Amplitudes at Weak Coupling as Polytopes in AdS_5

TL;DR

This paper reveals that one-loop scalar box functions in N=4 super Yang-Mills can be geometrically interpreted as volumes of polytopes in AdS_5, linking physics amplitudes to hyperbolic geometry and polytope volumes.

Contribution

It introduces a novel geometric interpretation of one-loop scalar box functions as volumes of polytopes in AdS_5, connecting scattering amplitudes to hyperbolic geometry.

Findings

Box functions correspond to volumes of geodesic tetrahedra in AdS_5.

These volumes are given by the Bloch-Wigner dilogarithm.

Amplitude combinations form boundaryless polytopes attached to null polygons.

Abstract

We show that one-loop scalar box functions can be interpreted as volumes of geodesic tetrahedra embedded in a copy of AdS_5 that has dual conformal space-time as boundary. When the tetrahedron is space-like, it lies in a totally geodesic hyperbolic three-space inside AdS_5, with its four vertices on the boundary. It is a classical result that the volume of such a tetrahedron is given by the Bloch-Wigner dilogarithm and this agrees with the standard physics formulae for such box functions. The combinations of box functions that arise in the n-particle one-loop MHV amplitude in N=4 super Yang-Mills correspond to the volume of a three-dimensional polytope without boundary, all of whose vertices are attached to a null polygon (which in other formulations is interpreted as a Wilson loop) at infinity.