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This is the second installment of a series of three papers in which wedescribe a method to determine higher-point correlation functions in one-loopopen-superstring amplitudes from first principles. In this second part, westudy worldsheet functions defined on a genus-one surface built from thecoefficient functions of the Kronecker--Einsenstein series. We construct twoclasses of worldsheet functions whose properties lead to several simplifyingfeatures within our description of one-loop correlators with the pure-spinorformalism. The first class is described by functions with prescribedmonodromies, whose characteristic shuffle-symmetry property leads to aLie-polynomial structure when multiplied by the local superfields from part Iof this series. The second class is given by so-called generalized ellipticintegrands (GEIs) that are constructed using the same combinatorial patterns ofthe BRST pseudo-invariant superfields from part I. Both of them lead to compactand combinatorially rich expressions for the correlators in part III. Theidentities obeyed by the two classes of worldsheet functions exhibit strikingparallels with those of the superfield kinematics. We will refer to thisphenomenon as a duality between worldsheet functions and kinematics.

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