THE SYMMETRY OF spinℂDIRAC SPECTRUMS ON RIEMANNIAN PRODUCT MANIFOLDS (original) (raw)

The symmetry of spinC Dirac spectrums on Riemannian

2016

It is well-known that the spectrum of a spin C Dirac operator on a closed Riemannian spin C manifold M 2k of dimension 2k for k ∈ N is symmetric. In this article, we prove that over an odd-dimensional Riemannian product M 2p 1 × M 2q+1 2 with a product spin C structure for p ≥ 1, q ≥ 0, the spectrum of a spin C Dirac operator given by a product connection is symmetric if and only if either the spin C Dirac spectrum of M 2q+1 2 is symmetric or (e 1 2 c1(L1)Â (M 1))[M 1 ] = 0, where L 1 is the associated line bundle for the given spin C structure of M 1 .

The spectrum of twisted Dirac operators on compact flat manifolds

2003

Let M be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of M , and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group Z k 2 , we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the η-series, in terms of values of Hurwitz zeta functions, and the η-invariant. We give the dimension of the space of harmonic spinors and characterize all Z k 2-manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs of Z k 2manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat n-manifolds, pairwise nonhomeomorphic to each other of the order of a n .

On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces

Journal of Geometry and Physics, 1996

The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates. These eigenfunctions are then used to derive the heat kernel of the iterated Dirac operator on these spaces. They are then studied as cross sections of homogeneous vector bundles, and a group-theoretic derivation of the spinor spherical functions and heat kernel is given based on Harish-Chandra's formula for the radial part of the Casimir operator.

Symmetry operators for Dirac's equation on two-dimensional spin manifolds

2008

It is shown that the second order symmetry operators for the Dirac equation on a general twodimensional spin manifold may be expressed in terms of Killing vectors and valence two Killing tensors. The role of these operators in the theory of separation of variables for the Dirac equation is studied.

The squares of the dirac and spin-dirac operators on a riemann-cartan space(time)

Reports on Mathematical Physics, 2007

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 in several topics of modern Mathematics, in particular in the study of the geometry of moduli spaces of a class of black holes, the geometry of NS-5 brane solutions of type II supergravity theories and BPS solitons in some string theories. We obtain a generalized Lichnerowicz formula, decompositions of the Dirac and spin-Dirac operators and their squares in terms of the standard Dirac and spin-Dirac operators and using the fact that spinor fields (sections of a spin-Clifford bundle) have representatives in the Clifford bundle we present also a noticeable relation involving the spin-Dirac and the Dirac operators.

The spin bfc^{\bf c}bfc Dirac operator on high tensor powers of a line bundle

Mathematische Zeitschrift, 2002

We study the asymptotic of the spectrum of the spin c Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin-Uribe [13, Theorem 2], which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin .

Generic Metrics and Connections on Spin- and Spin c -Manifolds

Communications in Mathematical Physics, 1997

We study the dependence of the dimension h 0 (g, A) of the kernel of the Atyiah-Singer Dirac operator D g,A on a spin c-manifold M on the metric g and the connection A. The main result is that in the case of spin-structures the value of h 0 (g) for the generic metric is given by the absolute value of the index provided dimM ∈ {3, 4}. In dimension 2 the mod-2 index theorems have to be taken into a account and we obtain an extension of a classical result in the theory of Riemann surfaces. In the spin c-case we also discuss upper bounds on h 0 (g, A) for generic metrics, and we obtain a complete result in dimension 2. The much simpler dependence on the connection A and applications to Seiberg-Witten theory are also discussed.

A note on metric connections for chiral and Dirac spinors

2006

It is known that the bundle of Dirac spinors is produced as a direct sum of two bundles-the bundle of chiral spinors and its Hermitian conjugate bundle. In this paper some aspects of metric connections for chiral and Dirac spinors are resumed and their relation is studied.

Dirac operator on spinors and diffeomorphisms

Classical and Quantum Gravity, 2013

The issue of general covariance of spinors and related objects is reconsidered. Given an oriented manifold M , to each spin structure σ and Riemannian metric g there is associated a space S σ,g of spinor fields on M and a Hilbert space H σ,g = L 2 (S σ,g , vol g (M )) of L 2 -spinors of S σ,g . The group Diff + (M ) of orientation-preserving diffeomorphisms of M acts both on g (by pullback) and on [σ] (by a suitably defined pullback f * σ). Any f ∈ Diff + (M ) lifts in exactly two ways to a unitary operator U from H σ,g to H f * σ,f * g . The canonically defined Dirac operator is shown to be equivariant with respect to the action of U , so in particular its spectrum is invariant under the diffeomorphisms.