Gradient Representations and Affine Structures in AE(n)

We study the indefinite Kac-Moody algebras AE(n), arising in the reduction of Einstein's theory from (n+1) space-time dimensions to one (time) dimension, and their distinguished maximal regular subalgebras sl(n) and affine A_{n-2}^{(1)}. The interplay between these two subalgebras is used, for n=3, to determine the commutation relations of the `gradient generators' within AE(3). The low level truncation of the geodesic sigma-model over the coset space AE(n)/K(AE(n)) is shown to map to a suitably truncated version of the SL(n)/SO(n) non-linear sigma-model resulting from the reduction Einstein's equations in (n+1) dimensions to (1+1) dimensions. A further truncation to diagonal solutions can be exploited to define a one-to-one correspondence between such solutions, and null geodesic trajectories on the infinite-dimensional coset space H/K(H), where H is the (extended) Heisenberg group, and K(H) its maximal compact subgroup. We clarify the relation between H and the corresponding subgroup of the Geroch group.Comment: 43 page

K(E10), Supergravity and Fermions

We study the fermionic extension of the E10/K(E10) coset model and its relation to eleven-dimensional supergravity. Finite-dimensional spinor representations of the compact subgroup K(E10) of E(10,R) are studied and the supergravity equations are rewritten using the resulting algebraic variables. The canonical bosonic and fermionic constraints are also analysed in this way, and the compatibility of supersymmetry with local K(E10) is investigated. We find that all structures involving A9 levels 0,1 and 2 nicely agree with expectations, and provide many non-trivial consistency checks of the existence of a supersymmetric extension of the E10/K(E10) coset model, as well as a new derivation of the `bosonic dictionary' between supergravity and coset variables. However, there are also definite discrepancies in some terms involving level 3, which suggest the need for an extension of the model to infinite-dimensional faithful representations of the fermionic degrees of freedom.Comment: 50 page

Counting supersymmetric branes

Maximal supergravity solutions are revisited and classified, with particular emphasis on objects of co-dimension at most two. This class of solutions includes branes whose tension scales with g_s^{-\sigma} for \sigma>2. We present a group theory derivation of the counting of these objects based on the corresponding tensor hierarchies derived from E11 and discrete T- and U-duality transformations. This provides a rationale for the wrapping rules that were recently discussed for \sigma<4 in the literature and extends them. Explicit supergravity solutions that give rise to co-dimension two branes are constructed and analysed.Comment: 1+33 pages. To the memory of Laurent Houart. v2: Published version with added reference

Supersymmetric quantum cosmological billiards

D=11 Supergravity near a space-like singularity admits a cosmological billiard description based on the hyperbolic Kac-Moody group E10. The quantization of this system via the supersymmetry constraint is shown to lead to wavefunctions involving automorphic (Maass wave) forms under the modular group W^+(E10)=PSL(2,O) with Dirichlet boundary conditions on the billiard domain. A general inequality for the Laplace eigenvalues of these automorphic forms implies that the wave function of the universe is generically complex and always tends to zero when approaching the initial singularity. We discuss possible implications of this result for the question of singularity resolution in quantum cosmology and comment on the differences with other approaches.Comment: 4 pages. v2: Added ref. Version to be published in PR

Spatial Patterns of Infant Mortality in Mali: The Effect of Malaria Endemicity

A spatial analysis was carried out to identify factors related to geographic differences in infant mortality risk in Mali by linking data from two spatially structured databases: the Demographic and Health Surveys of 1995-1996 and the Mapping Malaria Risk in Africa database for Mali. Socioeconomic factors measured directly at the individual level and site-specific malaria prevalence predicted for the Demographic and Health Surveys' locations by a spatial model fitted to the Mapping Malaria Risk in Africa database were examined as possible risk factors. The analysis was carried out by fitting a Bayesian hierarchical geostatistical logistic model to infant mortality risk, by Markov chain Monte Carlo simulation. It confirmed that mother's education, birth order and interval, infant's sex, residence, and mother's age at infant's birth had a strong impact on infant mortality risk in Mali. The residual spatial pattern of infant mortality showed a clear relation to well-known foci of malaria transmission, especially the inland delta of the Niger River. No effect of estimated parasite prevalence could be demonstrated. Possible explanations include confounding by unmeasured covariates and sparsity of the source malaria data. Spatial statistical models of malaria prevalence are useful for indicating approximate levels of endemicity over wide areas and, hence, for guiding intervention strategies. However, at points very remote from those sampled, it is important to consider prediction erro

Arguments for F-theory

After a brief review of string and $M$-Theory we point out some deficiencies. Partly to cure them, we present several arguments for ``$F$-Theory'', enlarging spacetime to $(2, 10)$ signature, following the original suggestion of C. Vafa. We introduce a suggestive Supersymmetric 27-plet of particles, associated to the exceptional symmetric hermitian space $E_{6}/Spin^{c}(10)$. Several possible future directions, including using projective rather than metric geometry, are mentioned. We should emphasize that $F$-Theory is yet just a very provisional attempt, lacking clear dynamical principles.Comment: To appear in early 2006 in Mod. Phys. Lett. A as Brief Revie

A dynamical inconsistency of Horava gravity

The dynamical consistency of the non-projectable version of Horava gravity is investigated by focusing on the asymptotically flat case. It is argued that for generic solutions of the constraint equations the lapse must vanish asymptotically. We then consider particular values of the coupling constants for which the equations are tractable and in that case we prove that the lapse must vanish everywhere -- and not only at infinity. Put differently, the Hamiltonian constraints are generically all second-class. We then argue that the same feature holds for generic values of the couplings, thus revealing a physical inconsistency of the theory. In order to cure this pathology, one might want to introduce further constraints but the resulting theory would then lose much of the appeal of the original proposal by Horava. We also show that there is no contradiction with the time reparametrization invariance of the action, as this invariance is shown to be a so-called "trivial gauge symmetry" in Horava gravity, hence with no associated first-class constraints.Comment: 28 pages, 2 references adde

G+++ Invariant Formulation of Gravity and M-Theories: Exact BPS Solutions

We present a tentative formulation of theories of gravity with suitable matter content, including in particular pure gravity in D dimensions, the bosonic effective actions of M-theory and of the bosonic string, in terms of actions invariant under very-extended Kac-Moody algebras G+++. We conjecture that they host additional degrees of freedom not contained in the conventional theories. The actions are constructed in a recursive way from a level expansion for all very-extended algebras G+++. They constitute non-linear realisations on cosets, a priori unrelated to space-time, obtained from a modified Chevalley involution. Exact solutions are found for all G+++. They describe the algebraic properties of BPS extremal branes, Kaluza-Klein waves and Kaluza-Klein monopoles. They illustrate the generalisation to all G+++ invariant theories of the well-known duality properties of string theories by expressing duality as Weyl invariance in G+++. Space-time is expected to be generated dynamically. In the level decomposition of E8+++ = E11, one may indeed select an A10 representation of generators Pa which appears to engender space-time translations by inducing infinite towers of fields interpretable as field derivatives in space and time.Comment: Latex 45 pages, 1 figure. Discussion on pages 19 and 20 altered. Appendix B amplified. 4 footnotes added. 2 references added. Acknowledgments updated. Additional minor correction

Eisenstein series for infinite-dimensional U-duality groups

We consider Eisenstein series appearing as coefficients of curvature corrections in the low-energy expansion of type II string theory four-graviton scattering amplitudes. We define these Eisenstein series over all groups in the E_n series of string duality groups, and in particular for the infinite-dimensional Kac-Moody groups E9, E10 and E11. We show that, remarkably, the so-called constant term of Kac-Moody-Eisenstein series contains only a finite number of terms for particular choices of a parameter appearing in the definition of the series. This resonates with the idea that the constant term of the Eisenstein series encodes perturbative string corrections in BPS-protected sectors allowing only a finite number of corrections. We underpin our findings with an extensive discussion of physical degeneration limits in D<3 space-time dimensions.Comment: 69 pages. v2: Added references and small additions, to be published in JHE