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Finite 2-group background coupling variation

Physical scope

Conditional finite change-of-variables result for Euclidean lattice models with finite 2-group background data on a closed oriented triangulated spacetime. It assigns no phase to an unspecified theory and proves no continuum limit.

Statement

Let $K$ be a finite ordered triangulation of a closed compact oriented smooth $d$ -manifold, $d \geq 2$ , with its chosen tangential structure and finite 2-group background gauge $2$ -groupoid as specified in the finite realization definition. For each background object $x$ , let the selected Euclidean finite-regulator model have a finite configuration set $Φ_{x}$ , dimensionless action $S_{x} : Φ_{x} \to R$ , measure weight $μ_{x} : Φ_{x} \to C$ , and total weight $W_{x} (ϕ) : = μ_{x} (ϕ) e^{- S_{x} (ϕ)}$ , in natural units $c = ℏ = 1$ . For an ordered tuple of $n \geq 0$ local observable insertions $O_{1}, \dots, O_{n}$ , let $o_{j, x} : Φ_{x} \to C$ be the model's insertion function and define the unnormalized correlator

C_{x} (O_{1}, \dots, O_{n}) : = \sum_{ϕ \in Φ_{x}} W_{x} (ϕ) \prod_{j = 1}^{n} o_{j, x} (ϕ),

where the empty product gives the partition function $Z_{x} = C_{x} ()$ . Suppose every $1$ -gauge arrow $f : x \to y$ has a model-specified bijection $T_{f} : Φ_{x} \to Φ_{y}$ , the observable transport satisfies $o_{j, y} (T_{f} ϕ) = o_{j, x} (ϕ)$ , and the adopted action-and-measure law is

W_{y} (T_{f} ϕ) = χ_{f} (x) W_{x} (ϕ),

with a model-specified phase $χ_{f} (x) \in U (1)$ independent of $ϕ$ . Assume these phases are $1$ on identity arrows, multiply under composition, and agree on $2$ -isomorphic arrows. Then for every such arrow and observable tuple,

C_{y} (α_{f} O_{1}, \dots, α_{f} O_{n}) = χ_{f} (x) C_{x} (O_{1}, \dots, O_{n}), Z_{y} = χ_{f} (x) Z_{x} .

If an allowed local background counterterm $L_{x}$ reweights the model by $e^{i L_{x}}$ , the phase changes to $χ_{f}^{'} (x) = e^{i L_{y}} χ_{f} (x) e^{- i L_{x}}$ . By the stated anomaly convention, the residual anomaly is absent exactly when one permitted counterterm makes every phase $1$ . The Postnikov cocycle $β$ fixes the allowed background cochains and gauge arrows but does not, by itself, specify $χ_{f}$ ; that phase is determined by the selected model's action and measure. The result is confined to the stated finite regulator and does not assert a continuum QFT or a universal anomaly classification.

Statement references

Benini, Córdova, and Hsin, §2.3 and §3 introduction

Schäfer-Nameki, §§2.8.1–2.8.2

Facts & Assumptions

Given: finite background objects and gauge arrows with the fixed left-action, normalized-cocycle, local-coefficient and associator conventions; the selected finite-regulator model, its finite configuration sets, weights, observable insertions, bijections, and coherent phase law stated above; and its declared set of allowed local background counterterms.

[A1]

Adopted model law: for each background $x$ the configuration set $Φ_{x}$ is finite; for each $1$ -arrow $f : x \to y$ , $T_{f}$ is a bijection, the observables satisfy $o_{j, y} (T_{f} ϕ) = o_{j, x} (ϕ)$ , and $W_{y} (T_{f} ϕ) = χ_{f} (x) W_{x} (ϕ)$ with $χ_{f} (x)$ independent of $ϕ$ . These are inputs from the selected model, not deductions from $β$ .

[F1]

Finite 2-group symmetry realization and background coupling: each gauge $1$ -arrow is assigned a configuration bijection and observable-algebra isomorphism transporting weights by a configuration-independent nonzero scalar; [A1] specifies the coherent phase-valued case; each $2$ -arrow $η : f \Rightarrow f^{'}$ is assigned a target-fiber pair $(J_{η}, j_{η})$ with $T_{f^{'}} = J_{η} \circ T_{f}$ , $α_{f^{'}} = j_{η} \circ α_{f}$ , and weight preservation. The comparison $2$ -morphisms and unit maps obey the stated pseudofunctor coherence equations. The residual phase is defined modulo the selected model's declared local counterterms.

[F2]

Finite 2-group background cochains: $G$ is finite, $A$ finite abelian, $ρ$ is a left action, $β$ is a normalized $3$ -cocycle with associator directed from $(g h) k$ to $g (h k)$ , and backgrounds satisfy $δ_{a} B = a^{*} β$ with the stated local-coefficient signs.

[F3]

Finite 2-group background gauge consistency: under those finite cochain hypotheses, the stated $(g, λ)$ transformation preserves flatness and the twisted tetrahedron equation. Its complete current proof and current-ready Step 1 record were inspected.

[F4]

Finite background gauge 2-groupoid: backgrounds are objects; raw vertex/edge labels $(g, λ)$ are $1$ -arrows; vertex cochains $t$ with $λ^{'} - λ = δ_{b} t$ are $2$ -arrows; composition uses the displayed $τ$ and associator uses $σ_{i} = β (k_{i}, h_{i}, g_{i})$ .

[F5]

Coherence and stabilizers of finite background gauge 2-groupoids: for every finite ordered complex and the displayed $(G, A, ρ, β)$ data, these operations form a finite $2$ -groupoid with the stated coherence and stabilizers. The result retains raw $1$ -gauge labels and vertex $2$ -gauge cochains.

Proof

1.1givenA1F1constructalgebra

Fix a gauge arrow $f : x \to y$ . Reindexing the finite sum over $Φ_{y}$ by the bijection $T_{f}$ gives $C_{y} (α_{f} O_{1}, \dots, α_{f} O_{n}) = \sum_{ϕ \in Φ_{x}} W_{y} (T_{f} ϕ) \prod_{j = 1}^{n} o_{j, y} (T_{f} ϕ)$ ; every target configuration appears exactly once because $T_{f}$ is bijective.

2.1A1step 1.1algebra

Substitute the adopted weight law and covariance of each insertion from [A1]: each summand equals $χ_{f} (x) W_{x} (ϕ) \prod_{j} o_{j, x} (ϕ)$ . The phase is independent of $ϕ$ , so it factors out of the finite sum and yields $C_{y} (α_{f} O_{1}, \dots, α_{f} O_{n}) = χ_{f} (x) C_{x} (O_{1}, \dots, O_{n})$ . Taking $n = 0$ gives $Z_{y} = χ_{f} (x) Z_{x}$ .

3.1A1F1F2F3F4F5step 2.1algebra∎

Reweighting by an allowed counterterm multiplies the source and target weights by $e^{i L_{x}}$ and $e^{i L_{y}}$ , respectively; applying step 2.1 to the reweighted sum therefore gives $χ_{f}^{'} (x) = e^{i L_{y}} χ_{f} (x) e^{- i L_{x}}$ . By the anomaly definition in [F1], the residual class is absent exactly when an allowed $L$ makes these phases all $1$ . The mathematical suppliers [F2]–[F5] specify which backgrounds and coherent gauge arrows are in this calculation, while [A1] separately supplies the action-and-measure phase; hence no anomaly value for another model follows from $β$ alone.

Finite 2-group background coupling variation

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources