Quantum Geometry and Quantum Gravity

The purpose of this contribution is to give an introduction to quantum geometry and loop quantum gravity for a wide audience of both physicists and mathematicians. From a physical point of view the emphasis will be on conceptual issues concerning the relationship of the formalism with other more traditional approaches inspired in the treatment of the fundamental interactions in the standard model. Mathematically I will pay special attention to functional analytic issues, the construction of the relevant Hilbert spaces and the definition and properties of geometric operators: areas and volumes.Comment: To appear in the AIP Conference Proceedings of the XVI International Fall Workshop on Geometry and Physics, Lisbon - Portugal, 5-8 September 200

Non-abelian gauge antisymmetric tensor fields

We construct the theory of non-abelian gauge antisymmetric tensor fields, which generalize the standard Yang-MIlls fields and abelian gauge p-forms. The corresponding gauge group acts on the space of inhomogeneous differential forms and it is shown to be a supergroup. The wide class of generalized Chern-Simons actions is constructed.Comment: 20 pages, Late

Quantum geometry from 2+1 AdS quantum gravity on the torus

Wilson observables for 2+1 quantum gravity with negative cosmological constant, when the spatial manifold is a torus, exhibit several novel features: signed area phases relate the observables assigned to homotopic loops, and their commutators describe loop intersections, with properties that are not yet fully understood. We describe progress in our study of this bracket, which can be interpreted as a q-deformed Goldman bracket, and provide a geometrical interpretation in terms of a quantum version of Pick's formula for the area of a polygon with integer vertices.Comment: 19 pages, 11 figures, revised with more explanations, improved figures and extra figures. To appear GER

Local Ugandan Production of Stable 0.2% Chlorhexidine Eye Drops

Purpose: The purpose of this study was to develop a protocol to prepare buffered chlorhexidine (CHX) eye drops (0.2% w/v) in the United Kingdom that can be reproduced at a production facility in Uganda. Buffered CHX eye drops can prevent CHX degradation and improve ocular tolerability during the treatment of fungal keratitis. Methods: Buffered CHX eye drops in amber glass containers were prepared using sodium acetate buffer at pH 5.90 to 6.75. Two commercial CHX solutions and CHX in water were used as controls. Eye drops were stored at 40°C (70% humidity, 21 months) in the United Kingdom and at ambient temperature in Uganda (30 months). High-performance liquid chromatography was used to determine CHX stability over time, and pH was monitored. Sterility was achieved using an autoclave (121°C, 15 minutes) and water bath (100°C, 30 minutes). Results: The pH of acetate-buffered CHX eye drops did not change over 21 months a40°C or at ambient temperature (30 months), whereas the pH of the unbuffered aqueouCHX displayed significant fluctuations, with an increase in acidity. The CHX concentration remained the same in both buffered and unbuffered eye-drop solutions. Eye dropsterilization was successful using an autoclave and a water bath. Conclusions: Stable, sterile, buffered CHX eye drops (pH 6.75) were successfully prepared first in the United Kingdom and then reproducibly in Uganda. This eye drops can be prepared in a hospital or pharmacy setting with limited resources, thus providing a cost-effective treatment for fungal keratitis. Translational Relevance: A protocol has been developed to prepare buffered CHX eydrops in low-and middle-income countries to treat fungal keratitis

Nonperturbative 2D Gravity, Punctured Spheres and Θ\Theta-Vacua in String Theories

We consider a model of 2D gravity with the coefficient of the Einstein-Hilbert action having an imaginary part $\pi/2$. This is equivalent to introduce a $\Theta$-vacuum structure in the genus expansion whose effect is to convert the expansion into a series of alternating signs, presumably Borel summable. We show that the specific heat of the model has a physical behaviour. It can be represented nonperturbatively as a series in terms of integrals over moduli spaces of punctured spheres and the sum of the series can be rewritten as a unique integral over a suitable moduli space of infinitely punctured spheres. This is an explicit realization \`a la Friedan-Shenker of 2D quantum gravity. We conjecture that the expansion in terms of punctures and the genus expansion can be derived using the Duistermaat-Heckman theorem. We briefly analyze expansions in terms of punctured spheres also for multicritical models.Comment: 12 pages + 2 figures available upon request. LaTeX Fil

Non-holomorphic terms in N=2 SUSY Wilsonian actions and RG equation

In this paper we first investigate the Ansatz of one of the present authors for K(\Psi,\bar\Psi), the adimensional modular invariant non-holomorphic correction to the Wilsonian effective Lagrangian of an N=2 globally supersymmetric gauge theory. The renormalisation group beta-function of the theory crucially allows us to express K(\Psi,\bar\Psi) in a form that easily generalises to the case in which the theory is coupled to N_F hypermultiplets in the fundamental representation of the gauge group. This function satisfies an equation which should be viewed as a fully non-perturbative ``non-chiral superconformal Ward identity". We also determine its renormalisation group equation. Furthermore, as a first step towards checking the validity of this Ansatz, we compute the contribution to K(\Psi,\bar\Psi) from instantons of winding number k=1 and k=2. As a by-product of our analysis we check a non-renormalisation theorem for N_F=4.Comment: 39 pages, LaTex file, no figure

Abelian gerbes as a gauge theory of quantum mechanics on phase space

We construct a U(1) gerbe with a connection over a finite-dimensional, classical phase space P. The connection is given by a triple of forms A,B,H: a potential 1-form A, a Neveu-Schwarz potential 2-form B, and a field-strength 3-form H=dB. All three of them are defined exclusively in terms of elements already present in P, the only external input being Planck's constant h. U(1) gauge transformations acting on the triple A,B,H are also defined, parametrised either by a 0-form or by a 1-form. While H remains gauge invariant in all cases, quantumness vs. classicality appears as a choice of 0-form gauge for the 1-form A. The fact that [H]/2i\pi is an integral class in de Rham cohomology is related with the discretisation of symplectic area on P. This is an equivalent, coordinate-free reexpression of Heisenberg's uncertainty principle. A choice of 1-form gauge for the 2-form B relates our construction with generalised complex structures on classical phase space. Altogether this allows one to interpret the quantum mechanics corresponding to P as an Abelian gauge theory.Comment: 18 pages, 1 figure available from the authors upon reques

Immunoblot analysis of the seroreactivity to recombinant Borrelia burgdorferi sensu lato antigens, including VlsE, in the long-term course of treated patients with Erythema migrans

Objective: We evaluated whether immunoblotting is capable of substantiating the posttreatment clinical assessment of patients with erythema migrans ( EM), the hallmark of early Lyme borreliosis. Methods: In 50 patients, seroreactivity to different antigens of Borrelia burgdorferi sensu lato was analyzed by a recombinant immunoblot test (IB) in consecutive serum samples from a minimum follow-up period of 1 year. Antigens in the IgG test were decorin- binding protein A, internal fragment of p41 (p41i), outer surface protein C (OspC), p39, variable major protein-like sequence expressed (VlsE), p58 and p100; those in the IgM test were p41i, OspC and p39. Immune responses were correlated with clinical and treatment-related parameters. Results: Positive IB results were found in 50% before, in 57% directly after therapy and in 44% by the end of the follow-up for the IgG class, and in 36, 43 and 12% for the IgM class. In acute and convalescence phase sera, VlsE was most immunogenic on IgG testing 60 and 70%), and p41i (46 and 57%) and OspC (40 and 57%) for the IgM class. By the end of the follow-up, only the anti-p41i lgM response was significantly decreased to 24%. Conclusions: No correlation was found between IB results and treatment-related parameters. Thus, immunoblotting does not add to the clinical assessment of EM patients after treatment. Copyright (c) 2008 S. Karger AG, Basel

Supramolecular materials: molecular packing of tetranitrotetrapropoxycalix[4]arene in highly stable films with second-order nonlinear optical properties

Highly stable films of tetranitrotetrapropoxycalix[4]arene (9) with second-order nonlinear optical (NLO) properties and a noncentrosymmetric structure were obtained by a novel crystallization process at 130-140 degrees C in a de electric field. The packing of 9 in these films was elucidated by a combination of X-ray diffraction, angle-dependent second- harmonic generation, and scanning force microscopy (SFM). The experimental results agree well with solid-state molecular dynamics calculations for these films. No crystalline phase was observed for nitrocalix[4]arene derivatives with longer or branched alkyl chains; this explains the limited NLO stability of films of these calixarenes. Scanning force microscopy o­n the aligned films of 9 showed two distinct surface lattice structures: a rectangular lattice (a = 9.3, b = 11.7 Angstrom) and a pseudohexagonal lattice (d approximate to 11.4 Angstrom). The combination of these data with the interlayer distance of 8.9 Angstrom (X-ray diffraction) allowed the packing of molecules of 9 in these structures to be fully elucidated at the molecular level