# Canonical flat C-fields on global ADE-orbifolds

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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.

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