On quantization of singular varieties and applications to D-branes

We calculate the ring of differential operators on some singular affine varieties (intersecting stacks, a point on a singular curve or an orbifold). Our results support the proposed connection of the ring of differential operators with geometry of D-branes in (bosonic) string theory. In particular, the answer does know about the resolution of singularities in accordance with the string theory predictions.Comment: LaTeX2e, 17 pages, misprints correcte

On Twistors and Conformal Field Theories from Six Dimensions

We discuss chiral zero-rest-mass field equations on six-dimensional space-time from a twistorial point of view. Specifically, we present a detailed cohomological analysis, develop both Penrose and Penrose-Ward transforms, and analyse the corresponding contour integral formulae. We also give twistor space action principles. We then dimensionally reduce the twistor space of six-dimensional space-time to obtain twistor formulations of various theories in lower dimensions. Besides well-known twistor spaces, we also find a novel twistor space amongst these reductions, which turns out to be suitable for a twistorial description of self-dual strings. For these reduced twistor spaces, we explain the Penrose and Penrose-Ward transforms as well as contour integral formulae.Comment: v4: 66 pages, typos fixed, appendix B revise

Differential Forms on Log Canonical Spaces

The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting. Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces.Comment: 72 pages, 6 figures. A shortened version of this paper has appeared in Publications math\'ematiques de l'IH\'ES. The final publication is available at http://www.springerlink.co

Felix Alexandrovich Berezin and his work

This is a survey of Berezin's work focused on three topics: representation theory, general concept of quantization, and supermathematics.Comment: LaTeX, 27 page

Entanglement of positive definite functions on compact groups

We define and study entanglement of continuous positive definite functions on products of compact groups. We formulate and prove an infinite-dimensional analog of Horodecki Theorem, giving a necessary and sufficient criterion for separability of such functions. The resulting characterisation is given in terms of mappings of the space of continuous functions, preserving positive definiteness. The relation between the developed group-theoretical formalism and the conventional one, given in terms of density matrices, is established through the non-commutative Fourier analysis.Comment: published versio

Mouse Embryonic Retina Delivers Information Controlling Cortical Neurogenesis

The relative contribution of extrinsic and intrinsic mechanisms to cortical development is an intensely debated issue and an outstanding question in neurobiology. Currently, the emerging view is that interplay between intrinsic genetic mechanisms and extrinsic information shape different stages of cortical development [1]. Yet, whereas the intrinsic program of early neocortical developmental events has been at least in part decoded [2], the exact nature and impact of extrinsic signaling are still elusive and controversial. We found that in the mouse developing visual system, acute pharmacological inhibition of spontaneous retinal activity (retinal waves-RWs) during embryonic stages increase the rate of corticogenesis (cell cycle withdrawal). Furthermore, early perturbation of retinal spontaneous activity leads to changes of cortical layer structure at a later time point. These data suggest that mouse embryonic retina delivers long-distance information capable of modulating cell genesis in the developing visual cortex and that spontaneous activity is the candidate long-distance acting extrinsic cue mediating this process. In addition, these data may support spontaneous activity to be a general signal coordinating neurogenesis in other developing sensory pathways or areas of the central nervous system

Rearrangement of Retinogeniculate Projection Patterns after Eye-Specific Segregation in Mice

It has been of interest whether and when the rearrangement of neuronal circuits can be induced after projection patterns are formed during development. Earlier studies using cats reported that the rearrangement of retinogeniculate projections could be induced even after eye-specific segregation has occurred, but detailed and quantitative characterization of this rearrangement has been lacking. Here we delineate the structural changes of retinogeniculate projections in the C57BL/6 mouse in response to monocular enucleation (ME) after eye-specific segregation. When ME was performed after eye-specific segregation, rearrangement of retinogeniculate axons in the dorsal lateral geniculate nucleus (dLGN) was observed within 5 days. Although this rearrangement was observed both along the dorsomedial-ventrolateral and outer-inner axes in the dLGN, it occurred more rapidly along the outer-inner axis. We also examined the critical period for this rearrangement and found that the rearrangement became almost absent by the beginning of the critical period for ocular dominance plasticity in the primary visual cortex. Taken together, our findings serve as a framework for the assessment of phenotypes of genetically altered mouse strains as well as provide insights into the mechanisms underlying the rearrangement of retinogeniculate projections

Otx2 Gene Deletion in Adult Mouse Retina Induces Rapid RPE Dystrophy and Slow Photoreceptor Degeneration

International audienceBACKGROUND: Many developmental genes are still active in specific tissues after development is completed. This is the case for the homeobox gene Otx2, an essential actor of forebrain and head development. In adult mouse, Otx2 is strongly expressed in the retina. Mutations of this gene in humans have been linked to severe ocular malformation and retinal diseases. It is, therefore, important to explore its post-developmental functions. In the mature retina, Otx2 is expressed in three cell types: bipolar and photoreceptor cells that belong to the neural retina and retinal pigment epithelium (RPE), a neighbour structure that forms a tightly interdependent functional unit together with photoreceptor cells. METHODOLOGY/PRINCIPAL FINDINGS: Conditional self-knockout was used to address the late functions of Otx2 gene in adult mice. This strategy is based on the combination of a knock-in CreERT2 allele and a floxed allele at the Otx2 locus. Time-controlled injection of tamoxifen activates the recombinase only in Otx2 expressing cells, resulting in selective ablation of the gene in its entire domain of expression. In the adult retina, loss of Otx2 protein causes slow degeneration of photoreceptor cells. By contrast, dramatic changes of RPE activity rapidly occur, which may represent a primary cause of photoreceptor disease. CONCLUSIONS: Our novel mouse model uncovers new Otx2 functions in adult retina. We show that this transcription factor is necessary for long-term maintenance of photoreceptors, likely through the control of specific activities of the RPE

Long-Distance Three-Color Neuronal Tracing in Fixed Tissue Using NeuroVue Dyes

Dissecting development of neuronal connections is critical for understanding neuronal function in both normal and diseased states. Charting the development of the multitude of connections is a monumental task, since a given neuron typically receives hundreds of convergent inputs from other neurons and provides divergent outputs for hundreds of other neurons. Although progress is being made utilizing various mutants and/or genetic constructs expressing fluorescent proteins like GFP, substantial work remains before a database documenting the development and final location of the neuronal pathways in an adult animal is completed. The vast majority of developing neurons cannot be specifically labeled with antibodies and making specific GFP-expressing constructs to tag each of them is an overwhelming task. Fortunately, fluorescent lipophilic dyes have emerged as very useful tools to systematically compare changes in neuronal networks between wild-type and mutant mice. These dyes diffuse laterally along nerve cell membranes in fixed preparations, allowing tracing of the position of a given neuron within the neuronal network in murine mutants fixed at various stages of development. Until recently, however, most evaluations have been limited to one, or at most, two color analyses. We have previously reported three color neuronal profiling using the novel lipophilic dyes NeuroVue (NV) Green, Red and Maroon (Fritzsch et al., Brain. Res. Bull. 66:249–258, 2005). Unfortunately such three color experiments have been limited by the fact that NV Green and its brighter successor, NV Emerald, both exhibit substantially decreased signal intensities when times greater than 48 hours at 37°C are required to achieve neuronal profile filling (unpublished observations). Here we describe a standardized test system developed to allow comparison of candidate dyes and its use to evaluate a series of 488 nm-excited green-emitting lipophilic dyes. The best of these, NV Jade, has spectral properties well matched to NV Red and NV Maroon, better solubility in DMF than DiO or DiA, improved thermostability compared with NV Emerald, and the ability to fill neuronal profiles at rates of 1 mm per day for periods of at least 5 days. Use of NV Jade in combination with NV Red and NV Maroon substantially improves the efficiency of connectional analysis in complex mutants and transgenic models where limited numbers of specimens are available