Spectral triples and Toeplitz operators

We give examples of spectral triples, in the sense of A. Connes, constructed using the algebra of Toeplitz operators on smoothly bounded strictly pseudoconvex domains in $C^n$, or the star product for the Berezin-Toeplitz quantization. Our main tool is the theory of generalized Toeplitz operators on the boundary of such domains, due to Boutet de Monvel and Guillemin.Comment: 31 page

Toeplitz Quantization and Asymptotic Expansions: Geometric Construction

For a real symmetric domain GR/KR, with complexification GC/KC, we introduce the concept of ''star-restriction'' (a real analogue of the ''star-products'' for quantization of Kähler manifolds) and give a geometric construction of the GR-invariant differential operators yielding its asymptotic expansion

Toeplitz operators on the domain {Z∈M2×2(C)∣ZZ∗<I}\{Z\in M_{2\times2}(\mathbb{C}) \mid Z Z^* < I\} with U(2)×T2\mathrm{U}(2)\times\mathbb{T}^2-invariant symbols

Let $D$ be the irreducible bounded symmetric domain of $2\times2$ complex matrices that satisfy $ZZ^* < I_2$. The biholomorphism group of $D$ is realized by $\mathrm{U}(2,2)$ with isotropy at the origin given by $\mathrm{U}(2)\times\mathrm{U}(2)$. Denote by $\mathbb{T}^2$ the subgroup of diagonal matrices in $\mathrm{U}(2)$. We prove that the set of $\mathrm{U}(2)\times\mathbb{T}^2$-invariant essentially bounded symbols yield Toeplitz operators that generate commutative $C^*$-algebras on all weighted Bergman spaces over $D$. Using tools from representation theory, we also provide an integral formula for the spectra of these Toeplitz operators

Emergent Geometry and Gravity from Matrix Models: an Introduction

A introductory review to emergent noncommutative gravity within Yang-Mills Matrix models is presented. Space-time is described as a noncommutative brane solution of the matrix model, i.e. as submanifold of \R^D. Fields and matter on the brane arise as fluctuations of the bosonic resp. fermionic matrices around such a background, and couple to an effective metric interpreted in terms of gravity. Suitable tools are provided for the description of the effective geometry in the semi-classical limit. The relation to noncommutative gauge theory and the role of UV/IR mixing is explained. Several types of geometries are identified, in particular "harmonic" and "Einstein" type of solutions. The physics of the harmonic branch is discussed in some detail, emphasizing the non-standard role of vacuum energy. This may provide new approach to some of the big puzzles in this context. The IKKT model with D=10 and close relatives are singled out as promising candidates for a quantum theory of fundamental interactions including gravity.Comment: Invited topical review for Classical and Quantum Gravity. 57 pages, 5 figures. V2,V3: minor corrections and improvements. V4,V5: some improvements, refs adde

Electroanalysis may be used in the Vanillin Biotechnological Production

This study shows that electroanalysis may be used in vanillin biotechnological production. As a matter of fact, vanillin and some molecules implicated in the process like eugenol, ferulic acid, and vanillic acid may be oxidized on electrodes made of different materials (gold, platinum, glassy carbon). By a judicious choice of the electrochemical method and the experimental conditions the current intensity is directly proportional to the molecule concentrations in a range suitable for the biotechnological process. So, it is possible to imagine some analytical strategies to control some steps in the vanillin biotechnological production: by sampling in the batch reactor during the process, it is possible to determine out of line the concentration of vanillin, eugenol, ferulic acid, and vanillic acid with a gold rotating disk electrode, and low concentration of vanillin with addition of hydrazine at an amalgamated electrode. Two other possibilities consist in the introduction of electrodes directly in the batch during the process; the first one with a gold rotating disk electrode using linear sweep voltammetry and the second one requires three gold rotating disk electrodes held at different potentials for chronoamperometry. The last proposal is the use of ultramicroelectrodes in the case when stirring is not possible