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:

- flat C-fields are classified by \(H^3(X/G_{\mathrm{ADE}}, U(1))\),
- there is a canonical map \(X/G_{\mathrm{ADE}} \longrightarrow B G_{\mathrm{ADE}}\) (classifying the \(G_{\mathrm{ADE}}\)-principal bundle \(X \to X/G_{\mathrm{ADE}}\))
- 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;
- 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.