Soft Particles and Infinite-Dimensional Geometry

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.