arXiv research

Study perturbs APS boundary conditions for Lorentzian Dirac operators.

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

Local index formula for Lorentzian Dirac operators on spacetimes.

New boundary conditions solve Cauchy problem for Dirac operators on spacetimes.

Study well-poses Dirac operator problem with APS boundary conditions.

On a compact globally hyperbolic Lorentzian spin manifold with smooth spacelike Cauchy boundary the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah-Patodi-Singer boundary conditions are imposed. In this paper we investigate to what extent these boundary conditions can be replaced by more general ones an…

We survey the correct definition of a generalized Dirac operator on a Space--Time and the classical result about propagation of singularities. This says that light travels along light--like geodesics. Finally we show this is also true for generalized Dirac operators.

Study of Dirac-Witten operator on Lorentzian manifolds under dominant energy condition.

Study shows Fredholmness of a Lorentzian Dirac operator on spacetimes.

Formula for index in Lorentzian spacetimes.

Geometric approach to Dirac operator evolution on spacetimes.

Paper proves rigidity for spin bands with specific conditions.

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…

We review some recent results on geometric equations on Lorentzian manifolds such as the wave and Dirac equations. This includes well-posedness and stability for various initial value problems, as well as results on the structure of these equations on black-hole spacetimes (in particular, on the Kerr solution), the ind…

Study finds the spectrum of a cubic Dirac operator on specific oscillator group manifolds.

Extends Atiyah-Singer Dirac operator study to non-compact spacetimes.

We consider a hyperbolic Dirac-type operator with growing potential on a a spatially non-compact globally hyperbolic manifold. We show that the Atiyah-Patodi-Singer boundary value problem for such operator is Fredholm and obtain a formula for this index in terms of the local integrals and the relative eta-invariant int…

We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemann…

Researchers prove Fredholm property for Dirac operator on specific spacetimes.

Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces im…

Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all th…

Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green's operators. The most prominent examples are wave operators and Dirac-type operators. This paper is devoted to a systematic study of this class of diffe…

Consider a formally self-adjoint first order linear differential operator acting on pairs (2-columns) of complex-valued scalar fields over a 4-manifold without boundary. We examine the geometric content of such an operator and show that it implicitly contains a Lorentzian metric, Pauli matrices, connection coefficients…

The paper deals with a formally self-adjoint first order linear differential operator acting on m-columns of complex-valued half-densities over an n-manifold without boundary. We study the distribution of eigenvalues in the elliptic setting and the propagator in the hyperbolic setting, deriving two-term asymptotic form…

We write the Dirac equation in curved 4-dimensional Lorentzian spacetime using concepts from the analysis of partial differential equations as opposed to geometric concepts.

In this paper we discuss the twistor equation in Lorentzian spin geometry. In particular, we explain the local conformal structure of Lorentzian manifolds, which admit twistor spinors inducing lightlike Dirac currents. Furthermore, we derive all local geometries with singularity free twistor spinors that occur up to di…

We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify …

We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.

We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived in a mathematically…

Classifies manifolds with specific spinors and constructs parallel spinors.

New spectral torsion defined for rescaled Dirac operators.

Study on symplectic Dirac operators on foliations, estimating eigenvalues.

Study on magnetic Dirac operators and their spectrum.

Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.

We propose a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive th…

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

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 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.

The study examines conditions for scalar curvature and Dirac operators on singular spaces.

We use the spectra of Dirac type operators on the sphere $S^{n}$ to produce sharp $L^{2}$ inequalities on the sphere. These operators include the Dirac operator on $S^{n}$, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers o…

The paper proves formulas and theorems for Dirac-Witten operators on manifolds with or without boundaries.

The paper introduces spectral Einstein functionals for Dirac operators on manifolds with boundary.

The paper calculates spectral torsion for rescaled Dirac operators on manifolds.

Dirac operator invertibility proven for specific manifolds.

Study properties of semi-symmetric Lorentzian spaces, foliated manifolds.