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I came across this paper by Horava (hep-th/9404101) where he discusses Chern-Simons theory on an orbifold. The upshot is that one needs to account for the orbifold singularities by inserting Wilson lines in a particular way and computing on the underlying topological space. This seems like a curious result to me, since I would expect a TQFT, such as CS, to not care about the orbifold singularities. There is a de Rham theorem for orbifolds where indeed the orbifold singularities do not survive. So my question is twofold. First, what’s the reason why CS is actually sensitive to orbifold singularities? And more broadly, how should one really deal with gauge theory on orbifolds, especially about the singularities? I don’t know if there hasn’t been much work on concrete examples or whether it is scarce and buried in the (sort of unrelated) orbifold as a target space literature.
Orbifold structure is certainly something to be regarded as affecting the “topological type” of a space.
Broadly, the idea is that in the vicinity of a $G$-orbi-singularity, all the usual structure (e.g. gauge field structure) is equipped with $G$-actions up to coherent higher gauge transformations.
For historical reasons, gauge fields on orbifolds happen to be most familiar in the case of the B-field and the C-field, where classes of flat such higher gauge fields go by the name “discrete torsion”, in string theory.
In mathematics this is most commonly discussed in terms of the twist in orbifold K-theory (that twist is the physicist’s B-field)
The general notion of higher gauge fields on orbifolds is also discussed in our articles/books Proper Orbifold Cohomology and Equivariant principal infinity-bundles.
(The next installation in this series, which will bring in the gauge potentials, keeps being “in preparation” since somehow we are being distracted by developing a topological quantum simulator now.)
Thanks, Urs. Will browse the cited articles when I get the chance. Also looking forward to the next article, over here we have a concrete application in mind that hopefully will be clearer to us with the appropriate general theory.
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