Introduces Chern-Dirac bundles on non-Kähler Hermitian manifolds.
problem No specific problem stated; focuses on new mathematical structures.
method Introduces Chern-Dirac bundles and operators, showing isomorphisms with cohomology.
result Spaces of harmonic spinors on V-spinor bundle isomorphic to Dolbeault cohomology.
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.
The paper constructs Dirac-harmonic maps using spinors.
problem Unclear conditions for Dirac-harmonic maps in previous work.
method Uses harmonic and twistor spinors to construct Dirac-harmonic maps.
result Shows conditions for existence of solutions under special assumptions.
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.
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.
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.
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 Mn admitting real Killing spinors (resp. parallel spinors), there exist warped product metrics ηˉ on Mn×R 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.
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.
Proof that stable minimal surfaces in 3D are flat.
problem Classification of stable minimal surfaces in R3. method Index theory for Dirac operators on twisted spinor bundles.
result Every complete two-sided stable minimal surface in R3 is flat. Investigates immersions in Sn using complex spinors.
problem Generalizing immersions of Riemann surfaces to Spin-manifolds.
method Uses complex spinors and the Dirac equation.
result Investigates submanifolds of SpinC-manifolds of constant curvature.
Extended superalgebras from twistor and Killing spinors in constant curvature and Einstein manifolds.
problem Constructing extended superalgebras from twistor and Killing spinors.
method Using twistor and Killing spinors, symmetry operators, and KY/CKY forms, extended superalgebras are constructed.
result Extended Killing and conformal superalgebras are constructed in constant curvature and Einstein manifolds.
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-c spinors and formalism of differential forms to simplify calculations. result Reproduces known normalisable zero modes of the twisted Eguchi-Hanson Dirac operator.
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…
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.
Eigenfunctions of the Dirac operator on spheres reveal complex nodal structures.
problem Finding eigenfunctions with specific nodal sets on spheres.
method Analyzing the Dirac operator on round spheres with arbitrary submanifolds.
result Eigenfunctions of the Dirac operator on spheres can have nodal sets corresponding to any given submanifolds.
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…
Paper constructs special solutions for symplectic Dirac operator.
problem No specific problem stated; focuses on mathematical construction.
method Symplectic analogue of Fueter theorem used to solve symplectic Dirac operator.
result Constructs polynomial solutions for symplectic Dirac operator.
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.
Smooth solutions and classification of Dirac-Einstein equations on 3-manifolds.
problem Analyzing solutions of Dirac-Einstein equations on R3. 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 −21-Killing spinors on S3. 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…
The paper studies harmonic forms and spinors on Taub-bolt space.
problem Analyzing harmonic forms and spinors on Taub-bolt, a Ricci-flat ALF space.
method Proving dimensions of harmonic 2-forms, constructing zero modes of Dirac operator, comparing with known results.
result Explicitly found a 2-parameter family of L2 zero modes of the Dirac operator. 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.
Study harmonic spinors on a family of Einstein manifolds, including Taub-NUT and Eguchi-Hanson.
problem Existence and explicit solutions of harmonic spinors on Einstein manifolds.
method Solve for spinors harmonic with respect to the Dirac operator twisted by a geometric connection.
result Explicit solutions and agreement with index theorem boundary conditions.
The article studies deformations of Z2-harmonic spinors on 3-manifolds.
problem Investigating the local structure of Z2-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 Z2-harmonic spinor with smooth singular set, the universal moduli space projects to a codimension 1 submanifold. 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.
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.
Proves short-term existence of heat flow for Dirac-harmonic maps on closed manifolds.
problem Existence of Dirac-harmonic maps on closed manifolds.
method Coupling heat flow for harmonic maps to a spinor.
result Short time existence of the heat flow for Dirac-harmonic maps on closed manifolds.