Noninvertible Symmetries, Fusion, and Junction Data

A system of topological defects defines a noninvertible symmetry only after fusion objects, junction vector spaces, associators, units, duals, and pivotal data satisfy their coherence equations. Integer fusion coefficients describe how channels decompose, but they do not specify how different decompositions are related. In the Ising category the duality line $\sigma$ is noninvertible because $\sigma\otimes\sigma\cong\mathbf1\oplus\psi$; its consistent action also depends on a nontrivial $F$-matrix.

Required background. Defects on Stratified Spacetimes supplies higher composition. Fusion Categories, Module Categories, and Bimodule Defects supplies rigidity and module data. Non-Invertible Topological Defects and Fusion supplies the physical operators. Helpful background. Fusion, Junctions, and Endpoints gives the physical junction interpretation, and Anomalies, RG Constraints, and Framework Limits explains why coherence is not the only possible obstruction.

Fusion objects and junction spaces

Let $\mathcal C$ be a fusion category with simple defect labels $a,b,c,\ldots$. A trivalent junction of incoming $a,b$ and outgoing $c$ lies in

$$V_{ab}^{c}=\operatorname{Hom}_{\mathcal C}(a\otimes b,c), \qquad N_{ab}^{\ c}=\dim V_{ab}^{c}.$$

Reassociating three defects gives an isomorphism

$$F^{abc}_{d}: \bigoplus_e V_{ab}^{e}\otimes V_{ec}^{d} \longrightarrow \bigoplus_f V_{bc}^{f}\otimes V_{af}^{d}.$$

For four defects, the five possible reassociation steps form a pentagon, and the two routes around it must give the same linear map. Units obey triangle identities. Rigidity supplies a dual $a^\vee$ with cap and cup junctions; a pivotal structure makes left and right duality compatible with planar isotopy. These data and equations are exactly what distinguish a fusion category from a fusion ring; Ostrik 2003, §§2.2–2.3, printed pp. 3–7 (PDF) gives the definitions.

An object is invertible when $a\otimes a^\vee\cong\mathbf1$. A simple object with quantum dimension $d_a>1$ cannot be invertible, since multiplicativity would give $d_a d_{a^\vee}=1$. Noninvertibility therefore has an immediate dimension diagnostic, although dimension alone does not determine the fusion or associator.

The Ising duality line

The Ising fusion category has simple objects $\mathbf1,\psi,\sigma$ and products

$$\psi\otimes\psi\cong\mathbf1, \qquad \psi\otimes\sigma\cong\sigma\otimes\psi\cong\sigma, \qquad \sigma\otimes\sigma\cong\mathbf1\oplus\psi.$$

The positive dimension solution is $d_{\mathbf1}=d_\psi=1$ and $d_\sigma=\sqrt2$. In a standard gauge, the only matrix-valued associator is

$$F^{\sigma\sigma\sigma}_{\sigma} =\frac1{\sqrt2} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix},$$

with rows and columns indexed by the intermediate channels $\mathbf1,\psi$. The matrix is unitary and squares to the identity. Together with the scalar $F$-symbols and compatible cap/cup maps, it solves the pentagon equations. Fröhlich, Fuchs, Runkel, and Schweigert derive Kramers–Wannier duality from the fusion of conformal defects and work out the Ising defect algebra in Fröhlich et al. 2004, “Defects in the critical Ising model” and “Duality defects,” printed pp. 2–5 (PDF).

The two terms in $\sigma\otimes\sigma$ have direct physical meaning: fusing two duality lines produces a superposition of the transparent line and the invertible spin-flip line. Thus no single defect can be the inverse of $\sigma$. The exact first application is returned to Interfaces, Folding, and Fusion, where the line acts on conformal correlators. The categorical statement does not by itself compute those correlators or establish an off-critical duality.

One can verify the dimension equation independently:

$$d_\sigma^2=d_{\mathbf1}+d_\psi=2.$$

The $F$-matrix also preserves the inner product on the two fusion channels. These checks are necessary, but a proposed category still requires every pentagon and triangle equation, not just this representative matrix.

Coherence failures

Start with nonnegative integers $N_{ab}^{\ c}$ that form an associative based ring. It may still be impossible to find invertible $F$-maps satisfying the pentagon. Even when solutions exist, inequivalent associators can categorify the same ring. Likewise, fusion coefficients can admit formal dual labels while no evaluation and coevaluation maps satisfy the snake identities.

The adversarial fixture chooses arbitrary phases in the Ising $F$-matrix so that one pentagon acquires a residual phase $\omega\ne1$. Fusion multiplicities and dimensions remain unchanged, but the two isotopic reductions of a four-junction network differ by $\omega$. The strongest surviving conclusion is a based fusion ring with chosen junction spaces. It is not a coherent category and therefore not a noninvertible symmetry action.

The matrix itself depends on bases in $V_{\sigma\sigma}^{\mathbf1}$ and $V_{\sigma\sigma}^{\psi}$. Rescaling those basis vectors conjugates $F$ by diagonal matrices and can move signs among several scalar $F$-symbols. Consequently an individual entry is not an invariant observable. Pentagon consistency, fusion multiplicities, quantum dimensions, and equivalence class of the category are invariant. A comparison that finds different $F$-matrices must first attempt the full gauge transformation before declaring two defect theories distinct.

Junction multiplicity matters equally. If $N_{ab}^{\ c}>1$, $F$ is a map between direct sums of higher-dimensional multiplicity spaces, not a scalar phase. Suppressing those indices can make a formally associative fusion ring look coherent while the actual pentagon fails. A reproducible specification therefore lists bases, duality maps, and every nontrivial multiplicity index alongside the fusion table.

Exercises

Explain why $\sigma$ cannot have an inverse even though it is self-dual.

Solution

Self-duality says $\sigma^\vee\cong\sigma$ and supplies cap and cup maps. Invertibility would require $\sigma\otimes\sigma^\vee\cong\mathbf1$. Instead $\sigma\otimes\sigma\cong\mathbf1\oplus\psi$, so an additional channel remains. Dualizability is weaker than invertibility.

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.