Scattering amplitudes of stable curves

TL;DR

The paper builds a rigorous bridge between algebraic geometry and high-energy physics by viewing leading singularities of n-point scattering amplitudes in N=4 Yang–Mills theory as probabilistic Brill–Noether data on stable curves in the MHV regime. It introduces a universal scattering amplitude map ${\bold\Lambda}$ from Picard varieties of genus g curves (with n=g+3 marked points, degree d=g+1) to the moduli space $M_{0,n}$, and proves that its degree is $2^g$ for general curves, with multiple complementary viewpoints (parabolic bundles, matrix models, hypertree combinatorics, and real-geometry). The framework is developed through compactifications via compactified Jacobians and moduli of parabolic bundles, real-geometry analyses on M-curves yielding localization of measures across $2^g$ components, and explicit genus-2/ hyperelliptic degenerations tied to Kummer/K3/dP4 geometries. A matrix-model description recasts the branch structure in terms of Lax flows and eigenlines, while the combinatorics of CT hypertrees and Tutte-type constructions encode maximally degenerate degenerations and hypertree divisors. Overall, the work reveals deep, multifaceted connections among moduli of curves, parabolic-bundle moduli, hypertree combinatorics, and amplitude leading singularities, with concrete real-geometry and degeneracy analyses complementing the complex-analytic theory.

Abstract

Equations of hypertree divisors on the Grothendieck-Knudsen moduli space of stable rational curves, introduced by Castravet and Tevelev, appear as numerators of scattering amplitude forms for n massless particles in N=4 Yang-Mills theory in the work of Arkani-Hamed, Bourjaily, Cachazo, Postnikov and Trnka. 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 amplitude forms 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 regime, which leads to a beautiful physics-inspired geometry for various classes of algebraic curves: smooth, stable, hyperelliptic, real algebraic, etc.

Figures (20)

• Figure 1: On-shell diagrams of MHV curves of genus $1$
• Figure 2: $\Xi:\,\operatorname{Bun}(\Bbb P^1;z_1,\ldots,z_n)\dashrightarrow M_{0,n}$, $(E;V_1,\ldots,V_n)\mapsto (q_1,\ldots,q_n).$
• Figure 3: MHV M-curves of type A (left) and type B (right)
• Figure 4: Scattering amplitude probability measures in genus $2$
• Figure 5: Two spherical CT hypertrees from the same triangulation CT_Crelle

Theorems & Definitions (105)

• Definition §1.6
• Definition §1.7
• Remark §1.9
• Example §1.10
• Example §1.12
• Theorem §1.13
• Example §2.1
• Remark §2.2
• Theorem §2.3
• proof