Scattering in Three Dimensions from Rational Maps

TL;DR

The paper develops a three-dimensional formulation of tree-level scattering by dimensional reduction from four-dimensional rational-map integrals, revealing a universal structure governed by four key integrand components. It shows how ABJM and BLG theories arise naturally in 3D and analyzes how SUSY truncation, infrared behavior, and KLT relations fit within this rational-map framework. The authors also provide detailed counting of solutions in 3D, linking their multiplicities to tangent/Euler numbers and showing a consistent picture across YM, SUGRA, and SCS theories. They propose a general SL(2,C)-invariant template for 3D amplitudes and discuss scattering equations as a foundational backbone, outlining future directions for extending the formalism to other dimensions and theories.

Abstract

The complete tree-level S-matrix of four dimensional ${\cal N}=4$ super Yang-Mills and ${\cal N} = 8$ supergravity has compact forms as integrals over the moduli space of certain rational maps. In this note we derive formulas for amplitudes in three dimensions by using the fact that when amplitudes are dressed with proper wave functions dimensional reduction becomes straightforward. This procedure leads to formulas in terms of rational maps for three dimensional maximally supersymmetric Yang-Mills and gravity theories. The integrand of the new formulas contains three basic structures: Parke-Taylor-like factors, Vandermonde determinants and resultants. Integrating out some of the Grassmann directions produces formulas for theories with less than maximal supersymmetry, which exposes yet a fourth kind of structure. Combining all four basic structures we start a search for consistent S-matrices in three dimensions. Very nicely, the most natural ones are those corresponding to ABJM and BLG theories. We also make a connection between the power of a resultant in the integrand, representations of the Poincaré group, infrared behavior and conformality of a theory. Extensions to other theories in three dimensions and to arbitrary dimensions are also discussed.