# Monodromy Relations in Higher-Loop String Amplitudes

## TL;DR

This paper derives new monodromy relations for one-loop open string amplitudes, revealing connections between different color orderings and elliptic zeta values, with implications for string scattering in flux backgrounds.

## Contribution

It introduces novel monodromy relations at one-loop level involving elliptic integrals and proposes a framework for understanding their physical interpretation and generalizations.

## Key findings

- Derived relations between planar and non-planar amplitudes.
- Identified connections between monodromy phases and elliptic zeta values.
- Highlighted issues in recent literature regarding monodromy structures.

## Abstract

New monodromy relations of loop amplitudes are derived in open string theory. We particularly study N-point one-loop amplitudes described by a world-sheet cylinder (planar and non-planar) and derive a set of relations between subamplitudes of different color orderings. Various consistency checks are performed by matching alpha'-expansions of planar and non-planar amplitudes involving elliptic iterated integrals with the resulting periods giving rise to two sets of multiple elliptic zeta values. The latter refer to the two homology cycles on the once-punctured complex elliptic curve and the monodromy equations provide relations between these two sets of multiple elliptic zeta values. Furthermore, our monodromy relations involve new objects for which we present a tentative interpretation in terms of open string scattering amplitudes in the presence of a non-trivial gauge field flux. Finally, we provide an outlook on how to generalize the new monodromy relations to the non-oriented case and beyond the one-loop level. Comparing a subset of our results with recent findings in the literature we find therein several serious issues related to the structure and significance of monodromy phases and the relevance of missed contributions from contour integrations.