Connes' Tangent Groupoid and Strict Quantization

We address one of the open problems in quantization theory recently listed by Rieffel. By developping in detail Connes' tangent groupoid principle and using previous work by Landsman, we show how to construct a strict, flabby quantization, which is moreover an asymptotic morphism and satisfies the reality and traciality constraints, on any oriented Riemannian manifold. That construction generalizes the standard Moyal rule. The paper can be considered as an introduction to quantization theory from Connes' point of view.Comment: LaTeX file, 22 pages (elsart.cls required). Minor changes. Final version to appear in J. Geom. and Phy

Groupoids and an index theorem for conical pseudo-manifolds

We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold $M$. A main ingredient is a non-commutative algebra that plays in our setting the role of $C_0(T^*M)$. We prove a Thom isomorphism between non-commutative algebras which gives a new example of wrong way functoriality in $K$-theory. We then give a new proof of the Atiyah-Singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds

A cohomological formula for the Atiyah-Patodi-Singer index on manifolds with boundary

International audienceWe give a cohomological formula for the index of a fully elliptic pseudodifferential operator on a manifold with boundary. As in the classic case of Atiyah-Singer, we use an embedding into an euclidean space to express the index as the integral of a cohomology class depending in this case on a noncommutative symbol, the integral being over a $C^\infty$-manifold called the singular normal bundle associated to the embedding. The formula is based on a K-theoretical Atiyah-Patodi-Singer theorem for manifolds with boundary that is drawn from Connes' tangent groupoid approach

Ultrathin 2 nm gold as ideal impedance-matched absorber for infrared light

Thermal detectors are a cornerstone of infrared (IR) and terahertz (THz) technology due to their broad spectral range. These detectors call for suitable broad spectral absorbers with minimalthermal mass. Often this is realized by plasmonic absorbers, which ensure a high absorptivity butonly for a narrow spectral band. Alternativly, a common approach is based on impedance-matching the sheet resistance of a thin metallic film to half the free-space impedance. Thereby, it is possible to achieve a wavelength-independent absorptivity of up to 50 %, depending on the dielectric properties of the underlying substrate. However, existing absorber films typicallyrequire a thickness of the order of tens of nanometers, such as titanium nitride (14 nm), whichcan significantly deteriorate the response of a thermal transducers. Here, we present the application of ultrathin gold (2 nm) on top of a 1.2 nm copper oxide seed layer as an effective IR absorber. An almost wavelength-independent and long-time stable absorptivity of 47(3) %, ranging from 2 $\mu$m to 20 $\mu$m, could be obtained and is further discussed. The presented gold thin-film represents analmost ideal impedance-matched IR absorber that allows a significant improvement of state-of-the-art thermal detector technology

A Short Survey of Noncommutative Geometry

We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the Riemann-Hilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geometric space, based on spectral considerations, which is central in the theory. As a simple illustration of the spectral formulation of geometry in the ordinary commutative case, we give a polynomial equation for geometries on the four dimensional sphere with fixed volume. The equation involves an idempotent e, playing the role of the instanton, and the Dirac operator D. It expresses the gamma five matrix as the pairing between the operator theoretic chern characters of e and D. It is of degree five in the idempotent and four in the Dirac operator which only appears through its commutant with the idempotent. It determines both the sphere and all its metrics with fixed volume form. We also show using the noncommutative analogue of the Polyakov action, how to obtain the noncommutative metric (in spectral form) on the noncommutative tori from the formal naive metric. We conclude on some questions related to string theory.Comment: Invited lecture for JMP 2000, 45

Quantized reduction as a tensor product

Symplectic reduction is reinterpreted as the composition of arrows in the category of integrable Poisson manifolds, whose arrows are isomorphism classes of dual pairs, with symplectic groupoids as units. Morita equivalence of Poisson manifolds amounts to isomorphism of objects in this category. This description paves the way for the quantization of the classical reduction procedure, which is based on the formal analogy between dual pairs of Poisson manifolds and Hilbert bimodules over C*-algebras, as well as with correspondences between von Neumann algebras. Further analogies are drawn with categories of groupoids (of algebraic, measured, Lie, and symplectic type). In all cases, the arrows are isomorphism classes of appropriate bimodules, and their composition may be seen as a tensor product. Hence in suitable categories reduction is simply composition of arrows, and Morita equivalence is isomorphism of objects.Comment: 44 pages, categorical interpretation adde