On generalised discrete torsion

A bstract 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 $$ {H}_G^2\\left(M;\\mathrm{U}(1)\\right) $$ H G 2 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 $$ {T}^6/{\\mathbb{Z}}_2^2 $$ T 6 / ℤ 2 2 and $$ {T}^7/{\\mathbb{Z}}_2^3 $$ T 7 / 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 $$ {T}^7/{\\mathbb{Z}}_2^3 $$ T 7 / 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.