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arXiv:2602.13782

Defect relative entropy in symmetric orbifold CFTs

Mostafa Ghasemi

hep-thcond-mat.stat-mechmath-phquant-phSubmitted Feb 14, 2026• Announced Feb 17, 2026• 0 citations• Revised Apr 20, 2026 (v2)
v1 · Feb 14, 2026v2 · Apr 20, 2026
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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.

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In this work, we compute the defect relative entropy between topological defects in the symmetric product orbifold CFT SymN(M)=M⊗N/SN\mathrm{Sym}^N(M) = M^{\otimes N}/S_NSymN(M)=M⊗N/SN​. Our analysis covers two distinct classes of defects: universal defects, which realize the Rep(SN)\mathrm{Rep}(S_N)Rep(SN​) 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 SNS_NSN​, and the other controlled by modular SSS-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.
Distinguishability is central to quantum information theory, but a quantitative measure for distinguishing topological defects--realizations of generalized symmetries in quantum field theory (QFT)--has been lacking. We introduce the notion of the \textit{defect relative entropy} to fill this gap for topological defects in two-dimensional conformal field theories (CFTs). For defects on a circle, we derive a universal formula that reduces defect relative entropy to a Kullback--Leibler divergence determined entirely by the modular S\mathcal{S}S-matrix and defect coefficients. Thus, the algebraic data governing modular transformations also determines defect distinguishability. A striking consequence is that certain distinct defects can have vanishing relative entropy when restricted to one side, implying that an observer confined to that side cannot distinguish them. This gives rise to information-theoretic equivalence classes of defects, which we term \textit{defect relative sectors}. We further introduce the sandwiched defect Rényi relative entropy and defect fidelity, derive general formulas for these quantities. Explicit calculations in the Ising model, tricritical Ising model, and su^(2)k\widehat{su}(2)_ksu(2)k​ WZW models illustrate our results.
The concept of distinguishability lies at the heart of quantum information theory. We introduce \textit{left-right relative entropy} as a quantitative measure of distinguishability within the space of boundary states in two-dimensional conformal field theories (CFTs). By tracing over either the left- or right-moving modes, we derive a universal formula for arbitrary regularized boundary states defined on a circle. Remarkably, the resulting quantity reduces to a Kullback--Leibler divergence, where the associated probability distribution is determined entirely by the modular S\mathcal{S}S-matrix and the boundary data. For diagonal CFTs, we obtain exact expressions for the left-right relative entropy in terms of modular data, and extend the framework to define \textit{Sandwiched Left-Right Rényi relative entropies} and \textit{left-right fidelity}. Applying this formalism to the Ising model, tricritical Ising model, and su^(2)k\widehat{su}(2)_ksu(2)k​ WZW model, we uncover a striking phenomenon: the left-right relative entropy between certain reduced boundary states vanishes even though the corresponding global boundary states are orthogonal. This observation motivates the introduction of \textit{relative entanglement sectors}, defined as equivalence classes of boundary states that are indistinguishable with respect to left-right relative entropy. These sectors transform as NIM-representations of global symmetries and exhibit level-dependent structures that mirror Z2\mathbb{Z}_2Z2​ 't Hooft anomalies. Our findings establish an unexpected bridge between quantum information measures, boundary conformal symmetry, and quantum anomaly constraints.
Abstract We study the entanglement entropy of gauged internal degrees of freedom in a two dimensional symmetric product orbifold CFT, whose configurations consist of N strands sewn together into “long” strings, with wavefunctions symmetrized under permutations. In earlier work a related notion of “entwinement” was introduced. Here we treat this system analogously to a system of N identical particles. From an algebraic point of view, we point out that the reduced density matrix on k out of N particles is not associated with a subalgebra of operators, but rather with a linear subspace, which we explain is sufficient. In the orbifold CFT, we compute the entropy of a single strand in states holographically dual in the D1/D5 system to a conical defect geometry or a massless BTZ black hole and find a result identical to entwinement. We also calculate the entropy of two strands in the state that represents the conical defect; the result differs from entwinement. In this case, matching entwinement would require finding a gauge-invariant way to impose continuity across strands.
We explore several aspects of categorical symmetry-resolved entanglement entropy (SREE) directly within two-dimensional rational conformal field theory (RCFT) (without invoking any SymTFT construction arXiv:2409.02806). We derive a general formula applicable whenever the action of the relevant topological defect lines on the annulus Hilbert space is known. This framework accommodates weakly and strongly symmetric boundaries, cloaking states, and fusion rings with multiplicities. We verify the formula in a range of diagonal unitary and non-unitary examples, including theories with generalized Haagerup-Izumi modular data. Furthermore, we extend the analysis to non-diagonal RCFTs. The 12E6\frac{1}{2}E_621​E6​ example demonstrates that closed-channel modular data and NIM-rep multiplicities alone do not suffice to determine the defect action on the complete open-channel Hilbert space.
We study the entanglement entropy of gauged internal degrees of freedom in a two dimensional symmetric product orbifold CFT, whose configurations consist of N strands sewn together into “long” strings, with wavefunctions symmetrized under permutations. In earlier work a related notion of “entwinement” was introduced. Here we treat this system analogously to a system of N identical particles. From an algebraic point of view, we point out that the reduced density matrix on k out of N particles is not associated with a subalgebra of operators, but rather with a linear subspace, which we explain is sufficient. In the orbifold CFT, we compute the entropy of a single strand in states holographically dual in the D1/D5 system to a conical defect geometry or a massless BTZ black hole and find a result identical to entwinement. We also calculate the entropy of two strands in the state that represents the conical defect; the result differs from entwinement. In this case, matching entwinement would require finding a gauge-invariant way to impose continuity across strands.