On Dijkgraaf-Witten Type Invariants

TL;DR

This paper constructs lattice models with subdivision invariance based on the gauge group Z_p, which recover Dijkgraaf-Witten invariants for 3-manifolds and extend to higher dimensions with multiple field types.

Contribution

It explicitly develops lattice models that realize Dijkgraaf-Witten invariants, including models with multiple field variables for higher-dimensional manifolds.

Findings

Models are subdivision invariant when coupling is quantized.

The simplest model reproduces the Dijkgraaf-Witten invariant for 3-manifolds.

Extensions to higher dimensions involve multiple field types.

Abstract

We explicitly construct a series of lattice models based upon the gauge group $Z_{p}$ which have the property of subdivision invariance, when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-$p$ flatness condition. The simplest model of this type yields the Dijkgraaf-Witten invariant of a $3$-manifold and is based upon a single link, or $1$-simplex, field. Depending upon the manifold's dimension, other models may have more than one species of field variable, and these may be based on higher dimensional simplices.