State Sum Models and Simplicial Cohomology

TL;DR

This paper investigates subdivision invariant lattice models based on Z_p gauge groups, focusing on four-dimensional cases, and explores their quantum properties through explicit partition function calculations.

Contribution

It introduces a class of subdivision invariant lattice models using simplicial cohomology with Z_p gauge groups, highlighting their quantum features and flatness conditions.

Findings

Partition function computed for RP^3 x S^1

Quantum Hilbert space differs from classical

Subdivision invariance achieved via quantized coupling

Abstract

We study a class of subdivision invariant lattice models based on the gauge group $Z_{p}$, with particular emphasis on the four dimensional example. This model is based upon the assignment of field variables to both the $1$- and $2$-dimensional simplices of the simplicial complex. The property of subdivision invariance is achieved when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-$p$ flatness condition. By explicit computation of the partition function for the manifold $RP^{3} \times S^{1}$, we establish that the theory has a quantum Hilbert space which differs from the classical one.