A Duality For The S Matrix

TL;DR

The paper proposes a dual, non-spacetime formulation for the S Matrix of ${\cal N}=4$ SYM in terms of leading singularities encoded as multidimensional residues on Grassmannians. It introduces the gauge-fixed integral ${\cal L}_{n;k}$ over a GL$(k)$-fixed matrix $C_{\alpha a}$, whose residues correspond to leading singularities across tree, 1-loop, and beyond, with a geometric interpretation via $G(k,n)$ and the Plücker (minors) coordinates. Through explicit NMHV and N$^2$MHV examples (including 6, 7, and 8-particle amplitudes), the work demonstrates how different contour choices yield equivalent representations, with the global residue theorem enforcing cyclicity, parity, and the cancellation of unphysical poles, as well as IR relations at 1-loop. The results provide substantial evidence that leading singularities determine amplitudes at all loop orders, and they reveal deep connections to algebraic geometry (Schubert calculus) and the potential for a universal generating function for Yangian invariants. The framework also offers a route toward understanding emergent spacetime and unitarity from a purely on-shell, contour-based perspective, with extensions to higher k and n and to gravity in prospects noted.

Abstract

We propose a dual formulation for the S Matrix of N = 4 SYM. The dual provides a basis for the "leading singularities" of scattering amplitudes to all orders in perturbation theory, which are sharply defined, IR safe data that uniquely determine the full amplitudes at tree level and 1-loop, and are conjectured to do so at all loop orders. The scattering amplitude for n particles in the sector with k negative helicity gluons is associated with a simple integral over the space of k planes in n dimensions, with the action of parity and cyclic symmetries manifest. The residues of the integrand compute a basis for the leading singularities. A given leading singularity is associated with a particular choice of integration contour, which we explicitly identify at tree level and 1-loop for all NMHV amplitudes as well as the 8 particle NNMHV amplitude. We also identify a number of 2-loop leading singularities for up to 8 particles. There are a large number of relations among residues which follow from the multi-variable generalization of Cauchy's theorem known as the "global residue theorem". These relations imply highly non-trivial identities guaranteeing the equivalence of many different representations of the same amplitude. They also enforce the cancellation of non-local poles as well as consistent infrared structure at loop level. Our conjecture connects the physics of scattering amplitudes to a particular subvariety in a Grassmannian; space-time locality is reflected in the topological properties of this space.