Finite non-Abelian lattice gauge data

Assume the axiom of countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$. Fix the compact noncommutative finite-dimensional real Lie group $G$ supplied by the earlier non-Abelian gauge-data definition. Its Lie algebra is $\mathfrak{g}=T_{e}G$, with bracket induced by left-invariant smooth vector fields. Fix also a finite-dimensional complex vector space $W$, a positive-definite Hermitian form $h$ on $W$, and a specified smooth group homomorphism $\rho :G\to U(W,h)$, where $U(W,h)$ is the group of invertible complex-linear maps preserving $h$. Thus multiplication and inversion on $G$ are smooth, $G$ is compact in its manifold topology, and every $\rho (g)$ preserves $h$. Every finite power $G^S$ below has the product smooth-manifold structure, with charts given by products of charts of $G$; for $S=\mathrm{\varnothing }$, set $G^S=\{()\}$ with its one-point smooth structure. No connection, principal bundle sector, action functional, or continuum theory is part of the lattice data below.

A finite oriented lattice two-complex is a finite set $\mathcal{V}$ of vertices, a finite set ${\mathcal{E}}^{+}$ containing one chosen orientation of each edge, an endpoint map $s,t:{\mathcal{E}}^{+}\to \mathcal{V}$, and a finite set $\mathcal{F}$ of faces. Each face $p\in \mathcal{F}$ is supplied with a based closed edge word $(e_1,\dots ,e_k)$ in the oriented edges ${\mathcal{E}}^{+}\bigsqcup \overline{{\mathcal{E}}^{+}}$, where the endpoint of each edge is the starting point of the next and the last endpoint is the first starting point. For the reverse orientation $\bar{e}$ of $e\in {\mathcal{E}}^{+}$, set $s(\bar{e})=t(e)$ and $t(\bar{e})=s(e)$.

A link configuration is a tuple $U=(U_e{)}_{e\in {\mathcal{E}}^{+}}\in G^{{\mathcal{E}}^{+}}$, with $U_{\bar{e}}:=U_e^{-1}$. The holonomy of a face $p=(e_1,\dots ,e_k)$ is the ordered product $U_p:=U_{e_1}\cdots U_{e_k}$ in traversal order, with the first traversed edge on the left. The vertex gauge group is $G^{\mathcal{V}}$ with coordinatewise multiplication; write $\mathbf{e}=(e{)}_{v\in \mathcal{V}}$ for its identity tuple. It acts by

$$(h\cdot U{)}_e=h_{s(e)}{U}_e{h}_{t(e)}^{-1},h=(h_v{)}_{v\in \mathcal{V}}\in G^{\mathcal{V}}\mathrm{.}$$

Choose a normalized bi-invariant finite Radon Borel probability measure ${\mu }_G$ on $G$; explicitly, ${\mu }_G(G)=1$ and ${\mu }_G(gA)={\mu }_G(Ag)={\mu }_G(A)$ for every $g\in G$ and Borel $A\subseteq G$. The link measure is the finite product ${\mu }_G^{{\mathcal{E}}^{+}}$; for ${\mathcal{E}}^{+}=\mathrm{\varnothing }$, this is the unit point mass on $G^{\mathrm{\varnothing }}=\{()\}$. Empty sums are zero. For each face choose a continuous dimensionless function ${\mathrm{\Phi }}_p:G\to \mathbb{R}$ satisfying ${\mathrm{\Phi }}_p(gz{g}^{-1})={\mathrm{\Phi }}_p(z)$ and ${\mathrm{\Phi }}_p(z^{-1})={\mathrm{\Phi }}_p(z)$. Define the dimensionless lattice action and its normalized expectation by

$$S_{\mathrm{\Lambda }}(U):=\sum _{p\in \mathcal{F}}{\mathrm{\Phi }}_p(U_p),Z_{\mathrm{\Lambda }}:={\int }_{G^{{\mathcal{E}}^{+}}}{e}^{-S_{\mathrm{\Lambda }}(U)}d{\mu }_G^{{\mathcal{E}}^{+}}(U),⟨O{⟩}_{\mathrm{\Lambda }}:=Z_{\mathrm{\Lambda }}^{-1}{\int }_{G^{{\mathcal{E}}^{+}}}O(U)e^{-S_{\mathrm{\Lambda }}(U)}d{\mu }_G^{{\mathcal{E}}^{+}}(U)$$

for bounded measurable $O:G^{{\mathcal{E}}^{+}}\to \mathbb{C}$. The finite-volume Gauss sector is the closed subspace of $L^2(G^{{\mathcal{E}}^{+}},{\mathbb{P}}_{\mathrm{\Lambda }})$ consisting of measurable functions with finite integral of $\mathrm{\mid }f{\mathrm{\mid }}^2$, modulo equality ${\mathbb{P}}_{\mathrm{\Lambda }}$-almost everywhere, whose equivalence classes are invariant under each element of $G^{\mathcal{V}}$. Here ${\mathbb{P}}_{\mathrm{\Lambda }}$ is the normalized probability with density $Z_{\mathrm{\Lambda }}^{-1}{e}^{-S_{\mathrm{\Lambda }}}$ relative to product link measure. For $v\in \mathcal{V}$ and $X\in \mathfrak{g}$, let $X^{(v)}$ be the tangent vector at the identity of $G^{\mathcal{V}}$ with component $X$ at $v$ and zero at other vertices. If $\alpha (h,U)=h\cdot U$, define the infinitesimal Gauss generator on $C^{\mathrm{\infty }}(G^{{\mathcal{E}}^{+}})$ by

$$(D_{v,X}f)(U):=d{f}_U\text{\hspace{0.17em}⁣}(d{\alpha }_{(\mathbf{e},U)}(X^{(v)},0))\mathrm{.}$$

The full Gauss sector uses invariance under every vertex gauge transformation; the infinitesimal equations $D_{v,X}f=0$ alone are not asserted to impose invariance under disconnected components of $G$. Link variables and the action are dimensionless. This finite mathematical regulator makes no continuum-limit or empirical claim.

Compact group, Lie algebra, and unitary representation input

Countable-choice assumption carried by the supplied Lie algebra construction

Differential as the tangent map of the smooth gauge action

Finite-dimensional real Lie group structure

Hausdorff and second-countable manifold topology

Borel sigma-algebra on the compact group

Borel sigma-algebra on finite products of the second-countable Lie group

Normalized bi-invariant Borel probability measure

Finite product sigma-algebra

Product measure on sigma-finite measure spaces

Existence, rectangle formula, and uniqueness of the finite product measure

Measure integral and integrability conventions

Measure defined by a nonnegative density

Complex $L^2$ pairing and norm convention

Well-definedness and norm properties of complex $L^2$

Meyer, §§1.2.3–1.2.4 and §1.4.1.1, printed pp. 11–14 and 22–24

Lohmayer, §§3.5.2–3.5.5 and §§4.1–4.2, printed pp. 44–51 and 59–63