  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.
