On Generalised Discrete Torsion

TL;DR

This paper generalizes discrete torsion in 2D gauged sigma models, allowing different local phases at singularities, and explores their geometric implications on orbifold Calabi-Yau and G2 manifolds.

Contribution

It introduces a new framework for local discrete torsion in orbifold theories and analyzes its effects on the topology of resulting geometries.

Findings

Local discrete torsion phases can vary but are not fully independent.

Different choices of generalized discrete torsion affect Betti numbers of G2 resolutions.

The orbifold CFTs realize only a subset of possible topologies in G2 resolutions.

Abstract

For a 2d gauged sigma model with target space $M$ and discrete gauge group $G$, we consider a generalisation of Vafa's discrete torsion $H^2(BG; U(1))$ that assigns different local discrete torsion phases to different singular loci of the orbifold $M/G$. Our generalised discrete torsion lives in $H^2_G(M; U(1))$, and gives a consistent implementation of Gaberdiel and Kaste's prescription for inserting such local discrete torsion phases by hand at higher genus. We revisit the original application to $T^6/\mathbb{Z}_2^2$ and $T^7/\mathbb{Z}_2^3$ orbifold CFTs, and determine what smooth Calabi-Yau and $G_2$ geometries result from different choices of the generalised discrete torsion. We find that the local discrete torsion phases can be different from each other, but are not completely independent either; in the $T^7/\mathbb{Z}_2^3$ case for example, the orbifold CFTs only realise 3 out of the 9 possible Betti numbers of $G_2$ resolutions constructed by Joyce.