Geometry of Higgs-branch superconformal primary bundles

TL;DR

This paper proves that Higgs-branch superconformal primaries in 4d N=2 theories form flat bundles over superconformal manifolds, indicating they have no Berry phase under exactly marginal deformations, and relates this to 2d chiral algebra structures.

Contribution

It establishes the flatness of Higgs-branch superconformal primary bundles, strengthening non-renormalization theorems and connecting to 2d chiral algebra correspondence.

Findings

Higgs-branch superconformal primary bundles are flat over N=2 superconformal manifolds.

Higgs-branch primaries have vanishing Berry phases under marginal deformations.

Provides a new proof of curvature vanishing for 1/2-BPS operators in 4d N=4 SYM.

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.