Defect Relative Sector in Quantum Field Theory

• Defect relative sector is defined as an equivalence class where the defect relative entropy vanishes, ensuring indistinguishable observables in quantum systems.
• It uses the modular S-matrix to compute a Kullback-Leibler divergence analogue, providing a rigorous statistical foundation in RCFTs.
• The framework applies to models such as the Ising and WZW models, offering insights into dualities, emergent symmetries, and RG flow classifications.

A defect relative sector is a mathematically and physically precise organizational structure within the study of topological and conformal defects in quantum field theory and statistical mechanics. It is defined as an equivalence class of defects that are indistinguishable with respect to a well-defined measure: the defect relative entropy. This concept features a rigorous statistical and information-theoretic foundation rooted in quantum information theory, modular tensor categories, and the representation theory underlying rational conformal field theories (RCFTs) (Ghasemi, 29 Jan 2026). The defect relative sector formalism refines the classification of defects beyond fusion and symmetry orbits, providing a physically meaningful grouping of defects that implement identical probabilistic superpositions of superselection sectors.

1. Definition and Foundational Formalism

Given two topological defects or interfaces $\mathcal{I}$, $\mathcal{I}'$ in a RCFT on a circle, the defect relative entropy is defined as

$$D\bigl(\mathcal{I}\|\mathcal{I}'\bigr) = \mathrm{tr}\bigl(\rho_{\mathcal{I}}\log\rho_{\mathcal{I}}\bigr) - \mathrm{tr}\bigl(\rho_{\mathcal{I}}\log\rho_{\mathcal{I}'}\bigr).$$

Here, $\rho_{\mathcal{I}}$ and $\rho_{\mathcal{I}'}$ denote the reduced density matrices associated to the respective defects. A defect relative sector is then the set of defects whose pairwise defect relative entropy vanishes: $$\mathcal{I} \sim \mathcal{I}' \quad\Longleftrightarrow\quad D(\mathcal{I} \| \mathcal{I}') = 0.$$ This equivalence is nontrivial: $D(\mathcal{I}\|\mathcal{I}')=0$ if and only if $\rho_{\mathcal{I}} = \rho_{\mathcal{I}'}$, meaning any observable (in particular, any correlation function) computed in the presence of either defect yields identical results (Ghasemi, 29 Jan 2026). The definition is universal and applies to rational CFTs; by extension, it is expected to apply in more general quantum field theoretic contexts as well.

2. Universal Formula and Vanishing Criterion

The defect relative entropy in diagonal RCFTs can be constructed using the modular $S$-matrix of the theory. Labeling topological defects by primaries $a$, $a'$: $$D\bigl(\mathcal{I}_a\|\mathcal{I}_{a'}\bigr) = \sum_{j} p^{(a)}_j \log\left(\frac{p^{(a)}_j}{p^{(a')}_j}\right),$$ where $p^{(a)}_j = |\mathcal{S}_{aj}|^2$ with $\mathcal{S}_{aj}$ the modular $S$-matrix. This formula is a direct analogue of the Kullback-Leibler divergence between the probability distributions associated to $\mathcal{I}_a$ and $\mathcal{I}_{a'}$ (Ghasemi, 29 Jan 2026).

The criterion for two defects to be in the same defect relative sector is thus: $$D(\mathcal{I}_a \| \mathcal{I}_{a'}) = 0 \Longleftrightarrow \forall j,\; |\mathcal{S}_{aj}| = |\mathcal{S}_{a'j}|,$$ i.e., their associated probability vectors are identical. In concrete models, this often means $a = a'$ or $a,a'$ are related by a discrete automorphism.

3. Exemplary Models and Explicit Sectors

The structure of defect relative sectors has been analyzed in several canonical RCFTs (Ghasemi, 29 Jan 2026):

• Ising Model ($c=1/2$): The set of defects splits into two sectors: $$\{\mathcal{I}_{\mathrm{id}}, \mathcal{I}_\epsilon\}$$, which are indistinguishable, and $\{\mathcal{I}_\sigma\}$, which is distinguished by nonzero defect relative entropy.
• Tricritical Ising Model ($c=7/10$): Nontrivial sectors arise such as $\{\mathcal{I}_{0}, \mathcal{I}_{3/2}\}$ and $\{\mathcal{I}_{1/10},\mathcal{I}_{3/5}\}$.
• $\widehat{su}(2)_k$ WZW Models: The sectors are of the form $\{j,\,k/2-j\}$ for each spin $j$.

This organization recovers known fusion-category automorphisms and symmetry orbits, such as the $\mathbb{Z}_2$ symmetry in Ising and charge conjugation in WZW models.

Model	Defect Relative Sectors
Ising ($c=1/2$)	$\{\mathrm{id},\,\epsilon\}$, $\{\sigma\}$
Tricritical Ising ($c=7/10$)	$\{0,\,3/2\}$, $\{1/10,\,3/5\}$, ...
$\widehat{su}(2)_k$ WZW	$\{j,\,k/2-j\}$ for all $j$

Zero defect relative entropy ensures all correlation functions in the presence of any defect in the same sector are identical.

4. Structural and Physical Significance

Grouping topological defects into defect relative sectors provides a physically robust and operationally meaningful classification. Unlike the full fusion ring, which encodes all fusion compatibilities, or the symmetry group, which classifies under automorphisms, relative sectors encode indistinguishability in terms of quantum information content. This is crucial when defects are used as generalized symmetry generators, for interface engineering in statistical mechanics, or for classifying renormalization group (RG) flows in the presence of defects (Ghasemi, 29 Jan 2026).

Mathematically, the structure relies only on the modular $S$-matrix and basic properties of probability theory, ensuring universal applicability within RCFT. The concept can be extended to non-rational settings and to conformal (non-topological) defects, with conjectured applications in the study of duality walls, RG interfaces, and higher-form symmetry defects.

5. Comparison with Other Defect Classification Schemes

The defect relative sector classification is coarser than the decomposition into simple objects of the fusion category, but more refined than group-theoretical symmetry orbits alone. For example, certain symmetry-related defects (such as charge-conjugate pairs) may lie in the same defect relative sector if and only if their modular $S$-matrix rows are identical in absolute value.

This framework also relates structurally to the theory of defect groups and relative QFTs in higher dimensions, where analogous equivalence relations emerge from mutual nonlocality or screening in topological field theories (Gårding, 2023, Bhardwaj et al., 2021). In all cases, the organizing principle is the physical indistinguishability of defects with respect to a suitable set of observables.

6. Generalizations and Outlook

The defect relative sector concept has an information-theoretic origin and is intimately tied to the modular data, suggesting its extension to settings such as non-unitary CFTs, logarithmic CFTs, boundary conformal field theory, and higher-dimensional TQFTs. The universal KL-divergence structure provides a rigorous basis for new invariants and measures involving non-invertible symmetries, duality defects, and higher-categorical generalizations.

A plausible implication is that grouping defects via vanishing defect relative entropy could enable the systematic identification of emergent symmetries and dualities not manifest in the original fusion or symmetry data. The conjecture is that this approach may provide new computational and conceptual tools for classifying and constructing RG interfaces, analyzing entropic RG flows, and exploring the landscape of non-invertible operator algebras (Ghasemi, 29 Jan 2026).