Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

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Deformation Quantization of Poisson Manifolds

Quantum field theory on noncommutative spaces

A model of Euclidean spacetime is presented in which at scales less than a certain length kappa the notion of a point does not exist. At scales larger then kappa the model resembles the 2-sphere S2. The algebra which determines the structure of the model, and which replaces the algebra of functions, is an algebra of matrices. The order of n of the matrices is connected with the length kappa and the radius r of the sphere by the relation kappa approximately r/n. The elements of differential calculus are sketched as well as the possible definitions of a metric and linear connection. A definition of the path integral is given and a few examples of field theory on a fuzzy sphere are finally referred to.

https://iopscience.iop.org/article/10.1088/0264-9381/9/1/008/pdf

The Fuzzy sphere

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

For general compact Kahler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of teh authors and Klimek and Lesniewski obtained for the tours and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finitedimensional matrix algebrasgl(N), N→∞.

/pdf/toeplitz-quantization-of-kahler-manifolds-ang-gl-n-n-limits-3qzxhkeh5e.pdf

Toeplitz quantization of Kähler manifolds ang gl(N), N→∞ limits

For general compact K\"ahler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras $gl(N)$, $N\to\infty$.

Toeplitz Quantization of K\"ahler Manifolds and $gl(N)$ $N\to\infty$

For general compact Kahler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras gl(N), N → ∞.

Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to\infty$

For arbitrary compact quantizable Kahler manifolds it is shown how a natural formal deformation quantization (star-product) can be obtained via Berezin-Toeplitz operators. Results on their semi-classical behaviour (their asymptotic expansion) due to Bordemann, Meinrenken, and Schlichenmaier are used in an essential manner. It is shown that the star-product is null on constants and fulfills parity. A trace is constructed and the relation to deformation quantization by geometric quantization is given.

/pdf/deformation-quantization-of-compact-kahler-manifolds-by-3383mnta0j.pdf

Deformation quantization of compact Kahler manifolds by Berezin-Toeplitz quantization

We give a complete identification of the deformation quantization which was obtained from the Berezin- Toeplitz quantization on an arbitrary compact Kahler manifold. The deformation quantization with the opposite star-product proves to be a differential deformation quantization with separation of variables whose dassifying form is explicitly calculated. Its characteristic dass (which dassifies star-products up to equivalence) is obtained. The proof is based on the microlocal description of the Szego kernel of a strictly pseudoconvex domain given by Boutet de Monvel and Sjostrand.

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Martin Schlichenmaier

Papers

Toeplitz quantization of Kähler manifolds ang gl(N), N→∞ limits

Toeplitz Quantization of K\"ahler Manifolds and $gl(N)$ $N\to\infty$

Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to\infty$

Deformation quantization of compact Kahler manifolds by Berezin-Toeplitz quantization

Identification of Berezin-Toeplitz Deformation Quantization