Noncommutative Riemann Surfaces

We compactify M(atrix) theory on Riemann surfaces Sigma with genus g>1. Following [1], we construct a projective unitary representation of pi_1(Sigma) realized on L^2(H), with H the upper half-plane. As a first step we introduce a suitably gauged sl_2(R) algebra. Then a uniquely determined gauge connection provides the central extension which is a 2-cocycle of the 2nd Hochschild cohomology group. Our construction is the double-scaling limit N\to\infty, k\to-\infty of the representation considered in the Narasimhan-Seshadri theorem, which represents the higher-genus analog of 't Hooft's clock and shift matrices of QCD. The concept of a noncommutative Riemann surface Sigma_\theta is introduced as a certain C^\star-algebra. Finally we investigate the Morita equivalence.Comment: LaTeX, 1+14 pages. Contribution to the TMR meeting ``Quantum aspects of gauge theories, supersymmetry and unification'', Paris 1-7 September 199

Remarks on the representation theory of the Moyal plane

We present an explicit construction of a unitary representation of the commutator algebra satisfied by position and momentum operators on the Moyal plane.Comment: 10 pages, minor changes, refs. adde

Ricci flow, quantum mechanics and gravity

It has been argued that, underlying any given quantum-mechanical model, there exists at least one deterministic system that reproduces, after prequantisation, the given quantum dynamics. For a quantum mechanics with a complex d-dimensional Hilbert space, the Lie group SU(d) represents classical canonical transformations on the projective space CP^{d-1} of quantum states. Let R stand for the Ricci flow of the manifold SU(d-1) down to one point, and let P denote the projection from the Hopf bundle onto its base CP^{d-1}. Then the underlying deterministic model we propose here is the Lie group SU(d), acted on by the operation PR. Finally we comment on some possible consequences that our model may have on a quantum theory of gravity.Comment: 8 page

On the noncommutative eikonal

We study the eikonal approximation to quantum mechanics on the Moyal plane. Instead of using a star product, the analysis is carried out in terms of operator-valued wavefunctions depending on noncommuting, operator-valued coordinates.Comment: 18 page

Primary osteoblast-like cells from patients with end-stage kidney disease reflect gene expression, proliferation, and mineralization characteristics ex vivo.

Osteocytes regulate bone turnover and mineralization in chronic kidney disease. As osteocytes are derived from osteoblasts, alterations in osteoblast function may regulate osteoblast maturation, osteocytic transition, bone turnover, and skeletal mineralization. Thus, primary osteoblast-like cells were cultured from bone chips obtained from 24 pediatric ESKD patients. RNA expression in cultured cells was compared with RNA expression in cells from healthy individuals, to RNA expression in the bone core itself, and to parameters of bone histomorphometry. Proliferation and mineralization rates of patient cells were compared with rates in healthy control cells. Associations were observed between bone osteoid accumulation, as assessed by bone histomorphometry, and bone core RNA expression of osterix, matrix gla protein, parathyroid hormone receptor 1, and RANKL. Gene expression of osteoblast markers was increased in cells from ESKD patients and signaling genes including Cyp24A1, Cyp27B1, VDR, and NHERF1 correlated between cells and bone cores. Cells from patients with high turnover renal osteodystrophy proliferated more rapidly and mineralized more slowly than did cells from healthy controls. Thus, primary osteoblasts obtained from patients with ESKD retain changes in gene expression ex vivo that are also observed in bone core specimens. Evaluation of these cells in vitro may provide further insights into the abnormal bone biology that persists, despite current therapies, in patients with ESKD

Dirac brackets from magnetic backgrounds

In symplectic mechanics, the magnetic term describing the interaction between a charged particle and an external magnetic field has to be introduced by hand. On the contrary, in generalised complex geometry, such magnetic terms in the symplectic form arise naturally by means of B-transformations. Here we prove that, regarding classical phase space as a generalised complex manifold, the transformation law for the symplectic form under the action of a weak magnetic field gives rise to Dirac's prescription for Poisson brackets in the presence of constraints.Comment: 9 page

Does quantum mechanics tell an atomistic spacetime?

The canonical answer to the question posed is "Yes." -- tacitly assuming that quantum theory and the concept of spacetime are to be unified by `quantizing' a theory of gravitation. Yet, instead, one may ponder: Could quantum mechanics arise as a coarse-grained reflection of the atomistic nature of spacetime? -- We speculate that this may indeed be the case. We recall the similarity between evolution of classical and quantum mechanical ensembles, according to Liouville and von Neumann equation, respectively. The classical and quantum mechanical equations are indistinguishable for objects which are free or subject to spatially constant but possibly time dependent, or harmonic forces, if represented appropriately. This result suggests a way to incorporate anharmonic interactions, including fluctuations which are tentatively related to the underlying discreteness of spacetime. Being linear and local at the quantum mechanical level, the model offers a decoherence and natural localization mechanism. However, the relation to primordial deterministic degrees of freedom is nonlocal.Comment: Based on invited talks at Fourth International Workshop DICE2008, held at Castello Pasquini / Castiglioncello, Italy, 22-26 September 2008 and at DISCRETE'08 - Symposium on Prospects in the Physics of Discrete Symmetries, held at IFIC, Valencia, Spain, 11-16 December 2008 - to appear in respective volumes of Journal of Physics: Conference Serie