arXiv research

For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of the Kostant's symplectic spinor bundle. Defining a Hilbert $C^*$-structure on this bundle for a suitable $C^*$-algebra, we o…

For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic ana…

Let $λ: \tilde{G}\to G$ be the non-trivial double covering of the symplectic group $G=Sp(V,ω)$ of the symplectic vector space $(V,ω)$ by the metaplectic group $\tilde{G}=Mp(V,ω).$ In this case, $λ$ is also a representation of $\tilde{G}$ on the vector space $V$ and thus, it gives rise to the representation of $\tilde{G…

We apply the geometric quantization method with real polarizations to the quantization of a symplectic torus. By quantizing with half-densities we canonically associate to the symplectic torus a projective Hilbert space and prove that the projective factor is expressible in terms of the Maslov-Kashiwara index. As in th…

Exterior differential forms with values in the (Kostant's) symplectic spinor bundle on a manifold with a given metaplectic structure are decomposed into invariant subspaces. Projections to these invariant subspaces of a covariant derivative associated to a torsion-free symplectic connection are described.

Paper constructs special solutions for symplectic Dirac operator.

Study symplectic spinors and Frobenius structures on manifolds.

We study symplectic manifolds $(M^{2l},ω)$ equipped with a symplectic torsion-free affine (also called Fedosov) connection $\nabla$ and admitting a metaplectic structure. Let $\mathcal{S}$ be the so called symplectic spinor bundle and let $R^S$ be the curvature tensor field of the symplectic spinor covariant derivative…

It is well known that spinors on oriented Riemannian manifolds cannot be defined as sections of a vector bundle associated with the frame bundle. For this reason spin and spin^c structures are often introduced. In this paper we prove that spin^c structures have a universal property among all other structures that enabl…

Study on symplectic Dirac operators on foliations, estimating eigenvalues.

In this paper, we prove that the "quantization commutes with reduction" phenomenon of Guillemin-Sternberg applies in the context of the metaplectic correction.

We give a classification of $1^{st}$ order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so called higher symplectic, harmonic …

Let $(M,ω)$ be a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure) and a torsion-free symplectic connection $\nabla.$ Symplectic Killing spinor fields for this structure are sections of the symplectic spinor bundle satisfying a certain first order partial dif…

This is the first of two articles aiming to introduce symplectic spinors into the field of symplectic topology and the subject of Frobenius structures. After exhibiting a (tentative) axiomating setting for Frobenius structures resp. 'Higgs pairs' in the context of symplectic spinors, we present immediate observations c…

By applying the symplectic cutting operation to cotangent bundles, one can construct a large number of interesting symplectic cones. In this paper we show how to attach algebras of pseudodifferential operators to such cones and describe the symbolic properties of the algebras.

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic stru…

We introduce the symplectic twistor operator $T_s$ in symplectic spin geometry, as a symplectic analogue of the twistor operator in Riemannian spin geometry. We focus on the real dimension 2 and compute the space of its solutions on ${\mathbb R}^2$. Our analysis is based on the techniques of metaplectic Howe duality.

Given a symplectic manifold $(M,ω)$ admitting a metaplectic structure, and choosing a positive $ω$-compatible almost complex structure $J$ and a linear connection $\nabla$ preserving $ω$ and $J$, Katharina and Lutz Habermann have constructed two Dirac operators $D$ and ${\wt{D}}$ acting on sections of a bundle of sympl…

These notes of a course given at IRMA in April 2009 cover some aspects of the representation theory of fundamental groups of manifolds of dimension at most 3 in compact Lie groups, mainly $\su$. We give detailed examples, develop the techniques of twisted cohomology and gauge theory. We review Chern-Simons theory and d…

We establish various results on the large level limit of projective quantum representations of surface mapping class groups obtained by quantizing moduli spaces of flat SU(n)-bundle. Working with the metaplectic correction, we proved that these projective representations lift to asymptotic representations. We show that…

We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. In particular, they may be used to give an explicit check that spherical braid grou…

This is the first in a series of papers devoted to an analogue of the metaplectic representation, namely, the minimal unitary representation of an indefinite orthogonal group; this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of Kirillov-Kostant. We begin by applying methods fro…

This paper shows connections between two complex mathematical theories are equivalent.

Let $K\subset S^3$ be a knot, $X:= S^3\setminus K$ its complement, and $\mathbb{T}$ the circle group identified with $\mathbb{R}/\mathbb{Z}$. To any oriented long knot diagram of $K$, we associate a quadratic polynomial in variables bijectively associated with the bridges of the diagram such that, when the variables pr…

Thick morphisms link quantum mechanics and supermanifolds.

Mathematical study supports connection between 3D manifolds and modular tensor categories.

The paper introduces new structures for left-symmetric algebroids.

We give a notion of compatibility between a Riemannian structure and a Jacobi structure. We prove that in case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, …

We give a notion of compatibility between a Riemannian metric and a Jacobi structure. We prove that in case of Poisson structures, contact structures and locally conformally symplectic structures, fundamental examples of Jacobi structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, $\fr…

Extending Jacobi and Riemannian compatibility to Lie algebroids.

Study projective and direct limits of Banach structures with connections to $G$-structures.

Defines structure constants for specific geometric structures on Lie groups.

Study on $G_2^*$ structures and almost para-contact structures in 7D.

Defines a new Poisson structure for generalized Sasakian spaces.

Study on types of generalized hypercomplex structures on tori and Kodaira-Thurston surface.

Classifies complex Dirac structures with invariants and local structure.

Extends corner structure study to general case, constructs normal Trans-Sasakian structures.

The paper explores geometric structures on Hom-Lie groups and algebras.

New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.

Study $GL(2)$-structures on manifolds leading to complex structures.

3D projective structures can be metrized with conformal structures.

Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one corres…

Spin-harmonic structures on low-dimensional manifolds.

Study equivalence between Hessian and Born structures on tangent bundles.

New geometric structure on surfaces generalizing complex structures.

Introduces compatibility between Dirac structures and Nijenhuis tensors.

New G2-structures found on Lie groups with strong structural conditions.

We introduce generalized almost contact structures which admit the $B$-field transformations on odd dimensional manifolds. We provide definition of generalized Sasakain structures from the view point of the generalized almost contact structures. We obtain a generalized Sasakian structure on a non-compact manifold which…