  Known examples of the holographic dictionary in asymptotically Anti-de Sitter
spacetimes equate moduli spaces of bulk vacua with conformal manifolds in the
dual quantum field theory. We demonstrate that the same identification holds
for gravity in asymptotically flat spacetimes in any dimension, in accord with
expectations derived from the celestial conformal field theory (CCFT)
formalism. Soft limits of moduli scalars described by the sigma model are
universal, and relate to parallel transport of $S$-matrix observables over the
moduli space of bulk vacua. The leading "soft moduli operator" is the shadow
transform of a dimension $\Delta=d$ marginal operator $M(x)$. The universal
form of the soft limit guarantees that $M(x)$ acts as a marginal deformation in
the CCFT$_d$, and coherent states of the soft scalars correspond to finite
deformations along the conformal manifold. This manifold typically has
curvature, which is captured by the antisymmetric double-soft theorem and which
reflects the Berry curvature in CCFT$_d$. We also compute the
Mellin-transformed four-point function in the sigma model and compare to a
formula of Kutasov for the curvature of the conformal manifold.
