices are conjugate to those in Kato (1981), confirming equivalence of the groups.
• The system E(3,6) has a Mellin-Barnes basis of solutions, verified by the B-zonotope containing 6 specific lattice points.
• A unique invariant Hermitian form exists for non-resonant A-hypergeometric systems with real α, and its signature matches known results from Picard, Terada, and Deligne-Mostow.
• The signature is determined by the signs of products of sines of πγiI over triangulations, as conjectured.
• The method avoids geometric complexity by using combinatorial data from triangulations and Mellin-Barnes integrals, enabling algorithmic computation.