Abstract
Two different constructions generating the low-energy expansion of genus-one configuration-space integrals appearing in one-loop open-string amplitudes have been put forward in refs. [1–3]. We are going to show that both approaches can be traced back to an elliptic system of Knizhnik-Zamolodchikov-Bernard(KZB) type on the twice-punctured torus.We derive an explicit all-multiplicity representation of the elliptic KZB system for a vector of iterated integrals with an extra marked point and explore compatibility conditions for the two sets of algebra generators appearing in the two differential equations.
Highlights
Introduction and summaryDuring the last years we have been experiencing a significant growth in understanding the mathematical concepts leading to recursion relations for scattering amplitudes in quantum field and string theory
The calculation of one-loop open-string amplitudes requires consideration of configurationspace integrals on a genus-one surface with boundary. The latter can be constructed by starting from a genus-one Riemann surface whose geometry is usually parametrized by a modular parameter τ ∈ C with Im τ > 0
While elliptic multiple zeta values (eMZVs) have been defined in eq (2.17) in terms of special values of iterated integrals, which featured repeated integration in insertion points zi, it is possible to write them in terms of τ -iterated integrals
Summary
During the last years we have been experiencing a significant growth in understanding the mathematical concepts leading to recursion relations for scattering amplitudes in quantum field and string theory. These integrals to be referred to as genus-one configuration-space integrals form the backbone of one-loop open-string amplitudes Both algorithms rely on differential equations of Knizhnik-Zamolodchikov-Bernard(KZB) type on a genus-one surface with boundaries. Vector of generating functions for planar n-point one-loop configuration-space integrals to be denoted by Zτ0,n with an auxiliary point z0: in particular a) we will find an all-multiplicity expression for the τ -derivative of Zτ0,n in order to connect with the approach in ref.
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