Scattering amplitudes of stable curves

Tevelev, Jenia

doi:10.2140/gt.2025.29.3063

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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Scattering amplitudes of stable curves

Jenia Tevelev

Geometry & Topology 29 (2025) 3063–3128

DOI: 10.2140/gt.2025.29.3063

Abstract

Hypertree divisors on the moduli space of stable rational curves were introduced by Castravet and Tevelev (2013). Their equations appear as numerators of scattering amplitude forms for $n$ particles in $N = 4$ Yang–Mills theory in the work of Arkani-Hamed, Bourjaily, Cachazo, Postnikov and Trnka (2016). Rather than being a coincidence, this is just the tip of the iceberg of an exciting relation between algebraic geometry and high energy physics. We interpret leading singularities of scattering amplitudes of massless particles as probabilistic Brill–Noether theory: the study of statistics of images of $n$ marked points under a random meromorphic function uniformly distributed with respect to the translation-invariant volume form of the Jacobian. We focus on the maximum helicity violating case, which leads to a beautiful physics-inspired geometry for various classes of complex algebraic curves: smooth, stable, hyperelliptic, real algebraic, etc.

Keywords

algebraic curves, stable algebraic curves, M-curves, scattering amplitudes, Brill–Noether theory

Mathematical Subject Classification

Primary: 14D20, 14E05, 14E20, 14F10, 14H10, 14H40, 14H81, 14P05

References

Publication

Received: 16 October 2020

Revised: 8 August 2023

Accepted: 7 September 2023

Published: 22 September 2025

Proposed: Richard P Thomas

Seconded: Mark Gross, Jim Bryan

Authors

Jenia Tevelev
Department of Mathematics and Statistics University of Massachusetts Amherst, MA United States

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Volume 29, issue 6 (2025)