### Long Live de Sitter!

**(Warning! This post uses MathML. The equations will probably look like garbage unless you are viewing it in Mozilla, with the requisite fonts installed.** Sorry, that’s just life.**)**

Kachru *et al* suggest a way to obtain classically-stable (and
quantum-mechanically long-lived) 4D de Sitter solutions of string theory.

The starting point is the class of compactifications introduced by Giddings *et al*. There, the flux-induced superpotential,

fixes the string coupling and the complex structure of the 3-fold $M$, leaving the Kahler modulus, $\rho $ (we’ll assume only one) as a flat direction.

The next stage (a bit of handwaving, but not implausible) is to assume that nonperturbative effects induce a superpotential for $\rho $ which lifts this remaining flat direction. In a fairly robust fashion, one ends up with a supersymmetric vacuum in 4D anti-de Sitter space.

Now comes the tricky step. We imagine changing the fluxes (a discrete choice) so that the tadpole cancellation condition now requires the presence of one (or a small number of) $\overline{\text{D3}}$ brane(s). This breaks supersymmetry and induces a term in the potential which would normally lead to a runaway behaviour for $\rho $. Naively, I might guess that the coefficient of this term would be large, and that it would totally overwhelm the nonperturbative superpotential which generated a minimum for $\rho $ in the first place.

Well, not according to these guys. They claim that the coefficient can be small (so as not to totally destabilize the minimum) and, moreover, can be fine-tuned (again, we have only discrete choices) to produce a minimum with a small (tiny!) positive cosmological constant.

If you swallow all of this, it’s not to hard to believe the last step: namely, while this minimum is only metastable (we still have $V\to 0$ for $\rho \to \mathrm{\infty}$, after all), it can be incredibly long-lived — more than ${10}^{10}$ years.

Some obvious points:

- If any of this makes sense,, it should have a description in terms of 4D supergravity, presumably related to the solutions that I blogged about previously.
- Supersymmetry breaking in the real world is much larger than
the scale of the cosmological constant. That’s always been a bit of
a problem, but may be less so here. Usually, the problem is
discussed assuming the supersymmetric vacuum is 4D flat space.
Here, the supersymmetric minimum would more naturally be thought of
as being anti-de Sitter. We have to lift the minimum of the
potential by a
*lot*(much more than its final value), thereby “explaining” why the supersymmetry breaking scale is so much larger than that of the cosmological constant. - Of course, the
*height*of the barrier is more directly related to the scale of supersymmetry breaking. So, as potential inhabitants of a false vacuum, we might, perhaps, have reason to be fearful of the next generation of accelerators.

As for the last, the possibility of vacuum decay triggered by anything less energetic than p-p collisions at 10

^{8}TeV seems to be excluded by cosmic ray collisions. See, e.g., hep-ph/9910333, p. 10.I haven’t checked out the paper yet – does this possibly falsify the particular construction? Of course, getting dS out of string theory is a good thing regardles.