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.
Highlights
We focus on four-dimensional superconformal quantum field theories (SCFTs) that possess nontrivial conformal manifolds M
In subsection II C, we argue quite generally that the existence of a nonrenormalization theorem for 3-point functions implies the integrability condition
Repeating the logic in [9] the authors of [22] argued that in a general four-dimensional N 1⁄4 2 SCFT with exactly marginal couplings ðλi; λiÞ ði 1⁄4 1; ...; dimC MÞ the 3-point functions CIJK of any triplet of Schur operators satisfy the nonrenormalization conditions
Summary
We focus on four-dimensional superconformal quantum field theories (SCFTs) that possess nontrivial conformal manifolds M. Repeating the logic in [9] the authors of [22] argued that in a general four-dimensional N 1⁄4 2 SCFT with exactly marginal couplings ðλi; λiÞ ði 1⁄4 1; ...; dimC MÞ the 3-point functions CIJK of any triplet of Schur operators (including the Higgs-branch superconformal primaries) satisfy the nonrenormalization conditions. The nonrenormalization theorem (1.12) plays a central rôle in this two-dimensional/fourdimensional correspondence since the two-dimensional chiral algebras are rigid structures, which are believed to be independent from the exactly marginal couplings of the corresponding four-dimensional theories.5 In this context, it is of interest to know the curvature of the Schur bundles, which determines whether the CIJK are just covariantly constant or (as a stronger statement) constant in a couplingconstant-independent basis The nonrenormalization theorem (1.12) plays a central rôle in this two-dimensional/fourdimensional correspondence since the two-dimensional chiral algebras are rigid structures, which are believed to be independent from the exactly marginal couplings of the corresponding four-dimensional theories. In this context, it is of interest to know the curvature of the Schur bundles, which determines whether the CIJK are just covariantly constant or (as a stronger statement) constant in a couplingconstant-independent basis
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