Equivariant quantization of orbifolds

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces... In this work, we prove existence of an equivariant quantization for orbifolds. Our construction combines an appropriate desingularization of any Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in \cite{PoRaWo}. Further, we suggest definitions of the common geometric objects on orbifolds, which capture the nature of these spaces and guarantee, together with the properties of the mentioned foliated resolution, the needed correspondences between singular objects of the orbifold and the respective foliated objects of its desingularization.Comment: 13 page

Equivariant symbol calculus for differential operators acting on forms

We prove the existence and uniqueness of a projectively equivariant symbol map (in the sense of Lecomte and Ovsienko) for the spaces $D_p$ of differential operators transforming p-forms into functions. These results hold over a smooth manifold endowed with a flat projective structure. As an application, we classify the Vect(M)-equivariant maps from $D_p$ to $D_q$ over any manifold M, recovering and improving earlier results by N. Poncin. This provides the complete answer to a question raised by P. Lecomte about the extension of a certain intrinsic homotopy operator.Comment: 14 page

The space of m-ary differential operators as a module over the Lie algebra of vector fields

The space of m-ary differential operators acting on weighted densities is a (m+1)-parameter family of modules over the Lie algebra of vector fields. For almost all the parameters, we construct a canonical isomorphism between this space and the corresponding space of symbols as sl(2)-modules. This yields to the notion of the sl(2)-equivariant symbol calculus for m-ary differential operators. We show, however, that these two modules cannot be isomorphic as sl(2)-modules for some particular values of the parameters. Furthermore, we use the symbol map to show that all modules of second-order m-ary differential operators are isomorphic to each other, except for few modules called singular.Comment: 20 pages; LaTeX2e; minor correction

A First Approximation for Quantization of Singular Spaces

Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces obtained as leaf closure spaces of regular Riemannian foliations on compact manifolds. These contain the orbit spaces of compact group actions and orbifolds. Our method uses foliation theory as a desingularization technique for such singular spaces. A quantization procedure on the orbit space of the symmetry group - that commutes with reduction - can be obtained from constructions which combine different geometries associated with foliations and new techniques originated in Equivariant Quantization. The present paper contains the first of two steps needed to achieve these just detailed goals.Comment: 30 page

Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

Let $M$ be an odd-dimensional Euclidean space endowed with a contact 1-form $\alpha$. We investigate the space of symmetric contravariant tensor fields on $M$ as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we consider symmetric tensor fields with coefficients in tensor densities, the vertical cotangent lift of contact form $\alpha$ is a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symmetric density valued tensor fields. This generalized Hamiltonian operator on the symbol space is invariant with respect to the action of the projective contact algebra $sp(2n+2)$. The preceding invariant operators lead to a decomposition of the symbol space (expect for some critical density weights), which generalizes a splitting proposed by V. Ovsienko

Projectively equivariant quantizations over the superspace Rp∣q\R^{p|q}

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields.Comment: 19 page

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Analysis by in Situ Hybridization of Cells Expressing mRNA for Tumor-Necrosis Factor in the Developing Thymus of Mice

We have used in situ hybridization to investigate the expression of TNF-α genes by thymic cells during fetal development in mice. In 14-day-old fetal thymuses, very scarce cells produce TNF-α mRNA. A second phase of cytokine gene expression starts on day 16. The density of positive cells progressively increases up to day 20. Thymuses at 15 days of gestation and after birth do not express detectable cytokine mRNA. In an attempt to identify the nature of the TNF-α mRNA-producing cells, acid phosphatase activity, which is characteristic of the macrophage lineage, was studied in the same thymuses. Acid phosphatase-positive cells only appear on day 15. Their frequency increases up to birth. However, no correlation can be established between acid phosphatase—and TNFα mRNA— positive cells. The results indicate that a small subset of thymic cells is responsible for TNF-α mRNA production during ontogeny: These cells are not yet identified. The possible role of TNF-α in thymic ontogeny is discussed