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The Einstein-Hilbert action on a manifold $M$ with boundary is

$$\frac{-1}{16\pi G}\int_M d^n x \sqrt{-g} R +\frac{1}{8\pi G} \int_{\partial M} d^{n-1}x \sqrt{|h|} K$$

where $K$ is the extrinsic curvature of $\partial M$ in the induced metric $h$. The Gibbons-Hawking boundary term is often justified by the fact that varying $g_{\mu\nu}\to g_{\mu\nu} + \delta g_{\mu\nu}$ would only give us the Einstein equation up to an annoying boundary term in its absence.

This is reminiscent of the anomaly inflow mechanism. For instance, Chern-Simons on an odd-d manifold $M$ with boundary is gauge anomaly free only if there is a chiral fermion living on $\partial M$. In another example, 11d M-theory on $M$ is only gauge + gravitational anomaly free if an $E_8$ supergauge theory lives on $\partial M$ (known as Horava-Witten theory).

The last example includes gravity, so this motivates my question: can the Gibbons-Hawking boundary term be framed as an anomaly cancellation mechanism? If so, which anomaly?

(Anyone who knows more about Horava-Witten theory: is there an answer in this special case, and does it generalize?)

Deschele Schilder
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Dwagg
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1 Answers1

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This is, as you say, analogous to Chern-Simons (CS) theory on a manifold $M$ with boundary. In this case, the non-gauge invariance is not interpreted as an anomaly, but rather an explicit breaking of the symmetry. The $3d$ Chern-Simons action under a gauge transformation $\delta A=dv+[A,v]$ reads $$\delta\int_M\text{tr}\big(AF-\frac{1}{3}A^3\big)=\int_{\partial M}\text{tr}\big(vdA\big)$$

The difference between this action and the Einstein-Hilbert (EH) action is that there does not exist a boundary term (at least that I can see in the literature or from what I can cook up in 30 minutes) in terms of the gauge field $A$ which can be added to the CS action which cancels the above term. Instead, to make the theory well defined you need to add anomalous chiral fermions on the boundary as you mentioned.

The EH action can be remedied without the introduction of anomalous fields on the boundary. If there were to be some hope of an anomaly inflow interpretation of the Gibbons-Hawking-York (GHY) term, one would need to find some anomalous field living on the boundary which cancels the non-covariance of EH. However, gravitational anomalies only occur in $4k+2$ dimensions, whereas the GHY term can be constructed in any dimension, so the relation with anomaly inflow is unlikely in general.

The only hope of a relation based on this would occur in dimensions $3,7,11,...$, but even in those cases the non-covariance of EH has a completely different form from all possible gravitational anomalies.

Edit

In Horava-Witten theory, the gravitational anomaly is coming from a chiral gravitino. As I mentioned the anomaly itself has a very different form (looks like traces of powers of R) from the 'explicit' non-covariance of the EH action.

fewfew4
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