Finite-graph U(1) phases for matrix kernels

Let $X$ and $E$ be finite sets, let $s,t:E\to X$ give the source and target of each directed edge, and assume $X\mathrm{\ne }\mathrm{\varnothing }$. Fix a positive integer $r$, a matrix $T_e\in M_r(\mathbb{C})$ for each $e\in E$, and an external link phase ${\zeta }_e\in U(1):=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$. The finite spin space is ${\mathcal{H}}_X:={⨁}_{x\in X}{\mathbb{C}}^r$, of dimension $d=r\mathrm{\mid }X\mathrm{\mid }$.

For a real vertex phase $\theta =({\theta }_x{)}_{x\in X}\in {\mathbb{R}}^X$, define ${\zeta }_e^{\theta }:=e^{i({\theta }_{s(e)}-{\theta }_{t(e)})}{\zeta }_e,D_{\theta }:={\mathrm{diag}}_{x\in X}(e^{i{\theta }_x}{I}_r)\in G{L}_d(\mathbb{C})\mathrm{.}$ Define the link-hopping matrix $V(\zeta )\in M_d(\mathbb{C})$ by its $x,y$ block indexed by $x,y\in X$ $V(\zeta {)}_{xy}:={\sum }_{\begin{array}{c}e\in E\\ s(e)=x,t(e)=y\end{array}}{T}_e{\zeta }_e\mathrm{.}$ An empty sum is the zero $r\times r$ matrix. Additional fixed diagonal blocks belong to a separate base matrix. These data are finite-dimensional algebraic data; the phases are external parameters, not rotor multiplication operators on $L^2({\mathbb{T}}^E)$.

The graph action uses the dimensionless phase angle $\theta$ as an independent external parameter. It does not choose an embedding of $G=(X,E,s,t)$ into spacetime or assert a continuum limit.