4.2 String theory

Unlike supergravity, string theory appears to be a consistent and well-defined theory of quantum gravity, and therefore calculating the value of the cosmological constant should, at least in principle, be possible. On the other hand, the number of vacuum states seems to be quite large, and none of them (to the best of our current knowledge) features three large spatial dimensions, broken supersymmetry, and a small cosmological constant. At the same time, there are reasons to believe that any realistic vacuum of string theory must be strongly coupled [136]; therefore, our inability to find an appropriate solution may simply be due to the technical difficulty of the problem. (For general introductions to string theory, see [137, 138]; for cosmological issues, see [139, 140]).

String theory is naturally formulated in more than four spacetime dimensions. Studies of duality symmetries have revealed that what used to be thought of as five distinct ten-dimensional superstring theories - Type I, Types IIA and IIB, and heterotic theories based on gauge groups E(8) x E(8) and SO(32) - are, along with eleven-dimensional supergravity, different low-energy weak-coupling limits of a single underlying theory, sometimes known as M-theory. In each of these six cases, the solution with the maximum number of uncompactified, flat spacetime dimensions is a stable vacuum preserving all of the supersymmetry. To bring the theory closer to the world we observe, the extra dimensions can be compactified on a manifold whose Ricci tensor vanishes. There are a large number of possible compactifications, many of which preserve some but not all of the original supersymmetry. If enough SUSY is preserved, the vacuum energy will remain zero; generically there will be a manifold of such states, known as the moduli space.

Of course, to describe our world we want to break all of the supersymmetry. Investigations in contexts where this can be done in a controlled way have found that the induced cosmological constant vanishes at the classical level, but a substantial vacuum energy is typically induced by quantum corrections [137]. Moore [141] has suggested that Atkin-Lehner symmetry, which relates strong and weak coupling on the string worldsheet, can enforce the vanishing of the one-loop quantum contribution in certain models (see also [142, 143]); generically, however, there would still be an appreciable contribution at two loops.

Thus, the search is still on for a four-dimensional string theory vacuum with broken supersymmetry and vanishing (or very small) cosmological constant. (See [144] for a general discussion of the vacuum problem in string theory.) The difficulty of achieving this in conventional models has inspired a number of more speculative proposals, which I briefly list here.

• In three spacetime dimensions supersymmetry can remain unbroken, maintaining a zero cosmological constant, in such a way as to break the mass degeneracy between bosons and fermions [145]. This mechanism relies crucially on special properties of spacetime in (2+1) dimensions, but in string theory it sometimes happens that the strong-coupling limit of one theory is another theory in one higher dimension [146, 147].

• More generally, it is now understood that (at least in some circumstances) string theory obeys the ``holographic principle'', the idea that a theory with gravity in D dimensions is equivalent to a theory without gravity in D - 1 dimensions [148, 149]. In a holographic theory, the number of degrees of freedom in a region grows as the area of its boundary, rather than as its volume. Therefore, the conventional computation of the cosmological constant due to vacuum fluctuations conceivably involves a vast overcounting of degrees of freedom. We might imagine that a more correct counting would yield a much smaller estimate of the vacuum energy [150, 151, 152, 153], although no reliable calculation has been done as yet.

• The absence of manifest SUSY in our world leads us to ask whether the beneficial aspect of canceling contributions to the vacuum energy could be achieved even without a truly supersymmetric theory. Kachru, Kumar and Silverstein [154] have constructed such a string theory, and argue that the perturbative contributions to the cosmological constant should vanish (although the actual calculations are somewhat delicate, and not everyone agrees [155]). If such a model could be made to work, it is possible that small non-perturbative effects could generate a cosmological constant of an astrophysically plausible magnitude [156].

• A novel approach to compactification starts by imagining that the fields of the Standard Model are confined to a (3+1)-dimensional manifold (or ``brane'', in string theory parlance) embedded in a larger space. While gravity is harder to confine to a brane, phenomenologically acceptable scenarios can be constructed if either the extra dimensions are any size less than a millimeter [157, 158, 159, 160, 161], or if there is significant spacetime curvature in a non-compact extra dimension [162, 163, 164]. Although these scenarios do not offer a simple solution to the cosmological constant problem, the relationship between the vacuum energy and the expansion rate can differ from our conventional expectation (see for example [165, 166]), and one is free to imagine that further study may lead to a solution in this context (see for example [167, 168]).

Of course, string theory might not be the correct description of nature, or its current formulation might not be directly relevant to the cosmological constant problem. For example, a solution may be provided by loop quantum gravity [169], or by a composite graviton [170]. It is probably safe to believe that a significant advance in our understanding of fundamental physics will be required before we can demonstrate the existence of a vacuum state with the desired properties. (Not to mention the equally important question of why our world is based on such a state, rather than one of the highly supersymmetric states that appear to be perfectly good vacua of string theory.)