Canonical flat C-fields on global ADE-orbifolds

Question

In the context of M-theory on \(G_2\)-manifolds one considers flat 3-form C-fields on \(G_{\mathrm{ADE}}\)-orbifolds, where \(G_{\mathrm{ADE}} \hookrightarrow SU(2)\) is a finite subgroup of \(SU(2) \simeq Spin(3) \to SO(3)\), hence a finite subgroup in the ADE-classification . Now for a global ADE-orbifold \(X/G_{\mathrm{ADE}}\) there is a canonical flat C-field. Namely:

1. flat C-fields are classified by \(H^3(X/G_{\mathrm{ADE}}, U(1))\),
2. there is a canonical map \(X/G_{\mathrm{ADE}} \longrightarrow B G_{\mathrm{ADE}}\) (classifying the \(G_{\mathrm{ADE}}\)-principal bundle \(X \to X/G_{\mathrm{ADE}}\))
3. there is a canonical equivalence \(H^3_{\mathrm{group}}(G, U(1)) \simeq H^3(B G , U(1))\) between the group cohomology of a group \(G\) and the cohomology of its classifying space;
4. there is a canonical cocycle in \(H^3_{\mathrm{group}}(G_{\mathrm{ADE}}, \mathbb{Z}/\vert G_{\mathrm{ADE}}\vert) \longrightarrow H^3_{\mathrm{group}}(G_{\mathrm{ADE}}, U(1))\) 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 \(G_{\mathrm{ADE}}\)-connection necessarily induces (since \(G_{\mathrm{ADE}}\) 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 \(X/G_{\mathrm{ADE}}\).

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.

Answer 1

While I still don't see that anyone made the connection of the Epa-Ganter universality to C-field physics, it should be said that, in physics lingo, those flat C-field configurations at ADE-singularities that I am asking about are known as "discrete torsion of the ABJ model" for black M2-branes, i.e. p. 8 of Aharony-Bergman-Jafferis's "Fractional M2-branes"  arXiv:0807.4924.