Non-Invertible Symmetry Lab

From Classical Group Actions to Fusion Categories on a 1+1D Spacetime Cylinder

Framework: Non-Invertible

1+1D Spacetime Cylinder & Defect Lines Space $S^1$ × Time $\mathbb{R}$

Insert Operator:
Direct Sum Splitting: Fusing Kramers-Wannier defect lines D × D branches into 1 + ψ. No inverse exists in group theory!
Topological Invariance & Defect Surgery Canonical Junction

Defect lines are topological: continuous deformations through spacetime do not alter quantum correlators. Splitting defects satisfy pentagon coherence equations and quantum bubble popping: $\langle D \text{ loop} \rangle = d_D = \sqrt{2}$.

Fusion Algebra & Proof Engine

D × D = 1 + ψ (Multi-channel Branching)
Defect Dim $d_a$
1.4142
Total Dim $\mathcal{D}$
2.0000
Algebraic Inverse
None (∄)

Fusion Multiplicity Matrix $N_a$

Why is this NOT an Oxymoron? In abstract algebra, a group demands $g \cdot g^{-1} = e$. But in quantum field theory, symmetries act on states via topological defect operators. When Kramers-Wannier duality maps order $\sigma$ to disorder $\mu$, fusing two defects maps $\sigma \to \mu \to \sigma$, returning a projector $(1 + \psi)$ rather than a pure isomorphism. Hence, symmetry is generalized from Group $G$ to a Fusion Category $\mathcal{C}$!

The Quora Question Reconciled

"Is a non-invertible symmetry an oxymoron in mathematics?"
Under the 19th-century definition of symmetry (Felix Klein's Erlangen program & group actions), yes: by definition, every group element has a unique inverse.

However, modern mathematics and quantum physics generalized this: symmetries are characterized by conservation laws and Ward identities arising from topological operators. When operators are allowed to sum into direct sums (vector spaces of defect lines), they form a semisimple tensor category (fusion category) where defects like $D$ lack an inverse yet generate exact selection rules.

Kramers-Wannier Duality ($d_D = \sqrt{2}$)

At the 2D critical Ising temperature, the high-temperature and low-temperature expansions are related by Kramers-Wannier duality. Inserting the defect line $D$ along the spatial circle enacts this duality.

Since $D \otimes D = 1 \oplus \psi$, the Perron-Frobenius eigenvalue of its adjacency matrix satisfies $d_D^2 = 1 + 1 = 2 \implies d_D = \sqrt{2}$. Because the quantum dimension is irrational, $D$ cannot be a conventional unitary symmetry (which must have integer dimension 1).