The Connes-Lott program on the sphere

We describe the classical Schwinger model as a study of the projective modules over the algebra of complex-valued functions on the sphere. On these modules, classified by $\pi_2(S^2)$, we construct hermitian connections with values in the universal differential envelope which leads us to the Schwinger model on the sphere. The Connes-Lott program is then applied using the Hilbert space of complexified inhomogeneous forms with its Atiyah-Kaehler structure. It splits in two minimal left ideals of the Clifford algebra preserved by the Dirac-Kaehler operator D=i(d-delta). The induced representation of the universal differential envelope, in order to recover its differential structure, is divided by the unwanted differential ideal and the obtained quotient is the usual complexified de Rham exterior algebra over the sphere with Clifford action on the "spinors" of the Hilbert space. The subsequent steps of the Connes-Lott program allow to define a matter action, and the field action is obtained using the Dixmier trace which reduces to the integral of the curvature squared.Comment: 34 pages, Latex, submitted for publicatio

Connes-Lott model building on the two-sphere

In this work we examine generalized Connes-Lott models on the two-sphere. The Hilbert space of the continuum spectral triple is taken as the space of sections of a twisted spinor bundle, allowing for nontrivial topological structure (magnetic monopoles). The finitely generated projective module over the full algebra is also taken as topologically non-trivial, which is possible over $S^2$. We also construct a real spectral triple enlarging this Hilbert space to include "particle" and "anti-particle" fields.Comment: 57 pages, LATE

Carnot-Caratheodory metric and gauge fluctuation in Noncommutative Geometry

Gauge fields have a natural metric interpretation in terms of horizontal distance. The latest, also called Carnot-Caratheodory or subriemannian distance, is by definition the length of the shortest horizontal path between points, that is to say the shortest path whose tangent vector is everywhere horizontal with respect to the gauge connection. In noncommutative geometry all the metric information is encoded within the Dirac operator D. In the classical case, i.e. commutative, Connes's distance formula allows to extract from D the geodesic distance on a riemannian spin manifold. In the case of a gauge theory with a gauge field A, the geometry of the associated U(n)-vector bundle is described by the covariant Dirac operator D+A. What is the distance encoded within this operator ? It was expected that the noncommutative geometry distance d defined by a covariant Dirac operator was intimately linked to the Carnot-Caratheodory distance dh defined by A. In this paper we precise this link, showing that the equality of d and dh strongly depends on the holonomy of the connection. Quite interestingly we exhibit an elementary example, based on a 2 torus, in which the noncommutative distance has a very simple expression and simultaneously avoids the main drawbacks of the riemannian metric (no discontinuity of the derivative of the distance function at the cut-locus) and of the subriemannian one (memory of the structure of the fiber).Comment: published version with additional figures to make the proof more readable. Typos corrected in this ultimate versio

Use of dual carbon–chlorine isotope analysis to assess the degradation pathways of 1,1,1-trichloroethane in groundwater

Compound-specific isotope analysis (CSIA) is a powerful tool to track contaminant fate in groundwater. However, the application of CSIA to chlorinated ethanes has received little attention so far. These compounds are toxic and prevalent groundwater contaminants of environmental concern. The high susceptibility of chlorinated ethanes like 1,1,1-trichloroethane (1,1,1-TCA) to be transformed via different competing pathways (biotic and abiotic) complicates the assessment of their fate in the subsurface. In this study, the use of a dual C-Cl isotope approach to identify the active degradation pathways of 1,1,1- TCA is evaluated for the first time in an aerobic aquifer impacted by 1,1,1-TCA and trichloroethylene (TCE) with concentrations of up to 20 mg/L and 3.4 mg/L, respectively. The reaction-specific dual carbon-chlorine (C-Cl) isotope trends determined in a recent laboratory study illustrated the potential of a dual isotope approach to identify contaminant degradation pathways of 1,1,1-TCA. Compared to the dual isotope slopes (Δδ13C/Δδ37CI) previously determined in the laboratory for dehydrohalogenation/hydrolysis (DH/HY, 0.33 ± 0.04) and oxidation by persulfate (∞), the slope determined from field samples (0.6 ± 0.2, r2 = 0.75) is closer to the one observed for DH/HY, pointing to DH/HY as the predominant degradation pathway of 1,1,1-TCA in the aquifer. The observed deviation could be explained by a minor contribution of additional degradation processes. This result, along with the little degradation of TCE determined from isotope measurements, confirmed that 1,1,1-TCA is the main source of the 1,1-dichlorethylene (1,1-DCE) detected in the aquifer with concentrations of up to 10 mg/L. This study demonstrates that a dual C-Cl isotope approach can strongly improve the qualitative and quantitative assessment of 1,1,1-TCA degradation processes in the field

On Pythagoras' theorem for products of spectral triples

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and provide non-unital counter-examples inspired by K-homology.Comment: Paper slightly shortened to match the published version; Lett. Math. Phys. 201

Subclinical coronary atherosclerosis identified by coronary computed tomographic angiography in asymptomatic morbidly obese patients

Obesity is a common public health problem and obese individuals in particular have a disproportionate incidence of acute coronary events. This study was undertaken to identify coronary artery lesions as well as associated clinical features, risk factors and demographics in patients with a body mass index (BMI) >40 kg/m2 without known coronary artery disease (CAD). Morbidly obese subjects were prospectively recruited to undergo coronary computed tomographic angiography (CCTA) using a dual-source computed tomography (CT) system. CAD was defined as the presence of any atherosclerotic lesion in any one coronary artery segment. The presence, location, and severity of atherosclerosis were related to patient characteristics. Forty-one patients (28 women, mean age, 50.4±10.0 years, mean BMI, 43.8±4.8 kg/m2) served as the study population. Of these, 25 patients (61%) had at least one coronary stenosis. All but 2 patients within the CAD cohort had coronary artery calcium (CAC) scores >0, and most plaques identified (75.4%) were non-calcified. There was a predilection of calcified and non-calcified atherosclerosis involving the left anterior descending (LAD) coronary artery compared with other coronary segments. Univariate predictors of CAD included older age, dyslipidemia, and diabetes. In this preliminary study of young morbidly obese patients, CCTA detected a high prevalence of calcified and non-calcified CAD, although the later predominated

Dirac Operators on Coset Spaces

The Dirac operator for a manifold Q, and its chirality operator when Q is even dimensional, have a central role in noncommutative geometry. We systematically develop the theory of this operator when Q=G/H, where G and H are compact connected Lie groups and G is simple. An elementary discussion of the differential geometric and bundle theoretic aspects of G/H, including its projective modules and complex, Kaehler and Riemannian structures, is presented for this purpose. An attractive feature of our approach is that it transparently shows obstructions to spin- and spin_c-structures. When a manifold is spin_c and not spin, U(1) gauge fields have to be introduced in a particular way to define spinors. Likewise, for manifolds like SU(3)/SO(3), which are not even spin_c, we show that SU(2) and higher rank gauge fields have to be introduced to define spinors. This result has potential consequences for string theories if such manifolds occur as D-branes. The spectra and eigenstates of the Dirac operator on spheres S^n=SO(n+1)/SO(n), invariant under SO(n+1), are explicitly found. Aspects of our work overlap with the earlier research of Cahen et al..Comment: section on Riemannian structure improved, references adde