Towards the n-point one-loop superstring amplitude II: Worldsheet functions and their duality to kinematics

TL;DR

This paper develops worldsheet functions on a genus-one surface for one-loop superstring amplitudes, revealing a duality with superfield kinematics and simplifying the calculation of correlators.

Contribution

It introduces two classes of worldsheet functions with specific properties that simplify one-loop correlator calculations and establish a duality with superfield kinematics.

Findings

Construction of worldsheet functions with prescribed monodromies

Introduction of generalized elliptic integrands (GEIs)

Discovery of a duality between worldsheet functions and kinematics

Abstract

This is the second installment of a series of three papers in which we describe a method to determine higher-point correlation functions in one-loop open-superstring amplitudes from first principles. In this second part, we study worldsheet functions defined on a genus-one surface built from the coefficient functions of the Kronecker--Einsenstein series. We construct two classes of worldsheet functions whose properties lead to several simplifying features within our description of one-loop correlators with the pure-spinor formalism. The first class is described by functions with prescribed monodromies, whose characteristic shuffle-symmetry property leads to a Lie-polynomial structure when multiplied by the local superfields from part I of this series. The second class is given by so-called generalized elliptic integrands (GEIs) that are constructed using the same combinatorial patterns of the BRST pseudo-invariant superfields from part I. Both of them lead to compact and combinatorially rich expressions for the correlators in part III. The identities obeyed by the two classes of worldsheet functions exhibit striking parallels with those of the superfield kinematics. We will refer to this phenomenon as a duality between worldsheet functions and kinematics.