Finite SU(2) graph holonomy model

A finite oriented graph consists of finite vertex and edge sets $V,E$, source and target maps $s,t:E\to V$, and a fixed-point-free involution $e\mapsto \bar{e}$ with $s(\bar{e})=t(e)$ and $t(\bar{e})=s(e)$. A finite SU(2) graph configuration assigns $g_e\in SU(2)$ to every oriented edge and requires $g_{\bar{e}}=g_e^{-1}$. A graph path is a finite composable edge word $e_1,\dots ,e_n$; its holonomy is the later-left product $g_{e_n}\cdots g_{e_1}$, and the empty path has holonomy $I_2$.

A vertex gauge assignment is a function $h:V\to SU(2)$. It acts on edge variables by

$g_e⟼g_e^{\mathrm{\prime }}=h_{t(e)}{g}_e{h}_{s(e)}^{-1}\mathrm{.}$

A finite-dimensional unitary representation is a group homomorphism $\rho :SU(2)\to U(W)$ on a finite-dimensional complex inner-product space $W$ that is strongly continuous, meaning $g\mapsto \rho (g)w$ is continuous for each $w\in W$. Here

$U(W):=\{T\in \mathrm{GL}(W):⟨Tv,Tw⟩=⟨v,w⟩\text{ for all }v,w\in W\},$

with the operator-norm topology inherited from $\mathrm{End}(W)$. In finite dimension, strong continuity is equivalent to continuity in this operator norm.

For a closed graph path $\mathrm{\ell }=(e_1,\dots ,e_n)$, the associated Wilson trace is the basis-independent trace of the endomorphism of $W$ given by

$\mathrm{Tr}(\rho (g_{e_n})\cdots \rho (g_{e_1}))\mathrm{.}$

Here the trace of a finite-dimensional endomorphism is the trace of its matrix in any basis.

SU(2) is a compact Hausdorff matrix group

For sets $A$ and $B$ the collection of all functions $A\to B$ is a set, being a subset of $\mathcal{P}(A\times B)$

Real and complex inner-product spaces and their induced length

Strongly continuous unitary representations, invariant linear subspaces and intertwiners

The basis-independent trace of an endomorphism of a finite-dimensional vector space

This definition is a finite graph configuration model. An SU(2) graph is not obtained from a Lorentz connection without an additional reduction choice. This item makes no continuum-limit, Hamiltonian-constraint, physical-Hilbert-space, or empirical-observable claim.