Generalized-Symmetry Sectors, Selection Rules, and Reconstruction

Generalized-symmetry sectors are reconstructed from a categorical action by decomposing the relevant state, boundary, or operator category into module sectors and reading allowed transitions from morphism spaces. This yields selection rules once the action is known. The inverse problem is not unique: one defect category can have several inequivalent module-category realizations, and none of them by itself determines the local QFT, its scaling data, or its positive representation.

Required background. Categorical Symmetries, Higher Representations, and Charges supplies module actions. Boundary Conditions and Interfaces in Functorial Field Theory supplies boundary sectors. Superselection Sectors and DHR Reconstruction supplies the stronger AQFT reconstruction theorem and its locality hypotheses. Helpful background. Superselection Rules and Accessible Entanglement gives the operational consequence of inaccessible coherences, while Actions, Generalized Charges, and Selection Rules owns the physical selection rules.

Sector decompositions from an action

Let $\rho:\mathcal C\to\operatorname{End}(\mathcal M)$ be an action of a fusion category on a finite semisimple category. Choose simple sector representatives $m_i$. The integers

$$(N_a)_{ij} =\dim\operatorname{Hom}_{\mathcal M}(m_i,a\triangleright m_j)$$

say which sector transitions a defect $a$ permits. Coherence gives

$$N_aN_b=\sum_c N_{ab}^{\ c}N_c.$$

For an ordinary finite abelian group, every $N_g$ is invertible and simultaneous character projectors can be written by Fourier transform. For a noninvertible category, $N_a$ may be noncommuting or singular. Sector projectors then belong naturally to an enlarged algebra such as the tube algebra, whose multiplication retains junction data. Diagonalizing one $N_a$ is not a categorical Fourier transform.

Selection rules are vanishing statements. A boundary-changing operator from $m_j$ to $m_i$ carrying endpoint label $a$ can exist only when the displayed hom-space is nonzero. Fusion of two such operators is controlled by both the $N_{ab}^{\ c}$ and the action associator. Thus multiplicity matrices decide which channels are allowed, while the higher maps decide how allowed channels compose.

Ostrik proves that module categories over a semisimple monoidal category can be described by algebra objects and develops their Morita theory in Ostrik 2003, §§2.3 and 3, printed pp. 5–14 (PDF). That result reconstructs categorical module data under its hypotheses; it does not reconstruct a spacetime net or a Hamiltonian.

Ising boundary transitions

Take the regular Ising module with boundary sectors $\mathbf1,\psi,\sigma$. The duality line has action matrix

$$N_\sigma= \begin{pmatrix} 0&0&1\\ 0&0&1\\ 1&1&0 \end{pmatrix}.$$

Hence a $\sigma$ endpoint connects $\mathbf1$ or $\psi$ to $\sigma$, and connects $\sigma$ to either $\mathbf1$ or $\psi$. It cannot connect $\mathbf1$ directly to $\psi$ or preserve either one. The invertible line $\psi$ exchanges $\mathbf1$ and $\psi$ and fixes $\sigma$. Combining the rules gives

$$N_\sigma^2=I+N_\psi,$$

so two duality endpoints may fuse through a transparent or $\psi$ channel. This reconstructs the allowed junction transitions from the categorical action.

The exact physical application belongs to Defect Representations, Transverse Spin, and Tensor Structures. A conformal boundary operator must also satisfy transverse-spin and conformal-selection rules; the topological module action does not fix its dimension or OPE coefficient. Fröhlich and collaborators show how Ising duality defects act on order and disorder correlators in Fröhlich et al. 2004, printed pp. 2–5 (PDF), providing a concrete realization beyond the fusion matrices.

Reconstruction functors and their kernels

Write $\mathfrak R$ for a realization procedure sending a QFT with chosen topological defects to a categorical action $(\mathcal C,\mathcal M,\rho)$. Three distinct questions must be asked:

1. Faithfulness: can different QFT maps become the same natural transformation?
2. Fullness: does every categorical module functor come from a physical interface?
3. Essential surjectivity: is every abstract module category realized by a local QFT?

None is automatic. Positivity, locality, energy bounds, and analytic sewing can remove abstract categorical realizations. Conversely, two QFTs with different local observables can share the same topological sector.

DHR reconstruction is stronger because it begins with a Haag–Kastler net, transportable localized endomorphisms, permutation statistics in the appropriate dimension, and a conjugate theory. Those hypotheses allow recovery of a compact gauge group and field net. An arbitrary defect fusion category lacks this spacetime-local input, so the DHR conclusion cannot simply be transplanted.

The adversarial claim reconstructs a unique QFT from the Ising fusion category alone. At minimum, the regular module and other module categories give inequivalent boundary realizations; moreover, different local theories can contain equivalent topological sectors. The strongest surviving result is the set of categorical selection rules for the specified action. Uniqueness requires a separately proved fully faithful and essentially surjective realization theorem.

Sector completeness is a separate hypothesis. Even if all simple objects of $\mathcal M$ have been found, the chosen defect category may omit non-topological interfaces or sectors visible only in another superselection criterion. Conversely, an abstract simple module object may fail to correspond to a finite-energy boundary condition. A reconstruction statement must therefore name both the category being completed and the physical criterion selecting its realizable objects.

Exercises

Which Ising sector transitions are allowed by two successive $\sigma$ endpoints?

Solution

Because $N_\sigma^2=I+N_\psi$, two endpoints either preserve a sector through the transparent channel or act by $\psi$. Thus $\mathbf1$ may end as $\mathbf1$ or $\psi$, $\psi$ may end as $\psi$ or $\mathbf1$, and $\sigma$ returns to $\sigma$ with multiplicity two, one from each intermediate fusion channel.

References

• Fröhlich, Jürg, Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert. “Kramers–Wannier Duality from Conformal Defects.” Physical Review Letters 93 (2004): 070601. DOI; Open PDF.
• Ostrik, Victor. “Module Categories, Weak Hopf Algebras and Modular Invariants.” Transformation Groups 8 (2003): 177–206. DOI; Open PDF.