Categorical Symmetries, Higher Representations, and Charges

A categorical symmetry acts through a monoidal functor from its defect category to endofunctors of a category of states, boundaries, or operators. Module objects replace representation vectors, module functors replace equivariant linear maps, and natural transformations replace charged intertwiners. Selection rules are then statements that particular morphism spaces vanish. Eigenvalues of one fusion matrix are at most a decategorified diagnostic; they are not a complete set of generalized charges.

Required background. Fusion Categories, Module Categories, and Bimodule Defects supplies the action axioms. Actions, Generalized Charges, and Selection Rules supplies their physical interpretation. Helpful background. Noninvertible Symmetries, Fusion, and Junction Data supplies the $F$-symbols, Representations, Intertwiners, Invariants, and Tensor Decomposition gives the ordinary-group analogy, Multiplets, Invariants, and Selection Rules gives ordinary charge selection, and Generalized Symmetries and Information Diagnostics explains what symmetry-resolved information can and cannot recover.

Actions as higher representations

For a fusion category $\mathcal C$, an action on a category $\mathcal M$ is a strong monoidal functor

$$\rho:\mathcal C\longrightarrow\operatorname{End}(\mathcal M).$$

Thus $\rho(a)$ is an endofunctor, and coherent natural isomorphisms identify $\rho(a\otimes b)$ with $\rho(a)\circ\rho(b)$ and $\rho(\mathbf1)$ with the identity. Equivalently, $\mathcal M$ is a left $\mathcal C$-module category. Ostrik gives this definition and its coherence diagram in Ostrik 2003, Definition 6, printed pp. 5–6 (PDF).

If simple objects $m,n\in\mathcal M$ label boundary conditions, a topological defect $a\in\mathcal C$ can terminate so as to change $n$ into $m$ only when

$$\operatorname{Hom}_{\mathcal M}(m,a\triangleright n)\ne0.$$

The multiplicity

$$(N_a)_{mn}=\dim\operatorname{Hom}_{\mathcal M}(m,a\triangleright n)$$

is a categorical selection rule. Module associativity implies $N_aN_b=\sum_cN_{ab}^{\ c}N_c$. But the matrices remember only dimensions of morphism spaces. They forget the actual intertwiners, associators, extension classes, and phases in junction composition.

At the next categorical level, a morphism between two actions is a module functor $F:\mathcal M\to\mathcal N$ equipped with coherent maps $F(a\triangleright m)\cong a\triangleright F(m)$. A morphism between module functors is a module natural transformation. This is why a categorical charge is not generally one number: it consists of a sector object together with coherent responses to every symmetry defect and junction.

Boundary selection in the Ising category

Use the regular module category $\mathcal M=\mathcal C_{\mathrm{Ising}}$. Its simple boundary labels are $\mathbf1,\psi,\sigma$, and action is fusion. Bringing the duality line $\sigma$ to the boundary gives

$$\begin{aligned} \sigma\triangleright\mathbf1&=\sigma,\\ \sigma\triangleright\psi&=\sigma,\\ \sigma\triangleright\sigma&=\mathbf1\oplus\psi. \end{aligned}$$

Therefore a $\sigma$ endpoint permits transitions $\mathbf1\to\sigma$, $\psi\to\sigma$, and $\sigma\to\mathbf1$ or $\psi$. It forbids $\mathbf1\to\mathbf1$ and $\psi\to\psi$ at that junction. In the ordered basis $(\mathbf1,\psi,\sigma)$,

$$N_\sigma= \begin{pmatrix} 0&0&1\\ 0&0&1\\ 1&1&0 \end{pmatrix}, \qquad N_\sigma^2=I+N_\psi.$$

This matrix equation checks the fusion rule $\sigma^2=\mathbf1+\psi$. The exact first application belongs to Defect Representations, Transverse Spin, and Tensor Structures: there the allowed boundary-changing operators also carry conformal and transverse-spin quantum numbers. The module category supplies the topological selection rule; it does not determine scaling dimensions or OPE coefficients.

Why spectra are incomplete charges

Diagonalizing $N_\sigma$ gives eigenvalues $\sqrt2,-\sqrt2,0$. These numbers are useful characters of the based module, but they do not reconstruct the action. A simultaneous change of basis can preserve all fusion-matrix spectra while changing which simple boundary object corresponds to a physical condition. More seriously, inequivalent module associators can act on the same matrices. Their junction amplitudes differ even though every eigenvalue agrees.

The adversarial test compares two module actions with identical $N_a$ but inequivalent coherence maps. Calling the common spectra “the complete charges” predicts that they are the same representation. A junction experiment sensitive to the associator distinguishes them. The strongest valid conclusion is equality of decategorified fusion multiplicities, not equivalence of categorical actions.

For nonsemisimple categories the loss is larger: matrices on the Grothendieck group also erase extensions and nilpotent endomorphisms. A complete charge analysis must specify the module category, action functor, natural transformations, and the physical realization map.

Ordinary character theory is recovered in a limiting case. If $\mathcal C=\operatorname{Rep}(G)$ and $\mathcal M=\mathrm{Vect}$ is acted on by the fiber functor, natural automorphisms compatible with tensor product reconstruct group elements under the usual Tannakian hypotheses. For a general fusion category there need be no fiber functor to vector spaces, and the relevant “representation space” is the category $\mathcal M$ itself. This is the precise reason categorical charges can label boundary conditions and functors rather than eigenvectors.

There is a practical stop rule. First verify the matrix fusion relations, then verify the module pentagon for the action associators, and finally check that the physical endpoint operator realizes each abstract morphism with the required locality and grading. Failure at the first stage rejects the based action. Failure at the second leaves only multiplicity data. Failure at the third leaves a valid abstract module category but no established QFT realization. These three conclusions should never be merged.

Exercises

Use the Ising rules to verify $N_\psi N_\sigma=N_\sigma$.

Solution

$N_\psi$ exchanges the $\mathbf1$ and $\psi$ rows and columns while fixing $\sigma$. Because the first two rows and columns of $N_\sigma$ are symmetric under that exchange, multiplication leaves $N_\sigma$ unchanged, matching $\psi\otimes\sigma\cong\sigma$.

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.