Monodromy Relations in Higher-Loop String Amplitudes
S. Hohenegger, S. Stieberger
TL;DR
The paper derives monodromy relations for orientable one-loop open-string amplitudes on the cylinder, connecting planar and non-planar color-orderings through contour-prescription phase factors and boundary terms. It reveals two distinct elliptic structures via elliptic iterated integrals, yielding two sets of elliptic multiple zeta values (eMZVs) associated to the A and B cycles of the elliptic curve, and demonstrates how planar and non-planar sectors relate within this elliptic framework. Explicit four-point (and general N-point) planar and non-planar monodromy relations are developed, with careful treatment of branch cuts, bulk contributions, and the field-theory limit, where results reproduce known gauge-theory relations and clarify discrepancies in prior literature. The work further extends the discussion to non-orientable (Möbius strip) amplitudes and sketches prospects for higher-loop generalizations, highlighting conceptual and technical challenges, such as base-point dependence and bulk terms. Overall, the paper advances a detailed string-theoretic foundation for monodromy at one loop, connects it to the algebra of eMZVs, and maps string results to gauge-theory relations while clarifying issues in existing literature and outlining paths to broader generalizations.
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.