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Gauge invariance and gluing in the finite symmetry model

Physical scope

The specified finite three-dimensional vertex-spin model with G = Z/2, trivial A, zero Postnikov cocycle, and unit background response; no continuum extension or empirical claim.

Statement

Let $K$ be a finite ordered oriented simplicial complex of dimension at most three in the relative extension of the selected model, with $V_{K}$ vertices and $E_{K}$ edges. For a closed physical regulator, $K$ is a finite ordered triangulation of a closed compact oriented smooth three-manifold. Use $G = Z / 2$ , the trivial group $A = {0}$ , the unique action $ρ$ , the zero normalized Postnikov cocycle $β$ , and the unit background response $W_{K} (x) = 1$ . The dynamical spin configurations for each background are $Φ_{x} = C^{0} (K; G)$ , with action $S_{x} = 0$ and normalized measure $μ_{x} (σ) = 2^{- V_{K}}$ . A background gauge arrow with vertex labels $g$ transports spins by $T_{g} (σ) = σ + g$ and transports observables by pullback.

For every background $x$ , its spin partition weight is $Z_{x} = 1$ . The normalized background-gauging amplitude is $Z_{K}^{(2)} = \sum_{[x] \in π_{0} B (K)} \frac{∣ π_{2} (B (K), x) ∣}{∣ π_{1} (B (K), x) ∣} = \frac{1}{2^{V_{K}}} ∣ Ob B (K) ∣ .$ For a subcomplex $L \subseteq K$ and fixed background $s$ on $L$ , the relative background amplitude is

Z_{K, L}^{(2)} (s) = 2^{- (V_{K} - V_{L})} ∣ {x \in Ob B (K) : x ∣_{L} = s} ∣ .

If $K = K_{1} \cup_{Σ} K_{2}$ is a union of ordered subcomplexes with compatible restrictions and opposite induced seam orientations, the relative boundary amplitudes are gauge-invariant and obey $Z_{K}^{(2)} = \sum_{[s] \in π_{0} B (Σ)} Z_{K_{2}, Σ}^{(2)} (s) Z_{K_{1}, Σ}^{(2)} (s) \frac{∣ π_{2} (B (Σ), s) ∣}{∣ π_{1} (B (Σ), s) ∣} .$ For a one-vertex complex with unit background weight, the absolute normalized background-gauging amplitude is $1 / 2$ . These conclusions concern only the specified finite model; they do not establish a continuum quantum field theory or empirical behavior.

Statement references

selected finite physical model

adopted finite-model assumptions

finite background-gauge measure and gluing theorem

Moore and Saxena, §23.3

Freed and Quinn, §2

Baez and Lauda, §8.3

Benini, Córdova, and Hsin, §2.3

Facts & Assumptions

Given: a finite ordered oriented complex $K$ in the selected model, its flat background cochain, model spin fields and normalized measure, a fixed subcomplex $L$ with background $s$ , and a decomposition $K = K_{1} \cup_{Σ} K_{2}$ as in the Statement.

[F1]

Finite two-group lattice spin state-sum model: for each background $x$ , $Φ_{x} = C^{0} (K; Z / 2)$ , $S_{x} = 0$ , $μ_{x} (σ) = 2^{- V_{K}}$ , and the per-background partition weight is $W_{K} (x) = 1$ .

[F2]

Finite two-group lattice spin state-sum model: a raw background gauge arrow with vertex labels $g$ maps spins by $T_{g} (σ) = σ + g$ and observables by pullback; relative boundary amplitudes use normalized spin measure and the finite background higher-gauge sum.

[A1]

Adopted finite two-group gauging model: the finite spin model, its unit background response, and its relative state-sum convention are adopted assumptions.

[F3]

Finite 2-group homotopy gauging and cochain gluing: for finite groups, normalized cocycle data, an invariant scalar weight, and a compatible ordered-subcomplex decomposition, the higher gauge sum equals the normalized field-object sum and glues using the seam factor $∣ π_{2} ∣ / ∣ π_{1} ∣$ .

Proof

Given: the selected spin model, finite backgrounds and gauge arrows, and the subcomplex decomposition.

1.1givenF1F2A1constructalgebra

For each background $x$ , the configuration set has $2^{V_{K}}$ spin assignments. Each has action $0$ and weight $2^{- V_{K}}$ , so $Z_{x} = \sum_{σ \in Φ_{x}} 2^{- V_{K}} = 1$ . A gauge arrow with labels $g$ acts by the bijection $σ \mapsto σ + g$ ; it preserves the uniform measure and transports every observable by pullback. Therefore reindexing each finite correlator sum gives gauge covariance with phase $1$ , and the induced scalar background weight is constant and lies in $U (1)$ .

2.1givenF1A1F3step 1.1algebra

The groups are finite, $ρ$ is the unique trivial action, $β = 0$ is a normalized cocycle, and $W_{K} = 1$ is invariant and factors as $1 \cdot 1$ on every decomposition. Thus the hypotheses of [F3] hold, and substitution of $∣ G ∣ = 2$ , $∣ A ∣ = 1$ gives $c (J) = 2^{- V_{J}}$ , the absolute formula and the relative formula in the Statement.

3.1givenF2F3step 2.1algebra

Applying the gluing conclusion of [F3] to the decomposition and the invariant unit weights gives the stated seam contraction. The relative convention in [F2] uses exact restriction to $Σ$ . Since $A$ is trivial, $∣ π_{2} ∣ = 1$ and the seam factor is the ordinary reciprocal automorphism order, exactly as displayed. The orientation has no effect because every action phase is $1$ .

4.1givenF1F3step 2.1algebra∎

On one vertex there are no edges, so there is one background object and $V = 1$ . The absolute formula in step 2.1 yields $Z_{K}^{(2)} = 1 / 2$ .

Gauge invariance and gluing in the finite symmetry model

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources