Absence of twisting for non-trivial discrete torsion
Primoz Moravec
TL;DR
This work formalizes the notion of untwisted discrete torsion for finite groups $G$ acting on an $n$-torus via Dijkgraaf--Witten theory. It introduces the groups $ ext{Utw}^n(G)$ and proves a universal-coefficient framework that relates these to $H_{0n}(G,Z)$, with $ ext{Utw}^n(G) o ext{Hom}(H_{0n}(G,Z),U(1))$ in the finite case; in degree $2$ this recovers the Bogomolov multiplier/unramified Brauer group. The paper develops efficient algorithms to compute $ ext{Utw}^n(G)$ and the torus partition functions, and provides extensive computational data for finite subgroups of $ ext{SU}(3)$ and $ ext{SU}(4)$, revealing a striking contrast: no known $ ext{SU}(4)$ subgroup has a nontrivial Bogomolov multiplier, yet many admit nontrivial higher-dimensional untwisted torsion ($n\,=\,3$). It also clarifies the relationship between $ ext{Sha}^n(G)$ and $ ext{Utw}^n(G)$, showing $ ext{Sha}^n(G)\subseteq ext{Utw}^n(G)$ with possible strict inclusion, and emphasizes the torus-background viewpoint as a diagnostic for genuine topological twisting in orbifolds.
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 $\SU(4)$.