### From One, Many

Been meaning to make some comments about this, but just hadn’t gotten around to it.

Lenny Susskind has taken the paper by Kachru *et al* as an “existence proof” for (meta)stable de Sitter vacua in string theory. I’ve written previously about some of the handwaving involved, and how one might try to improve upon it. But, even if you do believe it constitutes an existence proof, Lenny’s leap of faith — that the existence of one such solution means there must be gazillions of them — is nothing short of breathtaking. There are other things I could grumble about in his paper, but this is the heart of the matter.

Finding stable de Sitter solutions in supergravity is hard. Generically, one expects that supersymmetry-breaking will lead, not to a local de Sitter minimum, but to an instability. (In this class of theories, there’s always a “runaway” direction, where the scalar potential goes to zero at infinity; the problem is to produce a local minimum, with positive vacuum energy, at some finite value of the field.)

Mike Douglas has taken up the crusade, by embarking on a program to “count” supersymmetric “vacua” which, when supersymmetry-breaking is taken into account, would lead to de Sitter vacua. He then hopes to do statistics on the space of such vacua (analogous, I guess, to the Bayesian approach to the Anthropic Principle). But we know from many examples that supersymmetry-breaking can “lift” supersymmetric ground states. The non-supersymmetric theory may have fewer (even metastable) ground states than the supersymmetric one. Even granting all of Lenny’s assumptions, I don’t see the relevance of counting supersymmetric vacua.

Posted by distler at April 8, 2003 11:42 AM
## Re: From One, Many

In order for Shamit et al’s solutions to work for Lenny’s purposes you at least have to believe their claims that they should be able to tune their solutions with sufficient resolution that there might be quite a few of them.

Unfortunately, they don’t construct any explicit examples in which to investigate whether or not this is true. Now, all technical aspects of the problem aside, consider the one example they *do* provide. They plot their potential, demonstrating the existence of a DeSitter vacua. The SUSY breaking piece of the potential is proportional to the number of anti-D3branes. They call this parameter D; it also depends on the string coupling and the minimum size of the warp factor (where the anti-D3’s sit). To produce their plot they provide some numbers. As far as I can tell (and this is a casual observation) there is no way to translate these numbers into quantities like “number of anti-D3 branes”. But we want this mechanism to be robust under reasonable adjustment of parameters. So lets double D, or the number of anti-D3s. The minimum becomes very shallow. Now let’s triple D. The minimum vanishes.

Is this robust? You might say that, since I don’t know how many anti-D3 branes their value of D corresponds to, I don’t know if doubling or tripling D corresponds to a “reasonable tuning”. That is, maybe going from 1000000 to 1000001 anti-D3s is reasonable, but 1000000 to 3000000 isn’t. That’s possible. But in order for the mechanism to be robust you’d like to have a little wiggle room. I changed D by a factor of 3 and the minimum vanished. That’s not even an order of magnitude.

This is an extremely shallow argument (no pun intended). The notion of changing D and leaving everything else the same may not be okay. Also, since their example was probably just cooked up to draw a picture we probably shouldn’t take it as being a meaningful representation of their ideas. Still, without a concrete example I have to try to pull intuition from somewhere.

Another thing, and this just reflects technical ignorance on my part, is it okay to simply superimpose the tree level superpotential and the non-perturbative corrections like they do? I know there are examples where this is okay, but I’ve also been told there are examples where it isn’t. Morally speaking, I wouldn’t be worried if the potential, with corrections, still landed me in a SUSY vacuum. But we’re taught to be wary of superimposing things (though not always) when we don’t have SUSY to protect us. Should I be worried about whether or not this is okay, or is there some basic fact that I should be aware of that makes this a no-brainer?