Scattering in Three Dimensions from Rational Maps

TL;DR

This paper derives new formulas for three-dimensional supersymmetric gauge and gravity amplitudes using rational maps, revealing connections to known theories like ABJM and BLG, and exploring the structure of S-matrices.

Contribution

It introduces rational map-based formulas for 3D supersymmetric amplitudes by dimensional reduction, connecting them to known theories and analyzing their structural properties.

Findings

Formulas for 3D super Yang-Mills and supergravity amplitudes derived from 4D cases.

Identification of structures like Parke-Taylor factors, Vandermonde determinants, and resultants in the integrand.

Most natural 3D theories correspond to ABJM and BLG models.

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 Poincar\'e group, infrared behavior and conformality of a theory. Extensions to other theories in three dimensions and to arbitrary dimensions are also discussed.