We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-…
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48 results for “Dirac spinor”
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.
A closed spin Kähler manifold of positive scalar curvature with smallest possible first eigenvalue of the Dirac operator is characterized by holomorphic spinors. It is shown that on any spin Kähler-Einstein manifold each holomorphic spinor is a finite sum of eigenspinors of the square of the Dirac operator. Vanishing t…
Representations of Dirac-Hestenes and Dirac spinor fields via coordinates of surfaces conformally immersed into 4-dimensional complex space are proposed. A relation between time evolution of spinor fields and integrable deformations of surfaces is discussed.
Classifies manifolds with specific spinors and constructs parallel spinors.
problem Classifying manifolds with generalized Killing spinors.
method Explicitly constructs parallel spinors and considers modified Dirac currents.
result Classifies Riemannian spin^c manifolds with type I and II imaginary generalized Killing spinors.
We construct exact solutions of the Einstein-Dirac equation, which couples the gravitational field with an eigenspinor of the Dirac operator via the energy-momentum tensor. For this purpose we introduce a new field equation generalizing the notion of Killing spinors. The solutions of this spinorial field equation are c…
Study uncoupled solutions to Dirac-Yang-Mills equations on spin manifolds.
problem Condition for vanishing Dirac current on harmonic spinors.
method Perturbation theory and index theorem.
result Existence of uncoupled solutions, classification of connection forms.
Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.
Two explicit formulas for metric connections in the bundle of Dirac spinors are studied. Their equivalence is proved. The explicit formula relating the spinor curvature tensor with the Riemann curvature tensor is rederived.
Defines relations between Dirac structures and spinors using Courant algebroid relations.
problem Defines relations between Dirac structures and spinors using Courant algebroid relations.
method Uses Courant algebroid relations to define relations between Dirac structures and spinors.
result Proves existence results for T-dual structures and demonstrates compatibility with Type II supergravity equations.
Study quantum diffusion on spectral triples and spinor bundles.
problem Characterize quantum diffusion on almost commutative spectral triples.
method Spin geometry, C *-Dirichlet forms, quantum stochastic flows.
result Existence of covariant quantum stochastic flows on spinor bundles.
Clarifies Einstein-Cartan gravitation with Dirac spinor on generalized frame bundle.
problem Formulating Einstein-Cartan gravitation on a frame bundle.
method Integrates Dirac spinor into the Einstein-Cartan spacetime structure.
result Variational equations imply standard field equations under standard frame bundle condition.
New eigenvalue estimate for CR manifolds' Kohn-Dirac operator.
problem Estimating eigenvalues of the Kohn-Dirac operator on CR manifolds.
method Characterizing equality case by CR twistor spinor existence; classifying manifolds with specific Ricci tensor properties.
result Classifying CR manifolds with at most two Webster Ricci tensor eigenvalues.
This paper studies Sasakian quasi-Killing spinors on 3D Sasakian manifolds.
problem Characterizing Sasakian quasi-Killing spinors on 3D Sasakian manifolds.
method Detailed analysis and geometric properties of Sasakian quasi-Killing spinors.
result Almost all Sasakian quasi-Killing spinors solve the Einstein-Dirac system with a non-zero cosmological constant.
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…
We study the Einstein-Dirac equation as well as the weak Killing equation on Riemannian spin manifolds with codimension one foliation. We prove that, for any manifold admitting real Killing spinors (resp. parallel spinors), there exist warped product metrics on such that $(M^n \ti…
Classification of ground state solutions to critical Dirac equation on spheres.
problem Classifying ground state solutions of the critical Dirac equation.
method Exploiting conformal covariance and relating to the Yamabe equation.
result Ground state solutions are given by Killing spinors up to conformal diffeomorphisms.
Operator fields in the bundle of Dirac spinors and their conversion to spatial fields are considered. Some commutator equations are studied with the use of the conversion technique.
Some aspects of Dirac spinors are resumed and studied in order to interpret mathematically the P and T operations in a gravitational field.
Proof that stable minimal surfaces in 3D are flat.
problem Classification of stable minimal surfaces in .
method Index theory for Dirac operators on twisted spinor bundles.
result Every complete two-sided stable minimal surface in is flat.
Researchers prove an index formula for spinors on 3-manifolds branching along graphs.
problem Index formula for Dirac operators on 3-manifolds with branch points.
method Analyzes Dirac operator on two-valued spinors on a 3-manifold with a graph branch, with boundary conditions.
result Index formula vanishes when the branch is a smooth curve, extends to graphs with vertices.
The basic first-order differential operators of spin geometry that are Dirac operator and twistor operator are considered. Special types of spinors defined from these operators such as twistor spinors and Killing spinors are discussed. Symmetry operators of massless and massive Dirac equations are introduced and releva…
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…
Revisits zero modes of Dirac operator on Eguchi-Hanson space.
problem Determining zero modes of the Dirac operator on Eguchi-Hanson space.
method Uses spin- spinors and formalism of differential forms to simplify calculations.
result Reproduces known normalisable zero modes of the twisted Eguchi-Hanson Dirac operator.
We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on t…
Study on harmonic spinors on specific Lie groups.
problem Existence of left-invariant harmonic spinors on 3D Lie groups.
method Revised spin Dirac operator formula for left-invariant spinors, identified constraints on Lie algebras, and classified metrics with harmonic spinors.
result Identified conditions and metrics for left-invariant harmonic spinors on 3D Lie groups.
We provide explicit spinor representations for Clifford algebras.
problem Building explicit representations of Clifford algebras.
method Explicit construction of spinor modules and parallel spinor fields.
result Explicit spinor representations for all mixed signature Clifford algebras.
The aim of the present paper is to clarify the relationship between immersions of surfaces and solutions of the inhomogeneous Dirac equation. The main idea leading to the description of a surface M^2 by a spinor field is the observation that the restriction to M^2 of any parallel spinor phi on R^3 is (with respect to t…
Smooth solutions and classification of Dirac-Einstein equations on 3-manifolds.
problem Analyzing solutions of Dirac-Einstein equations on .
method Proving smoothness and asymptotic behavior, classifying ground state solutions.
result Scalar part is given by Aubin-Talenti functions, spinorial part is conformal image of -Killing spinors on .
Paper introduces new fractional Dirac operator and Q-curvature.
problem Fractional Dirac operator and Q-curvature in spinors.
method Caffarelli-Silvestre extension, energy inequalities, weighted Sobolev inequality.
result Introduction of conformal fractional Dirac operator and Q-curvature.
I begin by explaining how Riemannian geometry can be understood in terms of principal fibre bundles and connections thereon. I then introduce and motivate the definition of a spinor structure in terms of familiar geometrical ideas. The central result of this thesis is a complete and constructive classification of spino…
Asymptotic Spectral Flowmath.DG
Study the spectral flow of Dirac operators on spinor bundles.
problem Understanding the asymptotic behavior of spectral flow for Dirac operators.
method Variation of eta invariant and local index theory technique.
result Established a uniform estimate of the eta invariant for large parameter values.
The paper finds exact solutions to a complex Einstein-Dirac-Maxwell system on 4D Sasakian spacetimes.
problem Finding exact solutions to an Einstein-Dirac-Maxwell system with Sasakian quasi-Killing spinors.
method Constructing a family of exact solutions on four-dimensional static Sasakian spacetimes using the Sasakian frame.
result Closed and open universe models are found with specific energy conditions.
The paper shows connections can be uniquely determined by their boundary data.
problem Determining unique connections from boundary measurements.
method Defined a Dirichlet-to-Neumann map for twisted Dirac Laplacians and showed its pseudodifferential properties.
result Equal Dirichlet-to-Neumann maps imply locally gauge equivalent connections.
New spinor fields reveal local or global geometric properties of manifolds.
problem Characterizing spinor fields on manifolds.
method Analyzing generalized imaginary Spin^c-Killing spinors and their associated vector fields.
result Local or global geometric descriptions of manifolds based on spinor fields.
We study the clustering of the lowest non negative eigenvalue of the Dirac operator on a general Dirac bundle when the metric structure is varied. In the classical case we show that any closed spin manifold of dimension greater than or equal to four has a Riemannian metric admitting non trivial harmonic spinors.
The article studies deformations of -harmonic spinors on 3-manifolds.
problem Investigating the local structure of -harmonic spinors on 3-manifolds.
method Uses Nash-Moser Implicit Function Theorem to handle infinite-dimensional obstruction bundle and loss of regularity.
result Near a -harmonic spinor with smooth singular set, the universal moduli space projects to a codimension 1 submanifold.
Constructs spin hyperbolic surfaces with a spectral gap for Dirac operator.
problem Finding spectral gaps for Dirac operators on hyperbolic surfaces.
method Explicit construction of spin hyperbolic surfaces with increasing genus.
result Uniform spectral gap for Dirac operator on constructed surfaces.
In this paper, we describe the group SpinT (n) and give some properties of this group. We construct SpinT spinor bundle S by means of the spinor representation of the group SpinT (n) and define covariant derivative operator and Dirac operator on S. Finally, Schrodinger-Lichnerowicz-type formula is derived by using thes…
Investigates parallel spinors on Eguchi-Hanson metrics.
problem Analyzing parallel spinors on specific metrics.
method Investigated parallel spinors on Eguchi-Hanson metrics with harmonic spinors.
result Found complex 2-dimensional space of complex parallel spinors and solutions for metrics with zero scalar curvature.
Study invariant operators and vanishing theorems in CR geometry.
problem Analyzing invariant operators and vanishing theorems in CR geometry.
method Investigates Kohn-Dirac operators and derives CR invariant twistor operators.
result Proves vanishing theorems for harmonic spinors and Kohn-Rossi groups.
We characterize the Lie derivative of spinor fields from a variational point of view by resorting to the theory of the Lie derivative of sections of gauge-natural bundles. Noether identities from the gauge-natural invariance of the first variational derivative of the Einstein(--Cartan)--Dirac Lagrangian provide restric…
The paper studies Dirac operators and their solutions concentrating near singular sets.
problem Understanding concentration properties of solutions to Dirac equations.
method Analyzes Dirac operators of the form and their solutions.
result Solutions concentrate exponentially near the locus where the rank of jumps.
Classifies local boundary conditions for Dirac-type operators on manifolds.
problem Determining all local smooth boundary conditions for Dirac-type operators.
method Combining general theory of boundary value problems for Dirac operators and pointwise considerations.
result Classification of local self-adjoint regular boundary conditions for Dirac spinors in dimensions 3 and 4.
A general theory of quantum spinor structures on quantum spaces is presented, within the conceptual framework of the formalism of quantum principal bundles. Quantum analogs of all basic objects of the classical theory are constructed and analyzed. This includes Laplace and Dirac operators, quantum versions of Clifford …
We discuss a method to construct Dirac-harmonic maps developed by J.~Jost, X.~Mo and M.~Zhu in J.~Jost, X.~Mo, M.~Zhu, \emph{Some explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. \textbf{59} (2009), no. 11, 1512--1527.The method uses harmonic spinors and twistor spinors, and mainly applies to Dirac-harmon…
An integrated approach to Lie derivatives of spinors, spinor connections and the gravitational field is presented, in the context of a previously proposed, partly original formulation of a theory of Einstein-Carta-Maxwell-Dirac fields based on "minimal geometric data": all the needed underlying structure is geometrical…
We investigate the second Dirac eigenvalue on Riemannian manifolds admitting a Killing spinor. In small dimensions the whole Dirac spectrum depends on special eigenvalues on functions and 1-forms. We compute and discuss the formulas in dimension .