# Marginal deformations and defect anomalies

## TL;DR

This paper investigates how exactly marginal bulk deformations affect defect conformal field theories, revealing a coupling-dependent defect anomaly coefficient and analyzing behaviors across different defect dimensions with applications to Wilson loops and Rényi entropy.

## Contribution

It introduces a detailed analysis of defect anomaly coefficients' dependence on marginal couplings and extends the understanding of defect anomalies to arbitrary dimensions.

## Key findings

- Logarithmic divergence related to defect anomaly coefficient for even-dimensional spherical defects.
- Defect anomaly coefficient depends on the marginal coupling and is not invariant on the conformal manifold.
- Different behaviors observed for odd-dimensional defects, generalizing superconformal Wilson loop anomalies.

## Abstract

We deform a defect conformal field theory by an exactly marginal bulk operator and we consider the dependence on the marginal coupling of flat and spherical defect expectation values. For even dimensional spherical defects we find a logarithmic divergence which can be related to a $a$-type defect anomaly coefficient. This coefficient, for defect theories, is not invariant on the conformal manifold and its dependence on the coupling is controlled to all orders by the one-point function of the associated exactly marginal operator. For odd-dimensional defects, the flat and spherical case exhibit different qualitative behaviors, generalizing to arbitrary dimensions the line-circle anomaly of superconformal Wilson loops. Our results also imply a non-trivial coupling dependence for the recently proposed defect $C$-function. We finally apply our general result to a few specific examples, including superconformal Wilson loops and R\'enyi entropy.