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The Ising fusion object is simple and non-invertible

Example

A zero object is an object that is both initial and terminal. A monomorphism $u : Y \to X$ is a morphism such that, for every object $Z$ and morphisms $f, g : Z \to Y$ , $u \circ f = u \circ g$ implies $f = g$ . A subobject of $X$ is an isomorphism class of monomorphisms into $X$ , and $X$ is simple if it is nonzero and has no subobjects other than the zero subobject and $X$ itself. Let $I$ have as objects finite direct sums of the labels $1, ψ, σ$ . Its bilinear tensor product on labels is

1 \otimes a = a \otimes 1 = a, ψ \otimes ψ = 1, ψ \otimes σ = σ \otimes ψ = σ, σ \otimes σ = 1 \oplus ψ .

Morphisms are block-diagonal across the three labels, with an arbitrary finite complex matrix on each equal-label block and zero on unequal-label blocks; composition is blockwise matrix multiplication. Each label is simple by the stated criterion. The tensor product is bilinear, the unit is $1$ , and the unitors are strict. Choose one basis vector in each allowed one-dimensional fusion space. An object $X$ is tensor-invertible if some object $E$ satisfies both $X \otimes E ≅ 1$ and $E \otimes X ≅ 1$ ; otherwise it is non-invertible. A right tensor inverse of $σ$ means an object $E$ with $σ \otimes E ≅ 1$ . Encode intermediate labels by $ϵ (1) = 0$ and $ϵ (ψ) = 1$ . The associator $α_{a, b, c} : (a \otimes b) \otimes c \to a \otimes (b \otimes c)$ has coefficients $[F_{σ}^{σ σ σ}]_{a b} = 2^{- 1 / 2} (- 1)^{ϵ (a) ϵ (b)}$ for $a, b \in {1, ψ}$ , $[F_{σ}^{ψ σ ψ}]_{σ σ} = - 1$ , and $[F_{ψ}^{σ ψ σ}]_{σ σ} = - 1$ ; every other coefficient is $1$ when all required fusion channels are allowed and $0$ otherwise. Since $σ$ has no right tensor inverse, it has no two-sided tensor inverse and is non-invertible. This is an abstract algebraic example and makes no physical-realization claim.

Example references

Simon, Topological Quantum, §§9.2–9.3

Facts & Assumptions

Given: Objects of $I$ are finite direct sums of its three simple objects, so any object $E$ has multiplicities $n_{1}, n_{ψ}, n_{σ} \in Z_{\geq 0}$ .

[F1]

Finite semisimple fusion data for non-invertible defects: an object is tensor-invertible only if it has a two-sided tensor inverse.

[F2]

Skeletal Ising fusion-data package: $σ \otimes σ ≅ 1 \oplus ψ$ and $σ \otimes ψ ≅ σ$ .

[F3]

Coherence of the Ising fusion data: these fusion rules and associators define a finite semisimple fusion category, with $1, ψ, σ$ simple.

Verification

1.1givenF2algebra

Write an arbitrary object as $E = n_{1} 1 \oplus n_{ψ} ψ \oplus n_{σ} σ$ . Distributivity and [F2] give $σ \otimes E ≅ n_{σ} 1 \oplus n_{σ} ψ \oplus (n_{1} + n_{ψ}) σ$ . If $σ \otimes E ≅ 1$ , equality of simple-object multiplicities would require both $n_{σ} = 1$ and $n_{σ} = 0$ from the $1$ and $ψ$ coefficients. This is impossible.

2.1F1F3step 1.1∎

Thus $σ$ has no right tensor inverse and cannot have a two-sided tensor inverse by [F1]. By [F3], $σ$ is a simple object of $I$ ; it is therefore simple and non-invertible. The argument is purely categorical.

Depends on

Dependency tree · 2 levels

Sources

Steven H. Simon, Topological Quantum, §§9.2–9.3