Irreversibility of RG flows on three-dimensional defects
In plain words
Defects such as impurity lines or surfaces inside a critical material carry their own count of degrees of freedom. This count is proven to drop under coarse-graining for line, surface and four-dimensional defects, but not for three-dimensional ones.
Precise statement
For a conformal defect of dimension $p$ in a unitary d-dimensional CFT with the bulk held fixed, monotonicity under defect RG flows is proven for $p = 1$ (defect entropy; Cuomo, Komargodski, Raviv-Moshe 2021), $p = 2$ (b-theorem; Jensen and O'Bannon 2015, extended to surface defects in 2018) and $p = 4$ (defect a-theorem; Wang 2021). For $p = 3$, prove or refute that the defect contribution to the regularized sphere free energy decreases, as conjectured by Kobayashi, Nishioka, Sato and Watanabe (2018).
What would settle it
A proof of monotonicity of the defect sphere free energy for all unitary $p = 3$ defect flows, or a unitary counterexample.
Status in the literature
Unverified note
The naive $b$-theorem was found to fail when the bulk also flows (Shachar, Sinha, Smolkin 2024); the $p = 3$ case with fixed bulk is unproven.