Finite two-group lattice spin state-sum model

A finite ordered simplicial complex $K$ has a finite vertex set with a total order and a collection of nonempty finite vertex subsets, called simplices, closed under taking nonempty subsets. Its dimension is the maximum of $\mathrm{\mid }\sigma \mathrm{\mid }-1$ over its simplices. A closed physical regulator is a finite ordered triangulation of a closed compact oriented smooth three-manifold; relative amplitudes are also defined on its finite ordered subcomplexes. The increasing vertex order orients every simplex.

Choose $G=\mathbb{Z}\mathrm{/}2$, the trivial Abelian group $A=\{0\}$, the unique action $\rho :G\to \mathrm{Aut}(A)$, and the zero normalized Postnikov cocycle $\beta =0$. A background on $K$ is a flat edge cochain $a_{ij}\in G$, satisfying $a_{ij}+a_{jk}-a_{ik}=0$ on each ordered triangle; its face cochain $B$ is uniquely zero. A raw $1$-gauge arrow has vertex labels $g_i\in G$ and sends $a_{ij}$ to $a_{ij}^{\mathrm{\prime }}=g_i+a_{ij}-g_j$. The edge label $\lambda$ and vertex $2$-gauge label $t$ are necessarily zero, so this selected two-group is the ordinary finite $G$ symmetry regarded as a two-group with trivial $A$.

For each background $x=(a,0)$, the model's cell-value sets are $X_v=G$ at each vertex and a singleton at every positive-dimensional simplex. The configuration set is ${\mathrm{\Phi }}_x=C^0(K;G)$, the set of spin assignments $\sigma =({\sigma }_v{)}_{v\in K^{(0)}}$. There are no further constraints. Set the dimensionless action $S_x(\sigma )=0$, the finite measure weight ${\mu }_x(\sigma )=2^{-V_K}$, where $V_K$ is the number of vertices, and the observable algebra ${\mathcal{O}}_x=\mathrm{Fun}({\mathrm{\Phi }}_x,\mathbb{C})$. A function is local when it depends on spins in a specified subcomplex. The local spin observable $m_v(\sigma )=(-1{)}^{{\sigma }_v}$ transforms nontrivially under the constant global $G$ action, so this $0$-form symmetry is effective. For a closed one-dimensional subcomplex $\gamma$, the product ${\prod }_{v\in {\gamma }^{(0)}}{m}_v(\sigma )$ is a line-supported observable defined without an attached surface. The selected $1$-form group is trivial: its only defect is $U_0(\mathrm{\Sigma })=1$, and the line observable has only the trivial $A$-character.

The complete defect-bearing input domain consists of any finite list of admitted ordinary spin functions, with their subcomplex supports, and any finite decorated network of the identity defect $U_0$. Every stratum has the trivial $A$ label, every junction has scalar $1$, and the model admits all choices of orientation or framing only with this same trivial response; the incidence and ordering data are recorded even though their factors are $1$. For each such input $I$, set ${\mathrm{\Omega }}_x(I)={\mathrm{\Phi }}_x$ and

$$w_x(I;\sigma )=2^{-V_K}\prod _j{o}_{j,x}(\sigma ),{\widehat{C}}_x(I)=\sum _{\sigma \in C^0(K;G)}{2}^{-V_K}\prod _j{o}_{j,x}(\sigma )\mathrm{.}$$

Thus every identity-network factor is exactly $1$ for every spin configuration. The combined boundary datum in the relative convention is the fixed background restriction $s$; no boundary spin value is fixed, and there is no additional boundary datum for a closed regulator. Gauge transport sends each spin by $T_g$ and every insertion function by pullback, transports the boundary background when it is allowed to vary, and preserves identity-network labels and junctions. Boundary-fixed relative gauge arrows restrict to $g=0$ on the boundary subcomplex. These are bijections of the complete inputs and configurations and preserve the displayed integrands with phase $1$; all comparison maps and $2$-arrow maps are identities under the strict translation composition law.

An identity support may be moved through any smooth decorated isotopy preserving incidence and avoiding the other supports: the model evaluates every intermediate identity network by exactly the same spin sum, including when its support is not a subcomplex of the fixed regulator. The endpoint comparison is the identity on spins and on ordinary insertion functions. If a supplied compatible subdivision $K^{\mathrm{\prime }}$ is used to represent a support, retain the old ordinary observables by pullback along restriction of vertex spins to $K$. Summing over the $V_{K^{\mathrm{\prime }}}-V_K$ new spins gives $2^{V_{K^{\mathrm{\prime }}}-V_K}{2}^{-V_{K^{\mathrm{\prime }}}}=2^{-V_K}$ for each old spin assignment, hence exactly the same complete correlator, for any supplied compatible refined background. This is a comparison for this identity response and these transported observables, not an assertion of triangulation independence of the separate background-gauging amplitude.

For a background gauge arrow with vertex labels $g$, the configuration bijection is $T_g(\sigma {)}_v={\sigma }_v+g_v$ and the observable-algebra isomorphism is ${\alpha }_g(O)=O\circ T_g^{-1}$. These assignments respect identity and composition. The per-background partition weight is

$$W_K(x)=Z_x=\sum _{\sigma \in {\mathrm{\Phi }}_x}{2}^{-V_K}=1.$$

The boundary state space on a subcomplex $L$ is the finite-dimensional space $\mathcal{H}(L)=\mathrm{Fun}({\pi }_0\mathcal{B}(L),\mathbb{C})$. For a fixed background $s$ on $L$, the relative convention fixes only this background restriction and sums over all spins on $K$ with weight $2^{-V_K}$; it does not fix boundary spin values. Thus the spin sector contributes $1$ for each extending background and introduces no additional seam factor. The relative background amplitude then uses the relative higher-gauge sum. All fields, action phases, measures, and amplitudes are dimensionless in the convention $\mathrm{\hslash }=1$.

The closed-manifold data instantiate the finite-regulator two-group model interface. The relative subcomplex amplitudes and the background-only boundary state space are the additional boundary convention adopted here; they are not inferred from the closed-manifold interface. No continuum limit or empirical conclusion is part of this definition.

finite background-groupoid and relative higher-gauge sums

finite-regulator two-group model-data interface

Moore and Saxena, §23.3

Freed and Quinn, §2

Baez and Lauda, §8.3

Benini, Córdova, and Hsin, §2.3