Soft Particles and Infinite-Dimensional Geometry

TL;DR

This paper investigates how soft particle insertions in gauge theory and gravity relate to the geometry of infinite-dimensional vacuum moduli spaces, revealing connections between soft theorems and the curvature of these spaces.

Contribution

It extends the geometric interpretation of soft theorems from finite-dimensional sigma models to infinite-dimensional moduli spaces in gauge theories and gravity, linking soft insertions to curvature computations.

Findings

Double-soft gluon theorem computes the curvature of the vacuum space.

Abelian gauge theory and gravity have flat vacuum moduli spaces.

Feynman diagrams encode information about the vacuum manifold's geometry.

Abstract

In the sigma model, soft insertions of moduli scalars enact parallel transport of $S$-matrix elements about the finite-dimensional moduli space of vacua, and the antisymmetric double-soft theorem calculates the curvature of the vacuum manifold. We explore the analogs of these statements in gauge theory and gravity in asymptotically flat spacetimes, where the relevant moduli spaces are infinite-dimensional. These models have spaces of vacua parameterized by (trivial) flat connections at spatial infinity, and soft insertions of photons, gluons, and gravitons parallel transport $S$-matrix elements about these infinite-dimensional manifolds. We argue that the antisymmetric double-soft gluon theorem in $d+2$ bulk dimensions computes the curvature of a connection on the infinite-dimensional space Map$(S^d,G)/G$. The analogous metrics in abelian gauge theory and gravity are flat, as indicated by the vanishing of the antisymmetric double-soft theorems in those models. In other words, Feynman diagram calculations not only know about the vacuum manifold of Yang-Mills theory, they can also be used to compute its curvature. The results have interesting implications for flat holography.