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.

Type-B anomaly matching and the 6D (2,0) theory

“…3 ref. [4] argued that the states corresponding to HBOs have vanishing Berry phases under parallel transport on the superconformal manifold, 4 which is a fact that we will use. Second, the argument relies crucially on a superconformal Ward identity whose integrated form receives exactly the same contributions in the unbroken and broken phases.…”

“…Therefore, it is important to analyse to what extent coupling-constant-independent vevs v O can be consistently chosen. We propose that such a choice can be made on the Higgs branch because the connection on the bundle of Higgs-branch superconformal-primary operators is flat [4] and therefore one can trivially choose the vacuum so that ∂ λ i v O = 0. Under these conditions eq.…”

“…and from these equations we read off G CFT 22 = 2(k 2 −1) (2π) 4 . As a result, in this case we have verified explicitly the tree-level matching of type-B anomalies…”

The Coulomb and Higgs branches of $$ \mathcal{N} $$ = 1 theories of Class $$ {\mathcal{S}}_k $$

“…Therefore, it would be interesting to clarify if a similar phenomenon could take place also for the 4d N = 1 theories considered in this paper. As a possibly smaller step in this direction it would be interesting to clarify which of the properties, valid for N = 2 4d Higgs branches, discussed [30] have a counterpart for the N = 1 theories discussed in this paper.…”

“…Nonetheless, they can be determined from a simple deformation of Pestun's matrix model via [24][25][26][27]. 2 On the other hand, correlation functions of Higgs branch operators are independent of the coupling constants [29,30] and thus fixed, up to multiplet recombination, by a tree level calculation, which can be mapped to a 2d chiral algebra computation [29].…”