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Monodromy relations for string amplitudes on AdS space

Luis F. Alday ^1,*, Rodrigo S. Pitombo ^1,2,†, and Aurélie Strömholm Sangaré ^1,‡

¹Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
²ICTP South American Institute for Fundamental Research, Instituto de Física Teórica UNESP Rua Dr. Bento Teobaldo Ferraz 271, 01140-070, São Paulo, SP, Brazil

^*Contact author: luis.alday@maths.ox.ac.uk
^†Contact author: rs.pitombo@unesp.br
^‡Contact author: aurelie.sangare@maths.ox.ac.uk

Phys. Rev. D 113, L021903 – Published 26 January, 2026

DOI: https://doi.org/10.1103/mppb-xh1j

Abstract

We consider the Anti-de Sitter (AdS) Veneziano amplitude describing the scattering of four gluons in type IIB string theory on ${AdS}_{5} \times S^{3}$ . We propose a set of powerful monodromy relations between different color-ordered amplitudes. These relations arise as a consequence of the emergent worldsheet description of open string scattering on AdS. In flat space, they reduce to the usual monodromy relations for the Veneziano amplitude, and they hold order by order in the small curvature expansion. The relations hold for the results available in the literature, including the scattering of arbitrary Kaluza-Klein modes.

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References (27)

I. Heemskerk, J. Penedones, J. Polchinski, and J. Sully, Holography from conformal field theory, J. High Energy Phys. 10 (2009) 079.
L. F. Alday, C. Behan, P. Ferrero, and X. Zhou, Gluon scattering in AdS from CFT, J. High Energy Phys. 06 (2021) 020.
O. Aharony, A. Fayyazuddin, and J. M. Maldacena, The large N limit of $N = 2$ , $N = 1$ field theories from three-branes in F theory, J. High Energy Phys. 07 (1998) 013.
I. P. Ennes, C. Lozano, S. G. Naculich, and H. J. Schnitzer, Elliptic models, type IIB orientifolds and the AdS/CFT correspondence, Nucl. Phys. B591, 195 (2000).
A. Karch and E. Katz, Adding flavor to AdS/CFT, J. High Energy Phys. 06 (2002) 043.
L. F. Alday, S. M. Chester, T. Hansen, and D.-l. Zhong, The AdS Veneziano amplitude at small curvature, J. High Energy Phys. 05 (2024) 322.
L. F. Alday and T. Hansen, Single-valuedness of the AdS Veneziano amplitude, J. High Energy Phys. 08 (2024) 108.
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, J. High Energy Phys. 03 (2011) 025.
Multiple zeta-values also have weight, since they are multiple polylogarithms evaluated at $x = 1$ . For instance, $ζ (n)$ has weight $n$ .
E. Plahte, Symmetry properties of dual tree-graph n-point amplitudes, Nuovo Cimento A 66, 713 (1970).
N. E. J. Bjerrum-Bohr, P. H. Damgaard, and P. Vanhove, Minimal basis for gauge theory amplitudes, Phys. Rev. Lett. 103, 161602 (2009).
S. Stieberger, Open & closed vs. pure open string disk amplitudes, arXiv:0907.2211.
L. F. Alday, M. Nocchi, and A. S. Sangaré, Stringy KLT relations on AdS, arXiv:2504.19973.
H. N. Minh, M. Petitot, and J. van der Hoeven, L’algèbre des polylogarithmes par les séries génératrices, in Proceedings of FPSAC’99, 11-th International Conference on Formal Power Series and Algebraic Combinatorics (1999).
F. Brown, Single-valued hyperlogarithms and unipotent differential equations, http://www.ihes.fr/~brown/RHpaper5.pdf, 2004. Preprint available on the author’s webpage.
See Supplemental Material at http://link.aps.org/supplemental/10.1103/mppb-xh1j for a toy model illustrating the constraints imposed by the monodromy relations (Sec. I); properties of the monodromy linear operator (Sec. II); and the explicit expression for the first curvature correction to the AdS Veneziano amplitude for arbitrary KK modes (Sec. III).
The operators $τ_{i}$ from the left/right act on $f_{w}$ as $τ_{i} f_{j w} = δ_{i j} f_{w}, f_{w j} τ_{i} = δ_{i j} f_{w} .$
B. Wang, Bootstrap AdS Veneziano amplitude with arbitrary Kaluza-Klein modes, arXiv:2508.14968.
L. F. Alday, T. Hansen, and J. A. Silva, AdS Virasoro-Shapiro from dispersive sum rules, J. High Energy Phys. 10 (2022) 036.
L. F. Alday, T. Hansen, and J. A. Silva, AdS Virasoro-Shapiro from single-valued periods, J. High Energy Phys. 12 (2022) 010.
L. F. Alday, T. Hansen, and J. A. Silva, Emergent worldsheet for the AdS Virasoro-Shapiro amplitude, Phys. Rev. Lett. 131, 161603 (2023).
L. F. Alday and T. Hansen, The AdS Virasoro-Shapiro amplitude, J. High Energy Phys. 10 (2023) 023.
Y.-t. Huang, J.-Y. Liu, L. Rodina, and Y. Wang, Carving out the space of open-string S-matrix, J. High Energy Phys. 04 (2021) 195.
J. Berman, H. Elvang, and A. Herderschee, Flattening of the EFT-hedron: Supersymmetric positivity bounds and the search for string theory, J. High Energy Phys. 03 (2024) 021.
L.-Y. Chiang, Y.-t. Huang, and H.-C. Weng, Bootstrapping string theory EFT, J. High Energy Phys. 05 (2024) 289.
R. H. Boels, D. Marmiroli, and N. A. Obers, On-shell recursion in string theory, J. High Energy Phys. 10 (2010) 034.
R. H. Boels and T. Hansen, String theory in target space, J. High Energy Phys. 06 (2014) 054.

Phys. Rev. D 113, L021903– Published 26 January, 2026

Vol. 113, Iss. 2 — 15 January 2026

Received 15 October 2025
Accepted 17 December 2025

DOI: https://doi.org/10.1103/mppb-xh1j

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Monodromy relations for string amplitudes on AdS space

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