On Generalised Discrete Torsion

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.