Musings

We’re interested in M-theory on $\mathbb{R}^d\times T^{11-d}$. There are BPS $p$-branes in the $d$-dimensional theory, with tensions (masses, for $p=0$) given by the following table

Here, $M_{11}$ is the 11-dimensional Planck mass, and $R$ is a typical radius of the $T^n$. The last row corresponds to $\mathbb{R}^{d-3}\times (\text{Taub-NUT})\times T^{10-d}$, and is the D6-brane in $d=10$ and the KK monopole in $d=4$.

We’ll hold the rest of the moduli fixed, and consider varying just the overall size of the torus. To decouple the Kaluza-Klein modes, we want to take $R\to 0$, while holding the $d$-dimensional Planck mass, $$M_{(d)}^{d-2} = M_{(11)}^9 R^{11-d}$$ fixed. Rewriting the tensions in terms of $M_{(d)}$ gives the last column in the table above.

As you can see, for $d=2,3$, all of the particles and branes ($p\geq 0$) have masses that go to infinity in this limit. For $d=4$, however, the KK monopoles and the wrapped M5-brane go to zero mass, which is to say that decoupling fails. And similarly, for higher dimensions.

One can try approaching the boundary of moduli space differently, scaling different radii to zero at different rates (the limit that Green et al take is, in fact, one such asymmetrical limit), but the conclusion is the same.

The reason should be familiar. Consider a p-brane and its magnetic dual $p'=d-4-p$ brane (where KK modes are dual to KK monopoles and M5s are dual to M2s). The product of their tensions is $M_{(d)}^{d-2}$, which we are holding fixed. So, if you send one to infinity, the other goes to zero.

In $d=4$, particles are dual to particles. The BPS charges form a 56 of $\tilde{E}_7(\mathbb{Z})$ and we’ve chosen a particular way to send 28 “electric” states to infinite mass, while sending the 28 “magnetic” states to zero.

A loophole exists for $d=2,3$ because there isn’t electric-magnetic duality these dimensions (in $d=3$, particles are dual to instantons¹).

What can we conclude about $N=8$ supergravity in $d=4$?

That’s not so clear to me. I don’t think we want to take a limit where all the BPS masses go to infinity (even if that were possible). After all, there are charged blackhole solutions of the supergravity theory. And we don’t want to send their masses to infinity. That corresponds to sending the electric gauge coupling to infinity. To the contrary, starting with the supergravity theory and its charged blackhole solutions, we ought to be able to more-or-less bootstrap ourselves up to the full string theory, by demanding that they have a consistent quantum-mechanical interpretation.

Said differently, the relation $$T_p T_{4-d-p} = M_{(d)}^{d-2}$$ can be derived purely in the supergravity theory. There really wasn’t any stringy input. So, if the supergravity theory really is UV-complete, then all these degrees of freedom must be there. In other words, like it or not, the full quantum-mechanical theory of $N=8$ supergravity is M-theory, compactified on $T^7$, whether we put that in from the outset or not.

Of course, that’s something that’s not visible in perturbation theory…