Broken global symmetries and defect conformal manifolds

TL;DR

This paper explores how defect conformal manifolds arise from symmetry breaking in conformal defects, relating their geometry to correlation functions, and confirms the relations through examples in supersymmetric theories.

Contribution

It establishes an exact relation between the curvature of defect conformal manifolds and integrated 4-point functions of marginal operators, extending the understanding of defect deformations.

Findings

Derived an exact relation between curvature and 4-point functions.

Confirmed the relation using examples in supersymmetric theories.

Connected defect symmetry breaking to geometric structures on conformal manifolds.

Abstract

Just as exactly marginal operators allow to deform a conformal field theory along the space of theories known as the conformal manifold, appropriate operators on conformal defects allow for deformations of the defects. When a defect breaks a global symmetry, there is a contact term in the conservation equation with an exactly marginal defect operator. The resulting defect conformal manifold is the symmetry breaking coset and its Zamolodchikov metric is expressed as the 2-point function of the exactly marginal operator. As the Riemann tensor on the conformal manifold can be expressed as an integrated 4-point function of the marginal operators, we find an exact relation to the curvature of the coset space. We confirm this relation against previously obtained 4-point functions for insertions into the 1/2 BPS Wilson loop in ${\cal N} = 4$ SYM and 3d ${\cal N} = 6$ theory and the 1/2 BPS surface operator of the 6d ${\cal N} = (2, 0)$ theory.