### M2 Cosmology?

OK, perhaps I have missed the point of Easther, Greene, Jackson & Kabat, but I find it deeply puzzling. The intent, apparently, is to produce a modern (M-theory) counterpart to the old work of Brandenberger & Vafa ( Nucl. Phys. B316 (1989) 391).

The idea there was to imagine that space had the topology of T^{9}. If we populate it with a gas of free strings, they will be wound around the cycles of the T^{9} with some Boltzmann-like distribution. Turning the coupling to gravity back on, we obtain a cosmology in which some of the dimensions (those with zero winding number) expand, while others stay small.

The idea was to *explain* why we see 3 large dimensions.

Easther *et al* update this story to consider M-theory 2-branes wrapped on T^{10}. Actually, they consider only the latter half of the story — the cosmological solution in a sector with a fixed set of winding numbers.

But what’s puzzling is how the first part is supposed to go. There is no such thing as a gas of “free” M2-branes (for precisely the reasons that de Wit, Hoppe & Nicolai’s attempt (Nucl. Phys. B305 (1988) 545) to construct a world-volume theory of the M2-brane led, instead, to the Matrix description of all of M-theory). I don’t know how to do the thermodynamics of full-blown M-theory, so I don’t see how we’re supposed to “produce” the desired distribution of winding numbers.

Of course, if one of the dimensions is much smaller than all the rest, membranes wrapped around that dimension have much smaller effective tensions and dominate the thermodynamics. These guys (known as Type IIA strings) DO have a weakly-coupled description, and hence we can do their thermodynamics. But I don’t see how one might even get started in the general case.

**Update:** I’m even puzzled about the part of the story that they *do* try to tackle. Fix a total charge (set of wrapping numbers). At “late times”, as they would call it, you are dominated by the lowest-energy M2-brane configuration carrying that charge. I’m not sure they do that correctly.

Consider, as a baby example, strings wrapped on a rectangular T^{2} with side lengths R_{1},R_{2}. The charge is labelled by a pair of integers, (N_{1},N_{2}). Naively, you might think that the lowest energy configuration with that charge has energy

N

_{1}R_{1}+N_{2}R_{2}

(times the string tension). But that would be wrong. The lowest energy configuration has energy

k (p

_{1}^{2}R_{1}^{2}+p_{2}^{2}R_{2}^{2})^{1/2}

where k is the greatest common divisor of N_{1} and N_{1}: (N_{1},N_{2}) = (k p_{1}, k p_{2}).

Similar (though more complicated) formulae hold for higher dimensional tori and for the case of membranes wrapped on them. I could be wrong, but I *think* this makes a difference in their analysis.