Defects, Rigid Holography and $C$-theorems

TL;DR

This paper demonstrates that in scalar quantum field theories with defects, the RG flow is governed by a gradient of an entropy function, leading to a proof of monotonic $C$-functions in certain cases, with some limitations when fermions are included.

Contribution

It establishes the gradient flow property of defect RG flows using holography and Hamilton-Jacobi formalism, and proves the monotonicity of $C$-functions in conformal theories.

Findings

RG flow is the gradient of an entropy function.

$C$-functions decrease monotonically along RG flows in conformal cases.

Fermions introduce potential obstructions at two-loop order.

Abstract

We consider a general scalar QFT with a linear defect in $D=4-\epsilon$ and a surface defect in $D=6-\epsilon$. Using holography and the Hamilton-Jacobi formalism, we show that the $\beta$ functions controlling the defect RG flow are the gradient of the entropy function. In the case of conformal field theories, this allows the proof that the relevant $C$-functions decrease monotonically along the RG flow. We provide evidence that this property also holds in the full quantum theory for general scalar field theories. An obstruction to the gradient property seems to appear at two loop order when fermions are added.