Eisenstein series and automorphic representations

We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adeles A, thereby also paving the way for connections to number theory, representation theory and the Langlands program. Most of the results we present are already scattered throughout the mathematics literature but our exposition collects them together and is driven by examples. Many interesting aspects of these functions are hidden in their Fourier coefficients with respect to unipotent subgroups and a large part of our focus is to explain and derive general theorems on these Fourier expansions. Specifically, we give complete proofs of Langlands' constant term formula for Eisenstein series on adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic spherical Whittaker vector associated to unramified automorphic representations of G(Q_p). Somewhat surprisingly, all these results have natural interpretations as encoding physical effects in string theory. We therefore introduce also some basic concepts of string theory, aimed toward mathematicians, emphasising the role of automorphic forms. In addition, we explain how the classical theory of Hecke operators fits into the modern theory of automorphic representations of adelic groups, thereby providing a connection with some key elements in the Langlands program, such as the Langlands dual group LG and automorphic L-functions. Our treatise concludes with a detailed list of interesting open questions and pointers to additional topics where automorphic forms occur in string theory

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

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

Duality Symmetries and G^{+++} Theories

We show that the non-linear realisations of all the very extended algebras G^{+++}, except the B and C series which we do not consider, contain fields corresponding to all possible duality symmetries of the on-shell degrees of freedom of these theories. This result also holds for G_2^{+++} and we argue that the non-linear realisation of this algebra accounts precisely for the form fields present in the corresponding supersymmetric theory. We also find a simple necessary condition for the roots to belong to a G^{+++} algebra.Comment: 35 pages. v2: 2 appendices added, other minor corrections. v3: tables corrected, other minor changes, one appendix added, refs. added. Version published in Class. Quant. Gra

Malaria control – two years' use of insecticide treated bednets compared with insecticide house spraying in Kwazulu-Natal

Objectives_ The objective of this study was to produce data indicating whether insecticide-treated bednets should replac insecticide house spraying as a malaria control method in South Africa_ We report 2 years of preliminary data on malaria incidence comparing areas receiving insecticidetreated bednets and those subjected to house spraying in northern KwaZulu-Natal.Design, setting and subjects. In order to measure significant reductions in malaria incidence between the two interventions, a geographical information system (GIS) was used to identify and create seven pairs of geographical blood ; (areas) in the malaria high-risk areas of Ndumu and Makani in Ingwavuma magisterial district, KwaZulu-Natal, Individual blocks were then randomly allocated to either insecticide-treated bednets or house spraying with deltamethrin. Malaria cases were either routinely recorded by surveillance agents at home or were reported to the nearest health facility_Results and conclusions. The results show that 2 years' use of insecticide-treated bednets by communities in Ndumu and Makanis, KwaZulu-Natal, significantly reduced the malaria incidence both in 1997 (rate ratio (RR) =0_879, 95% confidence interval (Cn 0.80 - 0.95, P =0.04) and in 1998 (RR = 0.667, Cl 0_61 - 0.72, P = 0.0001). Using a t-test, these significant reductions were further confirmed by an assessment of the rate of change between 1996 and 1998, showing a 16% reduction in malaria incidence in blocks using bednets and an increase of 45% in sprayed areas (t = 2.534, P = 0.026 (12 df». In order to decide whether bednets : should replace house spraying in South Africa, we need more : data on the efficacy of treated bednets, their long-term acceptability and the cost of the two interventions

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

On the full, strongly exceptional collections on toric varieties with Picard number three

We investigate full strongly exceptional collections on smooth, com- plete toric varieties. We obtain explicit results for a large family of varieties with Picard number three, containing many of the families already known. We also describe the relations between the collections and the split of the push forward of the trivial line bundle by the toric Frobenius morphism

Flux compactification on smooth, compact three-dimensional toric varieties

Three-dimensional smooth, compact toric varieties (SCTV), when viewed as real six-dimensional manifolds, can admit G-structures rendering them suitable for internal manifolds in supersymmetric flux compactifications. We develop techniques which allow us to systematically construct G-structures on SCTV and read off their torsion classes. We illustrate our methods with explicit examples, one of which consists of an infinite class of toric CP^1 bundles. We give a self-contained review of the relevant concepts from toric geometry, in particular the subject of the classification of SCTV in dimensions less or equal to 3. Our results open up the possibility for a systematic construction and study of supersymmetric flux vacua based on SCTV.Comment: 27 pages, 10 figures; v2: references, minor typos & improvement

Selective Deuterium Ion Acceleration Using the Vulcan PW Laser

We report on the successful demonstration of selective acceleration of deuterium ions by target-normal sheath acceleration (TNSA) with a high-energy petawatt laser. TNSA typically produces a multi-species ion beam that originates from the intrinsic hydrocarbon and water vapor contaminants on the target surface. Using the method first developed by Morrison, et al.,$^{1}$ an ion beam with $>$99$\%$ deuterium ions and peak energy 14 MeV/nucleon is produced with a 200 J, 700 fs, $>10^{20} W/cm^{2}$ laser pulse by cryogenically freezing heavy water (D$_{2}$O) vapor onto the rear surface of the target prior to the shot. Within the range of our detectors (0-8.5$^{\circ}$), we find laser-to-deuterium-ion energy conversion efficiency of 4.3$\%$ above 0.7 MeV/nucleon while a conservative estimate of the total beam gives a conversion efficiency of 9.4$\%$.Comment: 5 pages, 5 figure