Monodromy of multiloop integrals in $d$ dimensions

TL;DR

This paper investigates the monodromy group of differential systems for multiloop integrals in various dimensions, revealing polynomial structures and bilinear relations that deepen understanding of their mathematical properties.

Contribution

It introduces a heuristic method to compute monodromy matrices as functions of space-time dimension and uncovers their Laurent polynomial structure with integer coefficients.

Findings

Monodromy matrices are Laurent polynomials in z with integer coefficients.

The monodromy group is a subgroup of GL(n, Z[z,1/z]).

Derived bilinear relations for monodromies in d and -d dimensions.

Abstract

We consider the monodromy group of the differential systems for multiloop integrals. We describe a simple heuristic method to obtain the monodromy matrices as functions of space-time dimension $d$. We observe that in a special basis the elements of these matrices are Laurent polynomials in $z=\exp(i\pi d)$ with integer coefficients, i.e., the monodromy group is a subgroup of $GL(n,\mathbb{Z}[z,1/z])$. We derive bilinear relations for monodromies in $d$ and $-d$ dimensions which follow from the twisted Riemann bilinear relations and check that the found monodromy matrices satisfy them.