A geometrical framework for amplitude recursions: bridging between trees and loops

TL;DR

This paper introduces a geometric framework using iterated integrals on Riemann surfaces to unify various recursive methods for calculating scattering amplitudes in quantum and string theories.

Contribution

It presents a novel geometric approach based on differential equations of KZ type to connect different amplitude recursions within a single unified framework.

Findings

Provides a geometric interpretation of amplitude recursions.

Unifies tree and loop amplitude calculations.

Uses differential equations to interpolate between geometries.

Abstract

Various methods for the recursive evaluation of scattering amplitudes in quantum field theory and string theory have been put forward during the last couple of years. In these proceedings we describe a geometrical framework, which is believed to be capable of treating many of these recursions in a unified way. Our recursive framework is based on manipulating iterated integrals on Riemann surfaces with boundaries. A geometric parameter appears as variable of a differential equation of KZ or KZB type. The parameter interpolates between two associated regularized boundary values, which contain iterated integrals closely related to scattering amplitudes defined on two different geometries.