arXiv research

Study on symplectic Dirac operators on foliations, estimating eigenvalues.

We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute those kernels for the complex projective spaces. We construct injections of subgroups of t…

Inspired by the results on symmetries of the symplectic Dirac operator, we realize symplectic spinor fields and the symplectic Dirac operator in the framework of (the double cover of) homogeneous projective structure in two real dimensions. The symmetry group of the homogeneous model of the double cover of projective g…

We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.

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…

We construct in projective differential geometry of the real dimension $2$ higher symmetry algebra of the symplectic Dirac operator ${D}\kern-0.5em\raise0.22ex\hbox{/}_s$ acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overde…

We compute the second coefficient of the composition of two Berezin-Toeplitz operators associated with the $\text{spin}^c$ Dirac operator on a symplectic manifold, making use of the full-off diagonal expansion of the Bergman kernel.

Let $G$ be a compact connected Lie group, and $M$ a compact Hamiltonian $G$-space, with moment map $J$. For each $G$-equivariant Hermitian vector bundle $E$ over $M$, one has an associated twisted Spin-C Dirac operator, whose equivariant index is a symplectic invariant of $E$. In the present paper, we study gluing prop…

In this paper we present a symplectic analogue of the Fueter theorem. This allows the construction of special (polynomial) solutions for the symplectic Dirac operator $D_s$, which is defined as the first-order $\mathfrak{sp}(2n)$-invariant differential operator acting on functions on ${\mathbb R}^{2n}$ taking values in…

For a Kähler Manifold $M$, the "symplectic Dolbeault operators" are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar\partial$ and $\bar\partial^*$, arise from Dirac operators on the canonical complex spinors on $M$. We give special atte…

We develop the theory of Berezin-Toeplitz operator on any compact symplectic prequantizable manifold from scratch. Our main inspiration is the Boutet de Monvel-Guillemin theory, that we simplify in several ways to obtain a concise exposition. A comparison with the spin-c Dirac quantization is also included.

We express the index of the Dirac operator on symplectic quotients of a Hamiltonian loop group manifold with proper moment map in terms of fixed point data.

Study Dirac operators on finite warped cylinders with gauge fields.

We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds with boundary. We emphasize var…

We establish the existence of the asymptotic expansion of the Bergman kernel associated to the spin-c Dirac operators acting on high tensor powers of line bundles with non-degenerate mixed curvature (negative and positive eigenvalues) by extending the paper " On the asymptotic expansion of Bergman kernel " (math.DG/040…

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 …

Suppose that $(M,E)$ is a compact contact manifold, and that a compact Lie group $G$ acts on $M$ transverse to the contact distribution $E$. In an earlier paper, we defined a $G$-transversally elliptic Dirac operator $\dirac$, constructed using a Hermitian metric $h$ and connection $\nabla$ on the symplectic vector bun…

We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we defi…

These letters, written in 1998-2000, contain various basic results about Courant algebroids (CAs), such as classification of exact and transitive CAs, reduction of CAs, description in terms of symplectic dg manifolds, a canonical generating Dirac operator, and a relation with Poisson-Lie T-duality.

We give a local normal form for Dirac structures. As a consequence, we show that the dimensions of the pre-symplectic leaves of a Dirac manifold have the same parity. We also show that, given a point $m$ of a Dirac manifold $M$, there is a well-defined transverse Poisson structure to the pre-symplectic leaf $P$ through…

Coupling Dirac structures are Dirac structures defined on the total space of a fibration, generalizing hamiltonian fibrations from symplectic geometry, where one replaces the symplectic structure on the fibers by a Poisson structure. We study the associated Poisson gauge theory, in order to describe the presymplectic g…

We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, …

Using tools from Dirac geometry and through an explicit construction, we show that every Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which relates, for a principal bundle $M\to M/H$, integrations of a Dirac structure o…

Consider a Hamiltonian action by a compact Lie group on a possibly noncompact symplectic manifold. We give a short proof of a geometric formula for decomposition into irreducible representations of the equivariant index of a Spin$^c$-Dirac operator in this context. This formula was conjectured by Michèle Vergne in 2006…

Several proofs have been published of the Mod Z gluing formula for the eta-invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the eta-invariant is left obscure in the literature. In this article we present a gluing formula for the eta-invariant which expresses the integer …

We construct an infinite-dimensional symplectic 2-groupoid as the integration of an exact Courant algebroid. We show that every integrable Dirac structure integrates to a "Lagrangian" sub-2-groupoid of this symplectic 2-groupoid. As a corollary, we recover a result of Bursztyn-Crainic-Weinstein-Zhu that every integrabl…

Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are decomposed into irreducible modules including finding the hidden symmetry (Schur--Weyl--Howe type duality) given by a represent…

We resolve Spin(7)-orbifolds using algebraic and symplectic techniques.

Generalizes Hamiltonian structures to Dirac structures for new mechanics models.

The paper constructs a symplectic groupoid for a specific Poisson structure.

New concept of coisotropic structures for differentiable stacks defined.

We study two quantization schemes for compact symplectic manifolds with almost complex structures. The first of these is the Spin-c quantization. We prove the analog of Kodaira vanishing for the Spin-c Dirac operator, which shows that the index space of this operator provides an honest (not virtual) vector space semicl…

New spectral torsion defined for rescaled Dirac operators.

Extends spectral number variance convergence to random matrix ensembles for twisted Laplacians.

A non-linear generalization of the Dirac operator in 4-dimensions, obtained by replacing the spinor representation with a hyperKahler manifold admitting certain symmetries, is considered. We show that the existence of a covariantly constant, generalized spinor defines a Kahler structure on the base 4-dimensional manifo…

Study on magnetic Dirac operators and their spectrum.

We explain the geometric origin of the $L_{\infty}$-algebra controlling deformations of pre-symplectic structures.

In this paper we introduce the Dirac and spin-Dirac operators associated to a connection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac operators associated with a Levi-Civita connection on a Riemannian (Lorentzian) space(time) and calculate the square of these operators, which play an important role i…

Mathematical proof of index equality for lattice Dirac operators and continuum operators.

Study perturbs APS boundary conditions for Lorentzian Dirac operators.

We extend our earlier work in [TZ1], where an analytic approach to the Guillemin-Sternberg conjecture [GS] was developed, to cases where the Spin$^c$-complex under consideration is allowed to be further twisted by certain natural exterior power bundles. The main result is a weighted quantization formula in the presence…

New spectral functionals for Dirac operators with inner fluctuations computed.

Study of Dirac-like operators on spin manifolds with large mass parameters.

The paper calculates a functional for a specific Dirac operator.

We study gauge transformations of Dirac structures and the relationship between gauge and Morita equivalences of Poisson manifolds. We describe how the symplectic structure of a symplectic groupoid is affected by a gauge transformation of the Poisson structure on its identity section, and prove that gauge-equivalent in…

We determine the structure of conformal powers of the Dirac operator on Einstein {\it Spin}-manifolds in terms of the product formula for shifted Dirac operators. The result is based on the techniques of higher variations for the Dirac operator on Einstein manifolds and spectral analysis of the Dirac operator on the as…

Defines spectral Einstein functionals for sub-Dirac operators on manifolds with boundary.

Study on Dirac operators on lightlike hypersurfaces in 4D Lorentzian manifolds.