[Paper Review] Scattering amplitudes of stable curves

Jenia Tevelev|arXiv (Cornell University)|Jul 8, 2020

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes a deep connection between scattering amplitudes in $\mathcal{N}=4$ Yang–Mills theory and algebraic geometry by interpreting leading singularities of amplitudes as probabilistic Brill–Noether theory on stable curves. It shows that residues of meromorphic forms on nodal rational curves—parametrized by the moduli space $M_{0,n}$—correspond to momentum-conserving configurations of massless particles, with the scattering amplitude form arising from a canonical volume form on a torsor over the Picard group of the curve.

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.

Motivation & Objective

• To establish a geometric framework linking scattering amplitudes in high-energy physics to algebraic geometry, particularly through the moduli space of stable rational curves.
• To interpret leading singularities of scattering amplitudes as statistics of residues of meromorphic forms under a random meromorphic function on a curve.
• To show that the momentum conservation laws in $\mathcal{N}=4$ Yang–Mills theory arise naturally from the residue theorem on nodal curves with marked points.
• To demonstrate that the scattering amplitude form on $X(k,n)$ (isomorphic to $M_{0,n}$ in the MHV case) is equivalent to the leading singularity form on the Grassmannian $G(k,n)$ via a canonical volume form on a torsor over the Picard group.
• To provide a geometric realization of the spinor helicity formalism in terms of line bundles and global sections on stable curves, with trivializations encoding momentum data.

Proposed method

• Uses a random tensor product factorization $\omega_C(p_1+\cdots+p_n) = L \otimes \tilde{L}$ on a stable curve $C$, where $L$ and $\tilde{L}$ are line bundles of degrees $d$ and $\tilde{d}$ with $d + \tilde{d} = 2g - 2 + n$.
• Imposes a uniform probability measure on $\operatorname{Pic}^d C$ via the translation-invariant volume form of the Jacobian, especially in the case of nodal rational curves where $\operatorname{Pic}^0 C \simeq (\mathbb{C}^*)^g$.
• Constructs a $2 \times 2$ matrix of log forms $\omega_{\alpha\tilde{\alpha}} = s_\alpha \otimes \tilde{s}_{\tilde{\alpha}}$ from global sections $s_\alpha \in H^0(C,L)$, $\tilde{s}_{\tilde{\alpha}} \in H^0(C,\tilde{L})$, with $|I| = |\tilde{I}| = 2$.
• Identifies the residue vectors $\operatorname{Res}_{p_i} \omega_{\alpha\tilde{\alpha}}$ as momenta of massless particles in $4$-dimensional Minkowski space, lying in the complexified light cone $\mathbb{L} \subset \operatorname{Mat}_{2,2}$.
• Defines a rational map $\hat{\Lambda}$ from the $\mathbb{C}^*$-torsor $\widehat{\operatorname{Pic}}^\vec{d} C$ to the Grassmannian $G(k,n)$, where the image corresponds to the row space of a matrix of spinor variables.
• Derives the scattering amplitude form as the pullback of the standard $\operatorname{dlog}$ form on $(\mathbb{C}^*)^{2v}/(\mathbb{C}^*)^i$, which encodes momentum conservation and on-shell conditions via delta functions in the amplitude.

Experimental results

Research questions

• RQ1 How can leading singularities of scattering amplitudes in $\mathcal{N}=4$ Yang–Mills theory be interpreted geometrically in terms of algebraic curves?
• RQ2 What is the role of the moduli space of stable rational curves in encoding momentum-conserving configurations of massless particles?
• RQ3 How does the random choice of line bundles and sections on a stable curve give rise to a probability distribution of scattering amplitudes?
• RQ4 In what way does the structure of the dual graph of a nodal curve correspond to the on-shell diagram of a scattering amplitude?
• RQ5 How is the scattering amplitude form on $X(k,n)$ related to the canonical volume form on the torsor $\widehat{\operatorname{Pic}}^\vec{d} C$ and the Grassmannian $G(k,n)$?

Key findings

• The scattering amplitude form on $X(k,n)$—which is isomorphic to $M_{0,n}$ in the MHV case ($k=2$)—is equivalent to the leading singularity form on $G(k,n)$, as shown via the canonical $\operatorname{dlog}$ form on $(\mathbb{C}^*)^{2v}/(\mathbb{C}^*)^i$.
• For a genus $4$ curve with $6$ marked points and a degree $6$ line bundle $L$, the rational map $\Lambda: \operatorname{Pic}^6 C \to X(3,6)$ has a well-defined degree, which is the solution to the '666 Puzzle' posed in the paper.
• The construction is reversible: given momentum data satisfying momentum conservation at each vertex of the dual graph (with no two adjacent momenta proportional), one can reconstruct a factorization $L \otimes \tilde{L}$ and sections $s_\alpha, \tilde{s}_{\tilde{\alpha}}$ such that the residues yield the given momenta.
• The spinor variables $\lambda_i, \tilde{\lambda}_i$ are encoded by trivializations of $L$ and $\tilde{L}$ at marked points, with the momentum $\mathbb{p}_i = \lambda_i \tilde{\lambda}_i^T$.
• The action of the little torus $(\mathbb{C}^*)^n$ on the Grassmannian corresponds to rescaling trivializations, and the quotient map $\pi: G(k,n) \to X(k,n)$ recovers the amplitude form via the $\operatorname{dlog}$ form.
• In the maximally degenerate case, the dimension of $\widehat{\operatorname{Pic}}^\vec{d} C$ is $2v - i$, where $v$ is the number of components and $i$ the number of internal edges, and the space is isomorphic to $(\mathbb{C}^*)^{2v}/(\mathbb{C}^*)^i$, with the $\operatorname{dlog}$ form matching the known amplitude formula from [Grass].