Throughout this Letter, we use $\mathsf{E}$ (and $\mathsf{S}$, $\mathsf{Z}$) to denote the non-supersymmetric Casimir energy (and Rényi entropy, partition function), while $E$ (and $S$, $Z$) refer to their supersymmetric counterparts.

We mainly focus on $A$-type theories labelled by Lie algebra $\mathfrak{g}=\mathfrak{su}(N)$.

The convention for ${\omega }_{1,2,3}$ and ${\sigma }_{1,2}$ follows [3], satisfying the supersymmetric condition ${\sigma }_1+{\sigma }_2={\omega }_1+{\omega }_2+{\omega }_3$.

In defect CFTs, $⟨T_{\mu }^{\mu }⟩=⟨T_{\mu }^{\mu }{⟩}_{\mathrm{bulk}}+{\delta }_D⟨T_{\mu }^{\mu }{⟩}_{\mathrm{def}}$, where ${\delta }_D$ represents the delta function localized on the defect submanifold.

The Wilson loop insertion modifies the saddle points of the matrix integral, depending on the representation $\mathcal{R}$ of the Wilson loop [33].

The surface defect must be orthogonal to the chiral plane [32, 41]. Since the defect is inserted on the ${\mathbb{R}}_{{\omega }_1}^2$ plane, we choose ${\mathbb{R}}_{{\omega }_3}^2$ as the chiral plane. The corresponding chiral limit is ${\sigma }_1-{\sigma }_2={\omega }_1+{\omega }_2-{\omega }_3$. Taking ${\omega }_1={\omega }_2=1$ then gives ${\sigma }_1{\sigma }_2=2{\omega }_3$.

It has been speculated that the surface-defect contribution to the supersymmetric Casimir energy depends only on $d_2$ [41, 43], but our result shows that this holds only in the chiral limit.

The surface defect contribution to non-SUSY EE is [22] $${\mathsf{S}}_1=\frac{1}{3}(b-\frac{d-3}{d-1}{d}_2)\log \ell /\epsilon .$$(24)