An Affine String Vertex Operator Construction at Arbitrary Level

An affine vertex operator construction at arbitrary level is presented which is based on a completely compactified chiral bosonic string whose momentum lattice is taken to be the (Minkowskian) affine weight lattice. This construction is manifestly physical in the sense of string theory, i.e., the vertex operators are functions of DDF ``oscillators'' and the Lorentz generators, both of which commute with the Virasoro constraints. We therefore obtain explicit representations of affine highest weight modules in terms of physical (DDF) string states. This opens new perspectives on the representation theory of affine Kac-Moody algebras, especially in view of the simultaneous treatment of infinitely many affine highest weight representations of arbitrary level within a single state space as required for the study of hyperbolic Kac-Moody algebras. A novel interpretation of the affine Weyl group as the ``dimensional null reduction'' of the corresponding hyperbolic Weyl group is given, which follows upon re-expression of the affine Weyl translations as Lorentz boosts.Comment: 15 pages, LaTeX2e, packages amsfonts, amssymb, xspace; final version to appear in J. Math. Phy

The Sugawara generators at arbitrary level

We construct an explicit representation of the Sugawara generators for arbitrary level in terms of the homogeneous Heisenberg subalgebra, which generalizes the well-known expression at level 1. This is achieved by employing a physical vertex operator realization of the affine algebra at arbitrary level, in contrast to the Frenkel--Kac--Segal construction which uses unphysical oscillators and is restricted to level 1. At higher level, the new operators are transcendental functions of DDF ``oscillators'' unlike the quadratic expressions for the level-1 generators. An essential new feature of our construction is the appearance, beyond level 1, of new types of poles in the operator product expansions in addition to the ones at coincident points, which entail (controllable) non-localities in our formulas. We demonstrate the utility of the new formalism by explicitly working out some higher-level examples. Our results have important implications for the problem of constructing explicit representations for higher-level root spaces of hyperbolic Kac--Moody algebras, and $E_{10}$ in particular.Comment: 17 pages, 1 figure, LaTeX2e, amsfonts, amssymb, xspace, PiCTe

On the Imaginary Simple Roots of the Borcherds Algebra gII9,1g_{II_{9,1}}

In a recent paper (hep-th/9703084) it was conjectured that the imaginary simple roots of the Borcherds algebra $g_{II_{9,1}}$ at level 1 are its only ones. We here propose an independent test of this conjecture, establishing its validity for all roots of norm $\geq -8$. However, the conjecture fails for roots of norm -10 and beyond, as we show by computing the simple multiplicities down to norm -24, which turn out to be remakably small in comparison with the corresponding $E_{10}$ multiplicities. Our derivation is based on a modified denominator formula combining the denominator formulas for $E_{10}$ and $g_{II_{9,1}}$, and provides an efficient method for determining the imaginary simple roots. In addition, we compute the $E_{10}$ multiplicities of all roots up to height 231, including levels up to $\ell =6$ and norms -42.Comment: 14 pages, LaTeX2e, packages amsmath, amsfonts, amssymb, amsthm, xspace, pstricks, longtable; substantially extended, appendix with new $E_{10}$ root multiplicities adde

Missing Modules, the Gnome Lie Algebra, and E10E_{10}

We study the embedding of Kac-Moody algebras into Borcherds (or generalized Kac-Moody) algebras which can be explicitly realized as Lie algebras of physical states of some completely compactified bosonic string. The extra ``missing states'' can be decomposed into irreducible highest or lowest weight ``missing modules'' w.r.t. the relevant Kac-Moody subalgebra; the corresponding lowest weights are associated with imaginary simple roots whose multiplicities can be simply understood in terms of certain polarization states of the associated string. We analyse in detail two examples where the momentum lattice of the string is given by the unique even unimodular Lorentzian lattice $II_{1,1}$ or $II_{9,1}$, respectively. The former leads to the Borcherds algebra $g_{1,1}$, which we call ``gnome Lie algebra", with maximal Kac-Moody subalgebra $A_1$. By the use of the denominator formula a complete set of imaginary simple roots can be exhibited, whereas the DDF construction provides an explicit Lie algebra basis in terms of purely longitudinal states of the compactified string in two dimensions. The second example is the Borcherds algebra $g_{9,1}$, whose maximal Kac-Moody subalgebra is the hyperbolic algebra $E_{10}$. The imaginary simple roots at level 1, which give rise to irreducible lowest weight modules for $E_{10}$, can be completely characterized; furthermore, our explicit analysis of two non-trivial level-2 root spaces leads us to conjecture that these are in fact the only imaginary simple roots for $g_{9,1}$.Comment: 31 pages, LaTeX2e, AMS packages, PSTRICK

Diabetes in Sub Saharan Africa 1999-2011: Epidemiology and Public Health Implications. A Systematic Review.

Diabetes prevalence is increasing globally, and Sub-Saharan Africa is no exception. With diverse health challenges, health authorities in Sub-Saharan Africa and international donors need robust data on the epidemiology and impact of diabetes in order to plan and prioritise their health programmes. This paper aims to provide a comprehensive and up-to-date review of the epidemiological trends and public health implications of diabetes in Sub-Saharan Africa. We conducted a systematic literature review of papers published on diabetes in Sub-Saharan Africa 1999-March 2011, providing data on diabetes prevalence, outcomes (chronic complications, infections, and mortality), access to diagnosis and care and economic impact. Type 2 diabetes accounts for well over 90% of diabetes in Sub-Saharan Africa, and population prevalence proportions ranged from 1% in rural Uganda to 12% in urban Kenya. Reported type 1 diabetes prevalence was low and ranged from 4 per 100,000 in Mozambique to 12 per 100,000 in Zambia. Gestational diabetes prevalence varied from 0% in Tanzania to 9% in Ethiopia. Proportions of patients with diabetic complications ranged from 7-63% for retinopathy, 27-66% for neuropathy, and 10-83% for microalbuminuria. Diabetes is likely to increase the risk of several important infections in the region, including tuberculosis, pneumonia and sepsis. Meanwhile, antiviral treatment for HIV increases the risk of obesity and insulin resistance. Five-year mortality proportions of patients with diabetes varied from 4-57%. Screening studies identified high proportions (> 40%) with previously undiagnosed diabetes, and low levels of adequate glucose control among previously diagnosed diabetics. Barriers to accessing diagnosis and treatment included a lack of diagnostic tools and glucose monitoring equipment and high cost of diabetes treatment. The total annual cost of diabetes in the region was estimated at US$67.03 billion, or US$8836 per diabetic patient. Diabetes exerts a significant burden in the region, and this is expected to increase. Many diabetic patients face significant challenges accessing diagnosis and treatment, which contributes to the high mortality and prevalence of complications observed. The significant interactions between diabetes and important infectious diseases highlight the need and opportunity for health planners to develop integrated responses to communicable and non-communicable diseases

AdS3_3 vacua and RG flows in three dimensional gauged supergravities

We study $AdS_3$ supersymmetric vacua in N=4 and N=8, three dimensional gauged supergravities, with scalar manifolds $(\frac{SO(4,4)}{SO(4)\times SO(4)})^2$ and $\frac{SO(8,8)}{SO(8)\times SO(8)}$, non-semisimple Chern-Simons gaugings $SO(4)\ltimes {\bf R}^6$ and $(SO(4)\ltimes {\bf R}^6)^2$, respectively. These are in turn equivalent to SO(4) and $SO(4)\times SO(4)$ Yang-Mills theories coupled to supergravity. For the N=4 case, we study renormalization group flows between UV and IR $AdS_3$ vacua with the same amount of supersymmetry: in one case, with (3,1) supersymmetry, we can find an analytic solution whereas in another, with (2,0) supersymmetry, we give a numerical solution. In both cases, the flows turn out to be v.e.v. flows, i.e. they are driven by the expectation value of a relevant operator in the dual $SCFT_2$. These provide examples of v.e.v. flows between two $AdS_3$ vacua within a gauged supergravity framework.Comment: 35 pages in JHEP form, 3 figures, typos corrected, references adde

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

Determination of the trap-assisted recombination strength in polymer light emitting diodes

The recombination processes in poly(p-phenylene vinylene) based polymer light-emitting diodes (PLEDs) are investigated. Photogenerated current measurements on PLED device structures reveal that next to the known Langevin recombination also trap-assisted recombination is an important recombination channel in PLEDs, which has not been considered until now. The dependence of the open-circuit voltage on light intensity enables us to determine the strength of this process. Numerical modeling of the current-voltage characteristics incorporating both Langevin and trap-assisted recombination yields a correct and consistent description of the PLED, without the traditional correction of the Langevin prefactor. At low bias voltage the trap-assisted recombination rate is found to be dominant over the free carrier recombination rate.

A lattice study of the two-dimensional Wess Zumino model

We present results from a numerical simulation of the two-dimensional Euclidean Wess-Zumino model. In the continuum the theory possesses N=1 supersymmetry. The lattice model we employ was analyzed by Golterman and Petcher in \cite{susy} where a perturbative proof was given that the continuum supersymmetric Ward identities are recovered without finite tuning in the limit of vanishing lattice spacing. Our simulations demonstrate the existence of important non-perturbative effects in finite volumes which modify these conclusions. It appears that in certain regions of parameter space the vacuum state can contain solitons corresponding to field configurations which interpolate between different classical vacua. In the background of these solitons supersymmetry is partially broken and a light fermion mode is observed. At fixed coupling the critical mass separating phases of broken and unbroken supersymmetry appears to be volume dependent. We discuss the implications of our results for continuum supersymmetry breaking.Comment: 32 pages, 12 figure