In the context of M-theory on G2-manifolds one considers flat 3-form C-fields on GADE-orbifolds, where GADE↪SU(2) is a finite subgroup of SU(2)≃Spin(3)→SO(3), hence a finite subgroup in the ADE-classification . Now for a global ADE-orbifold X/GADE there is a canonical flat C-field. Namely:
- flat C-fields are classified by H3(X/GADE,U(1)),
- there is a canonical map X/GADE⟶BGADE (classifying the GADE-principal bundle X→X/GADE)
- there is a canonical equivalence H3group(G,U(1))≃H3(BG,U(1)) between the group cohomology of a group G and the cohomology of its classifying space;
- there is a canonical cocycle in H3group(GADE,Z/|GADE|)⟶H3group(GADE,U(1)) which is the restriction of the second Chern-class.
The last statement is due to Prop. 4.1 in Epa-Ganter 16 , but even without that precise statement one sees that there should be such a cocycle: Consider the SU(2)-Chern-Simons action functional, restrict it to flat fields, and observe that a GADE-connection necessarily induces (since GADE is finite) a flat SU(2)-connection. The cocycle in question is the induced Chern-Simons invariant.
Taken together these statements give a canonical flat C-field on any global orbifold X/GADE.
My question is if any hints of these canonical flat C-fields on ADE-orbifolds have ever surfaced in the string theory literature?
One might expect this discussed in Atiyah-Witten 01, but it's not.