We argue that a generic instability afflicts vacua that arise in theories whose moduli space has large dimension. Specifically, by studying theories with multiple scalar fields we provide numerical evidence that for a generic local minimum of the potential the usual semiclassical bubble nucleation rate, Γ = A e −B , increases rapidly as function of the number of fields in the theory. As a consequence, the fraction of vacua with tunneling rates low enough to maintain metastability appears to fall exponentially as a function of the moduli space dimension. We discuss possible implications for the landscape of string theory. Notably, if our results prove applicable to string theory, the landscape of metastable vacua may not contain sufficient diversity to offer a natural explanation of dark energy.

We investigate the quantum behaviour of sigma models on coset superspaces G/H defined by Z 2n gradings of G. We find that, whenever G has vanishing Killing form, there is a choice of WZ term which renders the model quantum conformal, at least to one loop. The choice coincides with that for which the model is known to be classically integrable. This generalizes results for models associated to Z 4 gradings, including IIB superstrings in AdS 5 × S 5 .Ricci-flat, which guarantees that sigma models on them are conformal to one-loop; the result of Bershadsky, Zhukov and Vaintrob in [8] is that they are exactly conformal. (The representation theory of the relevant superalgebras [11] thus becomes important.) This remarkable fact should be contrasted with the more familiar sigma models on bosonic groups, where a correctly normalized WZ term (which may be thought of as a parallelizing torsion [12]) must be added to the action if the theory is to be quantum conformal [13].It was subsequently shown by Berkovits, Bershadsky, Hauer, Zhukov and Zwiebach [14] that sigma models on certain quotients G/H of Ricci-flat supergroups G by bosonic subgroups are again conformal to one loop -and can be expected to be so exactly -given a suitable WZ term. These targets include the coset superspace P SU (2, 2|4)/SO(1, 4) × SO (5), as well as P SU (1, 1|2)/U (1) × U (1), whose bosonic geometry is AdS 2 × S 2 . The vital property in each case is that the isotropy subgroup H is the fixed point set of a Z 4 grading of G. This, in particular, allows the addition of the required WZ term.What is striking is that the Z 4 grading was also a key element in the demonstration by Bena, Polchinski and Roiban [15] that the sigma model on P SU (2, 2|4)/SO(1, 4) × SO(5) is classically integrable. (See also [25], which contains references to the extensive recent work on this subject.) It was shown by one of the present authors [16] that a similar construction holds, more generally, for sigma models on spaces G/H defined by Z m gradings: for any m, the grading permits the addition of a certain preferred WZ term, and with this WZ term the equations of motion may be put into Lax form, ensuring integrability, just as in the Z 4 case.Given this result, and the dual role played by the Z 4 -grading, it is natural to hope that conformal invariance is also present in sigma models whose targets are quotients of (again, Ricci-flat) supergroups G by subgroups H defined by gradings of order greater than 4. In this paper, we show that this is indeed the case, at least at one loop.The structure of this paper is as follows: in section 2 we introduce some notation and discuss supergroups with vanishing Killing form and related coset superspaces. We also write down the sigma model action with WZ term, and recall the one-loop beta-functions. In section 3 we compute the Ricci curvature of a torsion connection on a reductive homogeneous superspace, and then in section 4 we demonstrate that, for the connection whose torsion is given by the preferred WZ term, this curvature vani...

We investigate flux vacua on a variety of one-parameter Calabi-Yau compactifications, and find many examples that are connected through continuous monodromy transformations. For these, we undertake a detailed analysis of the tunneling dynamics and find that tunneling trajectories typically graze the conifold point-particular 3-cycles are forced to contract during such vacuum transitions. Physically, these transitions arise from the competing effects of minimizing the energy for brane nucleation (facilitating a change in flux), versus the energy cost associated with dynamical changes in the periods of certain Calabi-Yau 3-cycles. We find that tunneling occurs only when warping due to back-reaction from the flux through the shrinking cycle is properly taken into account.

Non-local conserved charges in two-dimensional sigma models with target spaces SO(2n)/SO(n)×SO(n) and Sp(2n)/Sp(n)×Sp(n) are shown to survive quantization, unspoiled by anomalies; these theories are therefore integrable at the quantum level. Local, higher-spin, conserved charges are also shown to survive quantization in the SO(2n)/SO(n)×SO(n) models.Classical, two-dimensional sigma models on compact symmetric spaces G/H are integrable by virtue of conserved quantities which can arise as integrals of local or non-local functions of the underlying fields (the accounts in [1]-[5] contain references to the extensive literature). Since these models are asymptotically free and strongly coupled in the infrared, their quantum properties are not straightforward to determine. Nevertheless, following Lüscher [6], Abdalla, Forger and Gomes showed [7] that, in a G/H sigma model with H simple 1 , the first conserved non-local charge survives quantization (after an appropriate renormalization [6,7,8]), which suffices to ensure quantum integrability of the theory. By contrast, calculations using the 1/N expansion reveal anomalies that spoil the conservation of the quantum non-local charges in the CP N −1 = SU(N)/SU(N−1)×U(1) models for N > 2, and in the wider class of theories based on the complex Grassmannians SU(N)/SU(n)×SU(N−n)×U(1) for N > n > 1 [9].It was long suspected, therefore, that the G/H sigma models were quantum integrable only for H simple. So it was something of a surprise when exact S-Matrices were proposed 1 Here, and throughout this paper, we shall use 'simple' to mean that the corresponding Lie algebra has no non-trivial ideals. Hence U (1) is simple in our terminology, in addition to the usual non-abelian simple groups of the Cartan-Killing classification [13].

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