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Vacuum-channel exchange operator on its invariant subspaces

Example

Let $C$ be a unitary spherical fusion category over $C$ , and let $a, b, c$ be simple objects, with $a$ self-dual and $\dim {Hom}_{C} (1, a \otimes a) = 1$ . For $n \geq 2$ , set $T_{0} = b$ , $T_{j + 1} = T_{j} \otimes a$ , and $V = {Hom}_{C} (c, T_{n})$ . Equip $V$ with the split Hom inner product, linear in its first variable, defined by $g^{†} f = ⟨ f, g ⟩ {id}_{c}$ . Choose orthonormal split vertex bases in the fixed and pair-first trees, retaining every multiplicity coordinate, and fix $1 \leq i < n$ . Let $P_{i} = P_{i, 1} = F_{i}^{- 1} Π_{1} F_{i}$ be the vacuum-channel projector: $F_{i}$ is the matrix of postcomposition by associators to the tree fusing sites $i, i + 1$ first, and $Π_{1}$ keeps precisely the coordinates with that pair output $1$ . This is an orthogonal projection for the stated inner product. Supply the normalized vacuum-channel self-duality datum with Frobenius–Schur sign $κ_{a} \in {+ 1, - 1}$ , positive dimension $d_{a}$ , and signed loop parameter $δ_{a} = κ_{a} d_{a}$ . Suppose $q \in C^{\times}$ satisfies $q + q^{- 1} = δ_{a}$ . For $U_{i} = δ_{a} P_{i}$ and $B_{i} = q I_{V} - U_{i}$ , $B_{i}$ acts as $- q^{- 1} I_{V}$ on $ran P_{i}$ and as $q I_{V}$ on $\ker P_{i}$ . If either subspace is zero, its listed scalar is not an eigenvalue contributed by that subspace. The equation gives an algebraic channel action only; it does not identify $B_{i}$ with an experimentally realized adiabatic exchange.

Example references

Local anyonic channel projector

Frobenius–Schur sign in a normalized anyonic vacuum channel

The vacuum F-coefficient records the Frobenius–Schur sign

Local channel projections are orthogonal idempotents

Vacuum-channel anyonic projectors satisfy Temperley–Lieb relations

Vacuum projectors obey the reverse adjacent Temperley–Lieb relation

Temperley–Lieb projectors give a Hecke braid representation

Diagrammatic convention

All spectator strands are omitted. The left diagram is the positive braid generator $B_{i}$ , with the right input strand passing over the left and time running upward. The right diagram expresses its operator on the path space as $q I_{V} - U_{i}$ : $I_{V}$ is the pair of through-strands and $U_{i}$ is the formal cup-cap Temperley–Lieb generator representing $δ_{a} P_{i}$ . The picture records this algebraic representation and the cup-cap orientation; it does not claim a physical particle follows a path or that a microscopic Hamiltonian realizes the crossing. The proof uses the range/kernel calculation, not the picture.

diagram failed to render:
\begin{tikzpicture}[baseline=(current bounding box.center),>=stealth,thick]
  \node at (0.7,3.5) {$B_i$};
  \draw (0,0) .. controls (0,1.1) and (1.4,1.9) .. (1.4,3);
  \draw[white,line width=5pt] (1.4,0) .. controls (1.4,1.1) and (0,1.9) .. (0,3);
  \draw (1.4,0) .. controls (1.4,1.1) and (0,1.9) .. (0,3);
  \node at (0, -0.35) {$a$};
  \node at (1.4, -0.35) {$a$};
  \node at (0, 3.3) {$a$};
  \node at (1.4, 3.3) {$a$};
  \draw[->] (2.1,0.2) -- (2.1,2.8) node[midway,right] {$\text{time}$};

  \node at (3.7,1.5) {$=$};
  \node at (4.5,1.5) {$q$};
  \draw (5.1,0) -- (5.1,3);
  \draw (6.1,0) -- (6.1,3);
  \fill (5.1,0) circle (1.3pt); \fill (6.1,0) circle (1.3pt);
  \fill (5.1,3) circle (1.3pt); \fill (6.1,3) circle (1.3pt);
  \node at (7.0,1.5) {$-$};
  \node at (9.1,1.5) {$U_i=\delta_aP_i$};
  \draw (8.2,3) .. controls (8.2,2.3) and (9.2,2.3) .. (9.2,3);
  \draw (8.2,0) .. controls (8.2,.7) and (9.2,.7) .. (9.2,0);
  \fill (8.2,3) circle (1.3pt); \fill (9.2,3) circle (1.3pt);
  \fill (8.2,0) circle (1.3pt); \fill (9.2,0) circle (1.3pt);
  \node at (8.7,4.0) {$\text{cup}$};
  \node at (8.7,-1.0) {$\text{cap}$};
\end{tikzpicture}

Facts & Assumptions

Given: The category, normalized duality datum, split Hom path space with its stated inner product and orthonormal split vertex bases, vacuum-channel projector $P_{i}$ , and $q \in C^{\times}$ with $q + q^{- 1} = δ_{a}$ from the Example.

[F1]

Local anyonic channel projector: the local channel projector is $P_{i, s} = F_{i}^{- 1} Π_{s} F_{i}$ , retaining every multiplicity coordinate.

[F2]

Local channel projections are orthogonal idempotents: on $Hom (c, T_{n})$ with $g^{†} f = ⟨ f, g ⟩ {id}_{c}$ and orthonormal split vertex bases in both trees, each $P_{i, s}$ is a self-adjoint idempotent.

[F3]

Frobenius–Schur sign in a normalized anyonic vacuum channel: the normalized maps are $coev = \sqrt{d_{a}} v$ , $ev = κ_{a} \sqrt{d_{a}} v^{†}$ , and $κ_{a} \in {+ 1, - 1}$ ; this sign is not a braiding phase.

[F4]

The vacuum F-coefficient records the Frobenius–Schur sign: for the stated vacuum gauge, $δ_{a} = κ_{a} d_{a}$ .

[F5]

Vacuum-channel anyonic projectors satisfy Temperley–Lieb relations: on the repeated- $a$ path space $U_{i} = δ_{a} P_{i}$ with $δ_{a} = κ_{a} d_{a}$ ; its Statement gives the square relation, $U_{i} U_{i + 1} U_{i} = U_{i}$ , and distant commutation.

[F6]

Vacuum projectors obey the reverse adjacent Temperley–Lieb relation: for $n \geq 3$ and $1 \leq i \leq n - 2$ , the reverse relation $U_{i + 1} U_{i} U_{i + 1} = U_{i + 1}$ holds on this fixed-boundary repeated-charge space with its supplied normalized duality datum.

[F7]

Temperley–Lieb projectors give a Hecke braid representation: if $U_{i}$ satisfy the full displayed relations and $δ_{a} = q + q^{- 1}$ , then $B_{i} = q I_{V} - U_{i}$ has the stated braid and Hecke properties.

Verification

1.1F1givenalgebra

By [F1], $P_{i}^{2} = P_{i}$ . If $v \in ran P_{i}$ , then $P_{i} v = v$ and $B_{i} v = (q I_{V} - δ_{a} P_{i}) v = (q - δ_{a}) v = - q^{- 1} v$ , using $q + q^{- 1} = δ_{a}$ .

2.1F2step 1.1algebra

If $v \in \ker P_{i}$ , then $P_{i} v = 0$ and $B_{i} v = q v$ . Every $x \in V$ splits as $P_{i} x + (I_{V} - P_{i}) x$ , with the second summand in $\ker P_{i}$ by idempotence. The range and kernel intersect trivially, and for $u = P_{i} y$ and $w \in \ker P_{i}$ , self-adjointness from [F2] gives $⟨ u, w ⟩ = ⟨ y, P_{i} w ⟩ = 0$ . Thus $V = ran P_{i} \oplus \ker P_{i}$ orthogonally, so the two computed scalar actions exhaust $B_{i}$ and a zero subspace contributes no eigenvalue.

3.1F3F4F5F6F7step 1.1step 2.1∎

The source relations [F5] and, when $n \geq 3$ , the local reverse relation [F6] supply all Temperley–Lieb hypotheses for [F7]; for $n = 2$ both adjacent relations have no indices. The signed $δ_{a} = κ_{a} d_{a}$ from [F3]–[F4] is retained. Therefore [F7] identifies $B_{i}$ as the generator of the resulting braid/Hecke representation, with the scalar actions in steps 1.1–2.1 on the respective nonzero channel subspaces. No physical transport or measurement premise is used.

Remarks

The sign $κ_{a}$ is retained throughout. A choice of $q$ satisfying $q + q^{- 1} = κ_{a} d_{a}$ is a stated hypothesis; no unitarity of $q$ or physical realization is inferred.

Depends on

Dependency tree · 2 levels

Sources

Joshua Chen, The Temperley-Lieb categories and skein modules, ANU Honours Thesis, 2014