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It seems to be accepted that to produce chiral fermionic matter in a compactification $\mathbb{R}^4\times X$ of M-theory/11d SUGRA to four dimensions, we need the seven-manifold $X$ to have isolated (or "codimension seven") singularities.

The idea seems to have originated in Witten's "Anomaly Cancellation On Manifolds Of $G_2$ Holonomy", where the idea is that if the compactification manifold has isolated singularities, then the Chern-Simons-like term $$ I\propto \int_M C\wedge G\wedge G$$ for $C$ the SUGRA 3-form gauge field and $G=\mathrm{d}C$ its field strength develops an anomaly under a gauge transformation $C\mapsto C+\mathrm{d}\epsilon$ because we cannot use Stokes' theorem on the singular manifold, but only on a smooth manifold-with boundary $X'$ which is $X$ with small neighbourhoods of the singularities removed. The anomaly is $$ \delta I \propto \int_{\partial X'} \epsilon\wedge G\wedge G.\tag{1}$$

Now, Witten expands $\epsilon = \sum_i \epsilon^i w_i$ where the $w_i$ are harmonic forms on $X$ and the $\epsilon^i$ functions on $\mathbb{R}^4$. This is the first issue: Why can the a priori arbitrary 2-form $\epsilon$ in the gauge transformation be expanded in this manner?

The second issue arises from "assuming" that there is some additional phenomenon at the singularities in form of charged chiral fermions whose contribution to the gauge anomaly cancels (1). I completely agree that if there are such fermions with the charges required to cancel the anomaly, then it will cancel, but where do these actually come from? They are not obtained by Kaluza-Klein reduction from an M-theory field, and they are not moduli - their sole justification for being there seems to be because they are required to cancel the gauge anomaly, but why would we expect in the first place that the anomaly must cancel for some arbitrary singular manifold that we are forcing the theory onto?

The nature of these fermions seems indeed rather mysterious: They are parts of chiral superfields $\Phi^\sigma$, where the $\sigma$ is an index that "takes values in a set $T_\alpha$ that depends on $\alpha$", where $\alpha$ is Witten's label for the singularities. How this $T_\alpha$ depends on the $\alpha$ or what it actually is is never explained, and the only further thing we know about these fields is that $$ \sum_{\alpha\in T_\alpha} q^\alpha_i q^\alpha_j q^\alpha_k = \int_{Y_\alpha} w_i\wedge w_j \wedge w_k,$$ where $Y_\alpha$ is the boundary of the conical neighbourhood that was removed from $X$ to obtain $X'$ and the $q_i^\alpha$ are the $\mathrm{U}(1)$ charges of the field under the effective 4d gauge fields.

So, what is going on here? Can one see where the fermions arise from in the eleven-dimensional M-theory viewpoint alone without invoking duality to the heterotic string where one can see the emergence of the fermionic modes much more readily?

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  • $\begingroup$ For your first question concerning the decomposition of $\epsilon$, I think the answer is related to the Hodge decomposition (see e.g. en.wikipedia.org/wiki/Hodge_theory), since the exact parts of $\epsilon$ are irrelevant for the gauge transformation. $\endgroup$ – Antoine May 9 '17 at 10:48
  • $\begingroup$ M-Theory is free of gauge and gravitational anomalies in 11d. In the presence of a fivebrane or a defect, anomaly cancellation is through an adaption (to sugra/M-Theory) of the Callan-Harvey Inflow mechanism. In the presence of a fivebrane for example, there are fermions in the worldvolume theory of the fivebrane (which is anomalous). To cancel the anomaly, you add a Green-Schwarz term in the bulk, which cancels part of the anomaly and the remaining part is canceled by the Chern-Simons term (which you write too). $\endgroup$ – leastaction May 12 '17 at 0:37
  • $\begingroup$ To make sense of the Chern-Simons term in the presence of a brane or a singularity, you have to carve out the region just around the singularity. In this case, G = dC + (more) near the fivebrane and in fact dG = (a delta function 5-form) which is smoothed out by the regularization you have. The fermion zero modes on the fivebrane enter the anomaly polynomial for the theory on the fivebrane. In "morally" the same way, you have an anomaly cancellation condition in your system (without going into the nittygritties of the paper) which requires fermions to be present. Not sure if this helps... $\endgroup$ – leastaction May 12 '17 at 0:41

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