Logarithms and Volumes of Polytopes

TL;DR

This paper introduces a novel approach using contour integrals of logarithms to compute the volume of the dual amplituhedron and general polytopes, providing a triangulation-independent geometric description.

Contribution

It presents a new method for calculating polytope volumes via logarithmic contour integrals that are independent of triangulation, clarifying amplitude identities.

Findings

Contour integrals of logarithms effectively compute polytope volumes.

The method applies to the dual amplituhedron and general polytopes in any dimension.

It makes amplitude representation identities manifest.

Abstract

Describing the geometry of the dual amplituhedron without reference to a particular triangulation is an open problem. In this note we introduce a new way of determining the volume of the tree-level NMHV dual amplituhedron. We show that certain contour integrals of logarithms serve as natural building blocks for computing this volume as well as the volumes of general polytopes in any dimension. These building blocks encode the geometry of the underlying polytopes in a triangulation-independent way, and make identities between different representations of the amplitudes manifest.