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Finite gauge theory gluing is weighted push-pull

Statement

Fix an integer $d \geq 1$ and a finite group $G$ . The state spaces and linear maps defined for compatible collared triangulated $d$ -bordisms compose under gluing: if $M_{1} : Σ_{0} \to Σ_{1}$ and $M_{2} : Σ_{1} \to Σ_{2}$ , then

$Z_{M_{2} \cup_{Σ_{1}} M_{1}} = Z_{M_{2}} \circ Z_{M_{1}} .$

The cylinder on $Σ$ acts as the identity on $V_{G} (Σ)$ . Disjoint unions act by tensor products of state spaces and maps. Let $c_{Σ} : V_{G} (Σ) \to V_{G} (- Σ)$ be the anti-linear map $c_{Σ} (f) = \overline{f}$ under the common underlying groupoid of the orientation-reversed boundary. With the weighted inner products from the Definition, if $\overline{M} : - Σ_{1} \to - Σ_{0}$ denotes $M : Σ_{0} \to Σ_{1}$ with reversed orientation, then

$Z_{\overline{M}} \circ c_{Σ_{1}} = c_{Σ_{0}} \circ Z_{M}^{†},$

where $Z_{M}^{†}$ is the adjoint of $Z_{M}$ ; equivalently, after the pointwise-conjugation identifications, orientation reversal gives the weighted adjoint. For a connected closed $d$ -manifold $M$ , its scalar amplitude is $∣ Hom (π_{1} (M), G) ∣ / ∣ G ∣$ .

Statement references

Finite gauge state spaces and bordism spans

Finite groupoid push-pull satisfies Fubini and adjunction

Finite BF partition function is a topological invariant

Homotopy pullback of finite groupoids

Facts & Assumptions

Given: A fixed dimension $d \geq 1$ , finite group $G$ , compatible boundary triangulations, and composable collared triangulated bordisms.

[F1]

Finite gauge state spaces and bordism spans: $V_{G} (Σ)$ consists of invariant functions on boundary flat-connection isomorphism classes; $Z_{M} = (r_{1})_{!} r_{0}^{*}$ for the restriction span; the pairing weights each class by $1 / ∣ Aut ∣$ .

[F2]

Finite groupoid push-pull satisfies Fubini and adjunction: push-pull maps compose through homotopy-pullback spans, and transposing a span gives the adjoint for the weighted inner product.

[F3]

Finite BF partition function is a topological invariant: for a finite CW complex with connected components $K_{j}$ and chosen base vertices $v_{j}$ , $C_{G} (K)$ is equivalent to $\prod_{j} (Hom (π_{1} (K_{j}, v_{j}), G) / / G)$ ; restriction to a subcomplex corresponds to componentwise precomposition by its base-path-adjusted induced fundamental-group maps, up to the stated conjugating natural isomorphism. For connected $K$ , its cardinality is $∣ Hom (π_{1} (K, v), G) ∣ / ∣ G ∣$ .

[F4]

Flat-connection groupoids satisfy boundary descent: flat-connection groupoids on glued bordisms are equivalent to the homotopy pullback of the two restriction functors, with a boundary isomorphism included.

[F5]

Homotopy pullback of finite groupoids: the homotopy pullback has objects $(c, e, α : p (c) ≅ q (e))$ and arrows satisfying $q (v) \circ α = α^{'} \circ p (u)$ .

[F6]

Induced fundamental-group maps are well defined, functorial and invariant under based homotopy: a pointed continuous map induces a well-defined homomorphism on fundamental groups, and homotopic maps that fix the basepoint induce the same homomorphism.

Proof

1.1F1F4F5construct

By [F4], with the homotopy-pullback groupoid explicitly given by [F5], the groupoid on the glued bordism is equivalent to the homotopy-pullback composite of the restriction spans for $M_{1}$ and $M_{2}$ .

1.2F1F2F3F6construct

Write $Σ = ⨆_{j = 1}^{m} Σ_{j}$ and let $C_{j} = Σ_{j} \times [0, 1]$ . For each $j$ and $ϵ \in {0, 1}$ , choose $v_{j} \in Σ_{j}$ and base the end inclusion $i_{ϵ} : Σ_{j} \to C_{j}$ at $(v_{j}, ϵ)$ . The projection $p_{j} : C_{j} \to Σ_{j}$ satisfies $p_{j} i_{ϵ} = id$ , and $i_{ϵ} p_{j}$ is based-homotopic to the identity through $H_{s}^{(0)} (x, t) = (x, (1 - s) t)$ for $ϵ = 0$ and $H_{s}^{(1)} (x, t) = (x, (1 - s) t + s)$ for $ϵ = 1$ . By [F6], these maps induce inverse isomorphisms on fundamental groups. Precomposition with a group isomorphism is an equivalence of representation action groupoids, with inverse precomposition by its inverse and the identity map on conjugating arrows. The componentwise representation equivalence and restriction naturality in [F3] therefore make both end restriction functors $C_{G} (C_{j}) \to C_{G} (Σ_{j})$ equivalences; taking products gives the same conclusion for disconnected $Σ$ . Choose an edge path in each connected cylinder 1-skeleton joining its two end base vertices. The base-path naturality in [F3] gives a natural isomorphism between the two restriction functors. Hence the cylinder span is equivalent to the identity span. Each homotopy fiber of either end equivalence has one isomorphism class and trivial automorphism group, while the natural isomorphism identifies the two boundary values; its weighted push-pull map is therefore the identity on $V_{G} (Σ)$ .

1.3F1F2algebra

For a disjoint union, a flat assignment and a vertex gauge transformation are exactly independent assignments on its components. Hence the bulk and boundary groupoids are products. Since the component sets are finite, the indicator functions of pairs of isomorphism classes form the tensor-product basis of the functions on their product. The automorphism weights multiply and the finite fiber sums factor, so the state spaces and maps identify with tensor products.

1.4F1F2algebra

By the corrected orientation convention in [F1], reversing the manifold orientation leaves each fixed directed edge label and its gauge action unchanged; it only exchanges the incoming and outgoing restriction functors. Thus the span of $\overline{M}$ is the transposed span with the same underlying finite groupoids, so [F2] gives $Z_{\overline{M}} = Z_{M}^{†}$ on those function spaces. Every coefficient of these push-pull maps in the isomorphism-class indicator bases is a finite sum of positive reciprocal automorphism orders, hence is real; the identity on underlying groupoids therefore yields $Z_{\overline{M}} \circ c_{Σ_{1}} = c_{Σ_{0}} \circ Z_{M}^{†}$ . Pointwise conjugation is the only anti-linear operation, since this model has no cocycle phase.

2.1F1F2step 1.1

The push-pull composition identity in [F2] identifies the linear map of that composite span with $Z_{M_{2}} \circ Z_{M_{1}}$ . Equivalent finite groupoids give the same weighted fiber sums, so this map is $Z_{M_{2} \cup_{Σ_{1}} M_{1}}$ .

3.1F1F3step 2.1step 1.2step 1.3step 1.4∎

For empty boundary the amplitude is $∣ C_{G} (M) ∣$ by [F1]. Applying [F3] to connected $M$ gives $Z_{G} (M) = ∣ Hom (π_{1} (M), G) ∣ / ∣ G ∣$ , with all conjugation stabilizers retained. The gluing, identity, disjoint-union, and orientation statements above establish the asserted finite state-sum rules.

Gluing diagram

Write $r_{i} : C_{G} (M_{i}) \to C_{G} (Σ)$ for restriction to the common boundary and ${r e s}_{i}$ for restriction from the glued manifold to $M_{i}$ . The restriction square below commutes with the identity boundary isomorphism. It is a homotopy pullback: the comparison functor to $C_{G} (M_{2}) \times_{C_{G} (Σ)}^{h} C_{G} (M_{1})$ , sending a global field to its two restrictions and their identity boundary isomorphism, is the equivalence proved by descent. The proof is independent of the drawing.

\mathcal C_G(M_2\cup_\Sigma M_1) \arrow[r,"\mathrm{res}_1"] \arrow[d,"\mathrm{res}_2"'] &
\mathcal C_G(M_1) \arrow[d,"r_1"] \\
\mathcal C_G(M_2) \arrow[r,"r_2"'] & \mathcal C_G(\Sigma)

Remarks

This is an untwisted finite-group state sum. It does not define a continuous Lie-group BF integral or establish gravitational dynamics, and it gives no convergence theorem for the Ponzano–Regge state sum.