  It is known that the two- and three-point functions of Higgs-branch
superconformal primaries in 4d N=2 superconformal field theories obey
non-renormalization theorems on N=2 superconformal manifolds. In this paper we
prove a stronger statement, that the bundles of Higgs-branch superconformal
primaries over N=2 superconformal manifolds are endowed with a flat connection,
or equivalently that Higgs-branch superconformal primaries have vanishing Berry
phases under N=2 exactly marginal deformations. This statement fits well with
the proposed correspondence between the rigid structures of 2d chiral algebras
and the sector of Schur operators in 4d N=2 theories. We also discuss the
general interplay between non-renormalization theorems and the curvature of
bundles of protected operators and provide a new simpler proof of the vanishing
curvature of 1/2-BPS operators in 4d N=4 SYM theory that does not require the
use of the 4d tt* equations.
