Geometry of Higgs-branch superconformal primary bundles

Abstract

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.