Finite group gauge theory on graphs and gravity-like modes

Abstract

We study gauge theory with finite group $G$ on a graph $X$ using noncommutative differential geometry and Hopf algebra methods with $G$-valued holonomies replaced by gauge fields valued in a `finite group Lie algebra' subset of the group algebra $\mathbb{C}G$ corresponding to the complete graph differential structure on $G$. We show that this richer theory decomposes as a product over the nontrivial irreducible representations $\rho$ with dimension $d_{\rho }$ of certain noncommutative $U(d_{\rho })$-Yang-Mills theories, which we introduce. The Yang-Mills action recovers the Wilson action for a lattice but now with additional terms. We compute the moduli space ${\mathcal{A}}^{\times }/\mathcal{G}$ of regular connections modulo gauge transformations on connected graphs $X$. For $G$ Abelian, this is given as expected by phases associated to fundamental loops but with additional ${\mathbb{R}}_{>0}$-valued modes on every edge resembling the metric for quantum gravity models on graphs. For nonAbelian $G$, these modes become positive-matrix valued modes. We study the quantum gauge field theory in the Abelian case in a functional integral approach, particularly for $X$ the finite chain $A_{n+1}$, the $n$-gon ${\mathbb{Z}}_n$ and the single plaquette ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$. We show that, in stark contrast to usual lattice gauge theory, the Lorentzian version is well-behaved, and we identify novel boundary vs bulk effects in the case of the finite chain. We also consider gauge fields valued in the finite-group Lie algebra corresponding to a general Cayley graph differential calculus on $G$, where we study an obstruction to closure of gauge transformations.