Absence of twisting for non-trivial discrete torsion

TL;DR

This paper investigates discrete torsion on n-tori with finite symmetry groups, introducing the concept of untwisted classes, and provides algorithms for their computation with applications to subgroups of SU(4).

Contribution

It introduces the subgroup of untwisted classes in discrete torsion, derives a universal coefficient sequence, and develops algorithms for computing these classes and partition functions.

Findings

Defined the subgroup of untwisted classes in discrete torsion.

Derived a universal coefficient exact sequence involving these classes.

Implemented algorithms and computed examples for subgroups of SU(4).

Abstract

We study discrete torsion for the $n$--torus with finite symmetry group $G$ from the Dijkgraaf--Witten viewpoint. A class in $H^n(G,U(1))$ assigns a phase to each flat $G$--bundle, equivalently to each commuting $n$--tuple in $G$ up to conjugation. We introduce the subgroup $\Br^n(G)\subseteq H^n(G,U(1))$ of \emph{untwisted} classes, those whose Dijkgraaf--Witten phases are trivial on all commuting tuples, and derive a universal coefficient exact sequence involving this invariant. In degree $2$ this recovers the Bogomolov multiplier / unramified Brauer group. We implement algorithms for computing $\Br^n(G)$ and corresponding torus partition functions, and report on computations for families of finite subgroups of $\SU(4)$.