Subdivision Analysis of Topological $Z_{p}$ Lattice Gauge Theory

TL;DR

This paper studies the subdivision properties of 4D lattice gauge theories with $Z_{p}$ groups, showing invariance under subdivision moves and deriving boundary effective models with exact partition function calculations.

Contribution

It demonstrates subdivision invariance of Boltzmann weights at roots of unity and connects 4D models to boundary 3D models with explicit partition function evaluations.

Findings

Boltzmann weights are invariant under all $(k,l)$ subdivision moves at $p$th roots of unity.

Subdivision of manifolds induces boundary operators related to delta functions on trivial holonomies.

Partition functions for $S^{3}$ and $S^{2}\times S^{1}$ are explicitly computed.

Abstract

We analyze the subdivision properties of certain lattice gauge theories for the discrete abelian groups $Z_{p}$, in four dimensions. In these particular models we show that the Boltzmann weights are invariant under all $(k,l)$ subdivision moves, when the coupling scale is a $p$th root of unity. For the case of manifolds with boundary, we demonstrate analytically that Alexander type $2$ and $3$ subdivision of a bounding simplex is equivalent to the insertion of an operator which equals a delta function on trivial bounding holonomies. The four dimensional model then gives rise to an effective gauge invariant three dimensional model on its boundary, and we compute the combinatorially invariant value of the partition function for the case of $S^{3}$ and $S^{2}\times S^{1}$.