The Momentum Kernel of Gauge and Gravity Theories

TL;DR

This work derives an explicit n-point relation between closed and open string amplitudes through the momentum kernel ${\cal S}_{\alpha'}$, showing it generates monodromy relations and reduces to the field-theory kernel in the low-energy limit. It analyzes key properties of the kernel (reflection, factorization, annihilation, and j-independence) and demonstrates how amplitudes can be expressed in a minimal $(n-3)!$ basis. The authors also explore soft-graviton limits to obtain a more crossing-symmetric KLT form and discuss color–kinematics duality, constructing gravity numerators analogously to gauge-theory BCJ representations. The results illuminate deep connections between string theory factorization, gauge–gravity relations, and potential extensions to broader conformal-field-theory contexts and loop-level structures.

Abstract

We derive an explicit formula for factorizing an $n$-point closed string amplitude into open string amplitudes. Our results are phrased in terms of a momentum kernel which in the limit of infinite string tension reduces to the corresponding field theory kernel. The same momentum kernel encodes the monodromy relations which lead to the minimal basis of color-ordered amplitudes in Yang-Mills theory. There are interesting consequences of the momentum kernel pertaining to soft limits of amplitudes. We also comment on surprising links between gravity and certain combinations of kinematic and color factors in gauge theory.

Figures (1)

• Figure 1: The nested structure of the contours of integration for the variable $v^-_i$ corresponding to the ordering $0<v^+_2<v^+_3<\cdots<v^+_{n-2}<1$ of the $v_+$ variables.