Defect relative entropy in symmetric orbifold CFTs

Abstract: In this work, we compute the defect relative entropy between topological defects in the symmetric product orbifold CFT $\mathrm{Sym}^N(M) = M^{\otimes N}/S_N$. Our analysis covers two distinct classes of defects: universal defects, which realize the $\mathrm{Rep}(S_N)$ non-invertible symmetry, and non-universal defects. We show that the defect relative entropy reduces to a Kullback--Leibler (KL) divergence. The resulting expression decomposes naturally into two contributions: one governed by characters of the symmetric group $S_N$, and the other controlled by modular $S$-matrix elements of the seed RCFT. Remarkably, both sets of data appear as probability distributions, yielding an information-theoretic interpretation of permutation group data and modular data within the symmetric orbifold. The structure of the divergence depends sensitively on the defect class. For universal defects, only the permutation group data contributes; for maximally fractional defects, both permutation and modular data enter and together define the relevant probability distributions. This feature suggests that the maximally fractional defect can be understood as a kind of product of the RCFT defect and the symmetric orbifold defect.