Surprises in the Ordinary: $O(N)$ Invariant Surface Defect in the $\epsilon$-expansion

TL;DR

This paper investigates an $O(N)$ invariant surface defect in the Wilson-Fisher CFT using $ ext{epsilon}$-expansion, revealing unexpected non-renormalization properties and supporting a factorization conjecture in three dimensions.

Contribution

It provides the first perturbative computation of defect CFT data for an $O(N)$ surface defect in $d=4- ext{epsilon}$ dimensions, including anomalous dimensions and anomalies, and supports a factorization hypothesis in $d=3$.

Findings

Defect CFT data computed up to third order in $ ext{epsilon}$-expansion.

Observed non-renormalization of certain surface operator properties.

Supported the factorization of the defect into boundary conditions in $d=3$.

Abstract

We study an $O(N)$ invariant surface defect in the Wilson-Fisher conformal field theory (CFT) in $d=4-\epsilon$ dimensions. This defect is defined by mass deformation on a two-dimensional surface that generates localized disorder and is conjectured to factorize into a pair of ordinary boundary conditions in $d=3$. We determine defect CFT data associated with the lightest $O(N)$ singlet and vector operators up to the third order in the $\epsilon$-expansion, find agreements with results from numerical methods and provide support for the factorization proposal in $d=3$. Along the way, we observe surprising non-renormalization properties for surface anomalous dimensions and operator-product-expansion coefficients in the $\epsilon$-expansion. We also analyze the full conformal anomalies for the surface defect.