Dirac operators on Lagrangian submanifolds

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge–de Rham Laplacian provided the complex structure identifies the spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.     

La première valeur propre de l'opérateur de Dirac sur les variétés kählériennes compactes

K.D. Kirchberg [Ki1] gave a lower bound for the first eigenvalue of the Dirac operator on a spin compact Kähler manifoldM of odd complex dimension with positive scalar curvature. We prove that manifolds of real dimension 8l+6 satisfying the limiting case are twistor space (cf. [Sa]) of quaternionic Kähler manifold with positive scalar curvature and that the only manifold of real dimension 8l+2 satisfying the limiting case is the complex projective spaceCP ^4l+1.     

Versal Deformations of a Dirac Type Differential Operator

Abstract

If we are given a smooth differential operator in the variable x ∈ R/2πZ, its normal form, as is well known, is the simplest form obtainable by means of the Diff(S ¹)-group action on the space of all such operators. A versal deformation of this operator is a normal form for some parametric infinitesimal family including the operator. Our study is devoted to analysis of versal deformations of a Dirac type differential operator using the theory of induced Diff(S ¹)-actions endowed with centrally extended Lie-Poisson brackets. After constructing a general expression for tranversal deformations of a Dirac type differential operator, we interpret it via the Lie-algebraic theory of induced Diff(S ¹)-actions on a special Poisson manifold and determine its generic moment mapping. Using a Marsden-Weinstein reduction with respect to certain Casimir generated distributions, we describe a wide class of versally deformed Dirac type differential operators depending on complex parameters.     

S 1 actions and elliptic genera

A proof is given of Witten's conjectures for the rigidity of the index of the Dirac-Ramond operator on the loop space of a spin manifold which admits anS ¹ symmetry.Research supported in part by the National Science Foundation     

Optimal Eigenvalues Estimate for the Dirac Operator on Domains with Boundary

We give a lower bound for the eigenvalues of the Dirac operator on a compact domain of a Riemannian spin manifold under the MIT bag boundary condition. The limiting case is characterized by the existence of an imaginary Killing spinor. Mathematics Subject Classifications (2000). Differential Geometry, Global Analysis, 53C27, 53C40, 53C80, 58G25, 83C60.     

Connections on Clifford bundles and the Dirac operator

It is shown, how, in the setting of Clifford bundles, the spin connection (or Dirac operator) may be obtained by averaging the Levi-Civita connection (or Kähler-Dirac operator) over the finite group generated by an orthonormal frame of the base manifold.The familiar covariance of the Dirac equation under a simultaneous transformation of spinors and matrix representations emerges very naturally in this scheme, which can also be applied when the manifold does not possess a spin structure.     

On the Problem of Emergence of Classical Space—Time: The Quantum-Mechanical Approach

The Riemannian manifold structure of the classical (i.e., Einsteinian) space-time is derived from the structure of an abstract infinite-dimensional separable Hilbert space S. For this S is first realized as a Hilbert space H of functions of abstract parameters. The space H is associated with the space of states of a macroscopic test-particle in the universe. The spatial localization of state of the particle through its interaction with the environment is associated with the selection of a submanifold M of realization H. The submanifold M is then identified with the classical space (i.e., a space–like hypersurface in space-time). The mathematical formalism is developed which allows recovering of the usual Riemannian geometry on the classical space and, more generally, on space and time from the Hilbert structure on S. The specific functional realizations of S are capable of generating spacetimes of different geometry and topology. Variation of the length-type action functional on S is shown to produce both the equation of geodesics on M for macroscopic particles and the Schrödinger equation for microscopic particles.     

The Pfaffian line bundle

We analyze the holomorphic Pfaffian line bundle defined over an infinite dimensional isotropic Grassmannian manifold. Using the infinite dimensional relative Pfaffian, we produce a Fock space structure on the space of holomorphic sections of the dual of this bundle. On this Fock space, an explicit and rigorous construction of the spin representations of the loop groupsLO _n is given. We also discuss and prove some facts about the connection between the Pfaffian line bundle over the Grassmannian and the Pfaffian line bundle of a Dirac operator.Supported by a National Science Foundation Graduate Fellowship     

Locally Conformal Dirac Structures and Infinitesimal Automorphisms

We study the main properties of locally conformal Dirac bundles, which include Dirac structures on a manifold and locally conformal symplectic manifolds. It is proven that certain locally conformal Dirac bundles induce Jacobi structures on quotient manifolds. Furthermore we show that, given a locally conformal Dirac bundle over a smooth manifold M, there is a Lie homomorphism between a subalgebra of the Lie algebra of infinitesimal automorphisms and the Lie algebra of admissible functions. We also show that Dirac manifolds can be obtained from locally conformal Dirac bundles by using an appropriate covering map. Finally, we extend locally conformal Dirac bundles to the context of Lie algebroids.     

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝ ⁿ . They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres S ⁴ _θ. We construct the noncommutative algebras ?=C ^∞ (S ⁴ _θ) of functions on NC-spheres as solutions to the vanishing, ch _j (e) = 0, j < 2, of the Chern character in the cyclic homology of ? of an idempotent e∈M ₄ (?), e ²=e, e=e ^*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S ³ _θ distinct from quantum group deformations SU _q (2) of SU (2). We then construct the noncommutative geometry of S _θ ⁴ as given by a spectral triple ?, ℋ, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g _μν on S ⁴ whose volume form is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,

<Emphasis Type="Italic">θ</Emphasis>-Deformations as Compact Quantum Metric Spaces

Let M be a compact spin manifold with a smooth action of the n-torus. Connes and Landi constructed -deformations M of M, parameterized by n×n real skew-symmetric matrices . The Ms together with the canonical Dirac operator (D,) on M are an isospectral deformation of M. The Dirac operator D defines a Lipschitz seminorm on C(M), which defines a metric on the state space of C(M). We show that when M is connected, this metric induces the weak-* topology. This means that M is a compact quantum metric space in the sense of Rieffel.     

Nuclear Calabi-Yau space

In this note a Calabi-Yau manifold already found for⁹Li will be shown to carry an Euler number of six if Yang-Mills symmetry is broken. Not only does this specify the correct number of generations of quarks and leptons, but peaks on the manifold are associated with the lowest eigenvalues of aCP-invariant Dirac spin operatorC _[A].     

The First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces

We give a formula for the first eigenvalue of the Dirac operator acting on spinor fields of a spin compact irreducible symmetric space G/K.     

Curvature dependent lower bounds for the first eigenvalue of the Dirac operator

Using Weitzenböck techniques on any compact Riemannian spin manifold we derive inequalities that involve a real parameter and join the eigenvalues of the Dirac operator with curvature terms. The discussion of these inequalities yields vanishing theorems for the kernel of the Dirac operator D and lower bounds for the spectrum of D² if the curvature satisfies certain conditions.     

Function groups associated with constraint submanifolds

Let M be a submanifold of a manifold Q which has a generalized symplectic form ω. The submanifold M of Q is locally Hamiltonian if its vector fields are locally generated as a C^∞_ω- module by Hamiltonian vector fields of Q. If M is a first class submanifold, i.e., M is defined by first class constraints, then it is shown that M is locally Hamiltonian. It follows that if M is first class then M is a leaf of the singular foliation associated with the function group of all first class functions.     

Creating a monopole in 4D gauge theories

The problem of defining the second quantized monopole creation operator in non-Abelian gauge theories is discussed and exemplified by the (3 + 1)-dimensional Georgi-Glashow model. We construct the “coherent state” operator M(x) that creates the Coulomb magnetic field in terms of the Dirac singular electromagnetic potential. Our calculation of the vacuum expectation value of this operator 〈M(x)〉 in the confining phase indicates that it is free from the singularity along the Dirac string and in the leading order of perturbation theory the 〈M(x)〉 vanishes as a power of the volume of the system. This supports the conception that inclusion of the nonperturbative effects introduces an effective infrared cutoff on the calculation providing the finiteness of vacuum expectation value 〈M(x)〉. The text was submitted by the authors in English.     

One-Loop β Functions of a Translation-Invariant Renormalizable Noncommutative Scalar Model

In this paper we explain how to define “lower dimensional” volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential way noncommutative geometry and the analysis of Dirac operators on spin manifolds. However, the ultimate definitions of the lower dimensional volumes do not involve noncommutative geometry or spin structures at all.