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Nonzero Postnikov class with unit background response

Physical scope

One explicit finite background on a closed triangulated 3-sphere and its unit-weight response for a finite 2-group cochain model; no continuum QFT or empirical claim is asserted.

Example

Let $K = \partial Δ^{4}$ be the ordered boundary complex of a $4$ -simplex with vertices $0 < 1 < 2 < 3 < 4$ , a finite triangulation of the closed oriented $3$ -sphere. Let $G = A = Z_{2}$ under addition with trivial action, and define $β (g, h, k) = g h k$ modulo $2$ . On vertices set $q = (0, 1, 0, 1, 0)$ , put $a_{i j} = q_{j} - q_{i}$ , and define $B_{i j k} = q_{i} a_{i j} a_{j k} + c_{j k} - c_{i k} + c_{i j}$ modulo $2$ , where the edge cochain $c$ has $c_{01} = c_{03} = c_{23} = 1$ and all other edge labels zero. Then $β$ is a normalized $3$ -cocycle and is not the coboundary of any normalized $2$ -cochain, so its Postnikov class is nonzero under the cohomology convention of the background definition. The edge cochain is flat, and $δ_{a} B = a^{*} β$ on every tetrahedron. On tetrahedron $0123$ , $B_{012} = 1$ and $B_{013} = B_{023} = B_{123} = 0$ , so $δ_{a} B = 1$ . For the $0$ -form gauge labels $g_{0} = 1$ , $g_{1} = g_{2} = g_{3} = g_{4} = 0$ and all $1$ -form labels zero, the transformed edges on $0123$ are $a_{01}^{'} = 0$ , $a_{12}^{'} = a_{23}^{'} = a_{02}^{'} = 1$ , and $a_{13}^{'} = a_{03}^{'} = 0$ ; all four transformed face labels on that tetrahedron vanish. In the finite unit-response model, the partition weight remains $1$ and the gauge phase is trivial despite the nonzero Postnikov class. This is a finite response example, not a continuum QFT realization.

Example references

Benini, Córdova, and Hsin, §2.3 and §1.3

Schäfer-Nameki, §§2.8.1–2.8.2

Facts & Assumptions

Given: $K = \partial Δ^{4}$ with its ordered simplices, $G = A = Z_{2}$ with trivial action, $β (g, h, k) = g h k$ , vertex potential $q = (0, 1, 0, 1, 0)$ , the stated edge cochain $c$ , the resulting face cochain $B$ , the gauge parameters, and the finite unit-response model.

[A1]

Adopted finite model data: every allowed background has one configuration with measure $1$ and action $0$ , and every gauge arrow has the unique configuration bijection between singleton fibers; hence the model weight is $1$ on every background. Each $2$ -arrow intertwiner and each composition or unit comparison is the unique identity map on its singleton target fiber and constant observable algebra.

[F1]

Finite 2-group background cochains: a normalized group $3$ -cocycle is identified modulo coboundaries of normalized $2$ -cochains; the background equation on this tetrahedron is $B_{123} - B_{023} + B_{013} - B_{012} = β (a_{01}, a_{12}, a_{23})$ for the trivial action.

[F2]

Finite 2-group background gauge consistency: the prescribed $(g, λ)$ formula transforms admissible backgrounds to admissible backgrounds. Its complete current proof and current-ready Step 1 record were inspected.

[F3]

Finite background gauge 2-groupoid: vertex labels $g_{i}$ and edge labels $λ_{i j}$ define raw gauge $1$ -arrows, with the displayed $β$ -correction to $B$ .

[F4]

Finite 2-group symmetry realization and background coupling: the finite model response and its action/measure are independent inputs on the allowed background groupoid; unit weight is an admissible selected response.

[F5]

Finite 2-group background coupling variation: under a bijective gauge-arrow action and the specified phase law, the partition function changes by exactly that phase. Its hypotheses are the specified finite gauge bijections and coherent model phase law.

Verification

1.1givenF1algebra

The cochain $β (g, h, k) = g h k$ vanishes whenever any input is $0$ , so it is normalized. For the trivial action, its four-variable coboundary is $h k l - (g + h) k l + g (h + k) l - g h (k + l) + g h k$ ; in $Z_{2}$ minus equals plus, and expansion makes each of $h k l$ , $g k l$ , $g h l$ , and $g h k$ occur twice, so $δ β = 0$ . Any normalized $2$ -cochain $c$ on $Z_{2}$ can be nonzero only at $(1, 1)$ , and $(δ c) (1, 1, 1) = c (1, 1) - c (0, 1) + c (1, 0) - c (1, 1) = 0$ , whereas $β (1, 1, 1) = 1$ ; thus $β$ is not a coboundary and its class is nonzero.

2.1givenF1step 1.1algebra

For any tetrahedron $i < j < k < l$ , set $u = a_{i j}$ , $v = a_{j k}$ , and $w = a_{k l}$ . Flatness from the vertex potential gives $a_{i k} = u + v$ , $a_{j l} = v + w$ , and $q_{j} = q_{i} + u$ . For $b_{i j k} = q_{i} a_{i j} a_{j k}$ , expansion in $Z_{2}$ gives $(δ b)_{i j k l} = (q_{i} + u) v w + q_{i} (u + v) w + q_{i} u (v + w) + q_{i} u v = u v w$ , since every term containing $q_{i}$ occurs twice. Expanding $(δ^{2} c)_{i j k l} = (δ c)_{j k l} - (δ c)_{i k l} + (δ c)_{i j l} - (δ c)_{i j k}$ with $(δ c)_{i j k} = c_{j k} - c_{i k} + c_{i j}$ shows each edge value occurs twice, so $δ^{2} c = 0$ . Thus $δ_{a} B = δ b + δ^{2} c = a_{i j} a_{j k} a_{k l} = β (a_{i j}, a_{j k}, a_{k l})$ on every tetrahedron. On $0123$ , direct substitution gives $B_{012} = 1$ and $B_{013} = B_{023} = B_{123} = 0$ , hence the displayed tetrahedron value is $1$ . Therefore $(a, B)$ is an allowed background on the closed triangulation.

3.1F1F2F3step 2.1algebra

With $g = (1, 0, 0, 0, 0)$ , the formula $a_{i j}^{'} = g_{i} + a_{i j} - g_{j}$ gives the listed edge labels on $0123$ . On its face $012$ , $ζ_{012}^{g} = - β (1, 1, 1) = 1$ in $Z_{2}$ ; on faces $013$ , $023$ , and $123$ , every term in $ζ^{g}$ has a zero argument, so those corrections vanish. With $λ = 0$ and the original local $B$ values, this gives $B_{012}^{'} = 1 + 1 = 0$ and $B_{013}^{'} = B_{023}^{'} = B_{123}^{'} = 0$ . Hence $δ_{a^{'}} B^{'} = 0 = β (a_{01}^{'}, a_{12}^{'}, a_{23}^{'}) = β (0, 1, 1)$ on $0123$ . The theorem [F2] ensures the gauge transform also satisfies the equation on the other tetrahedra.

4.1A1F4F5step 3.1algebra∎

By [A1], both configurations have weight $1$ and the gauge bijection fixes the unique configuration, so the partition functions agree with phase $1$ . The coupling theorem [F5] yields the same relation. Thus this selected response has no anomaly phase although its background Postnikov class is nonzero; this conclusion uses the separate unit-response model input, not $β$ alone.

Remarks

The singleton unit-response assignment here specifies a background-response functor and its phase only. It does not construct an effective nontrivial defect action on dynamical observables or claim an effective realization of the full selected physical symmetry. The cochain and response computations remain meaningful without that additional realization data.

The diagram shows tetrahedron $0123$ , one of the five 3-simplices in $K$ . Arrows follow the increasing vertex order, and each $B_{i j k}$ label belongs to the corresponding oriented face. It records incidence only; the equations and calculations above establish the background and gauge-transform claims.

Nonzero Postnikov class with unit background response

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources