Can/has string theory solved cosmological constant problem? Can someone explain to me what the KKLT paper says, and what has and hasn't it achieved regarding the ability to construct solutions with a small positive or negative cosmological constant in string theory?
 A: Dear gob,
the KKLT 2003 paper
http://arxiv.org/abs/hep-th/0301240
is constructing a large number of stabilized vacua with a negative cosmological constant - anti de Sitter or AdS vacua - and, with a somewhat smaller certainty, many positive-cosmological-constant de Sitter or dS vacua derived from them.
The precise paper is nontrivial because of the particular technical context how this is achieved; interpretation will be left to the end. It considers type IIB string theory on Calabi-Yau manifolds with a topology. This topology has typically lots of "cycles" - noncontractible submanifolds - and there can be an integer-valued generalized (NS-NS and/or R-R) magnetic flux through each "cycle". By taking combinations, one may obtain googols (or powers of googols) of different stable points in the space of configurations of the "string fields".
The vacua they obtain at the beginning are AdS vacua. They're supersymmetric and the cosmological constant is negative. KKLT also show that there exist related vacua in which some "antibranes" are added. Those break supersymmetry but are metastable, with lifetimes that vastly exceed the current age of the Universe, so these vacua are potential candidates to correspond to our world. The KKLT computation of the de Sitter vacua's lifetime has come under some scrutiny and physicists differ in their opinion whether the existence of a large number of de Sitter vacua has been established; the situation is much clearer with the unbroken-supersymmetric AdS part of their construction.
Interpretation
These vacua are stable - that's a good thing because they don't include any exactly massless scalar fields whose value could spontaneously change, thus producing new (unobserved) long-range forces (and allowing the fundamental constants such as the fine-structure constant to change rapidly). The instabilities are nonperturbative - essentially quantum tunneling into other vacua. The precise choice which tunneling directions are favored has been updated by some newer papers, too. It remains a controversial technical question.
It seems very likely that string theory's equations have many solutions - people often say $10^{500}$ although the number is not known at any accuracy - that qualitatively look like the Universe around us. By their sheer number, some of them will have a small value of the cosmological constant, comparable to the observed value. The smallness was only guaranteed by having many sufficiently (seemingly) "random" solutions to choose from. Some of them will have a small cosmological constant.
Anthropic selection
A highly disputed question is whether we are living in a randomly chosen Universe or whether there exists a cosmological mechanism or an equation that picks a privileged vacuum (or at least a much smaller subset). As Weinberg argued decades ago, the existence of galaxies - which seems needed for the existence of intelligent beings - may act as a selection criterion because galaxies only arise if the cosmological constant is tiny, comparable to the observed one. According to the anthropic principle, we don't need to explain why the constants of Nature don't take values incompatible with life; if they did, there would be no one who could complain that the hopes for life were doomed.
The physicists defending the so-called "anthropic principle" usually say that there is no other selection criterion aside from the condition that galaxies and life may arise in the Universe. Moreover, they often assume that all the Universes with equal chances to produce life have the same "prior probability" - which means that we should live in a typical or random or average Universe among those that are compatible with life. As they usually admit, the actual probability distribution on the Universes is unknown and even in principle, we don't know an algorithm to calculate it. Even the most hardcore anthropic physicists know that they don't know whether the chances of a particular vacuum are increased by its longer lifetime, bigger volume, bigger expectation value of planets with life, and other factors. All these things are unknown and arguments favoring one answer over others are, so far, philosophical in character, not scientific.
If the KKLT or related construction is right and we live in a rather random Universe selected from the string theory solutions, then string theory solves the cosmological constant problem. String theory surely admits solutions with nonzero values of the cosmological constant. And a sufficient number of candidates that produce a cosmological constant of a random size is predicted and the emergence of life is a sufficient criterion to choose a solution that agrees with the real world. In practice, it could be difficult to locate the right solution if there are googols (or powers of googol) of candidates and ours is pretty much random - and the other parameters of particle physics depend on the properties of the compactification and fluxes somewhat randomly, too.
Other physicists, such as myself, are convinced that a deeper understanding of the structure of the space of vacua - the landscape - and the cosmological mechanisms that may related or comparing different places of the landscape will show that our vacuum is much more special and physics may ultimately become able to determine in which one we live. We're not there yet but it is fair to say that string theory has produced an internally consistent solution that may produce stable vacua with all the qualitative particle species and interactions we know as well as a reasonable value of the cosmological constant. Needless to say, it's the only framework known to modern physics that can offer such a realistic candidate description.
We won't know for some time whether this candidate description is valid. In particular, even if string theory is valid which is almost certainly the case, we won't know for a long time whether our Universe is close to one of the type IIB compactifications or whether it can only be described by another category of string models such as heterotic string theory; heterotic M-theory; type IIA braneworlds; M-theory on $G_2$ holonomy manifolds, or others. A much more detailed comparison of the vacua with the observed reality is needed to settle such questions.
Best wishes
Lubos
