In this note we compare the spinor bundle of a Riemannian manifold with the spinor bundles of the Riemannian factors . We show, that - without any holonomy conditions - the spinor bundle of for a special class of metrics is isomorphic to a bundle obtained by tensoring t…
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Spinor bundle constructed on loop space for string manifolds.
problem Constructing spinor bundles for string manifolds.
method Using ideas from Stolz and Teichner, a spinor bundle and fusion product are defined for a string manifold.
result Existence of spinor bundle with fusion product on a manifold X if and only if X admits a string structure.
This dissertation explores Clifford bundles and spinor fields in geometric and algebraic contexts.
problem Understanding spinor fields and their classification in geometric frameworks.
method Combines algebraic and geometric approaches to study Clifford structures on bundles and spinor fields.
result Identifies new spinor field classes in warped flux compactifications.
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.
Study geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
problem Geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
method Groupoid approach to pseudodifferential calculus, rescaled bundle.
result Rescaled bundle provides geometric characterization to asymptotic pseudodifferential calculus on spinor bundles.
The paper extends Strichartz's conjecture to spinor bundles over real hyperbolic spaces.
problem Extending Strichartz's conjecture to spinor bundles.
method Characterization of Poisson transform for spinor bundles and uniform estimates.
result Strichartz's conjecture is extended to spinor bundles over real hyperbolic spaces.
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-…
Spinor formalism is the formalism induced by solutions of the Clifford equation (the connecting operators). For the space-time manifold (n = 4), these operators, connecting the tangent and spinor bundle, are operators that are represented by the Dirac matrices in the special basis. Reduced connecting operators are repr…
Supposing that X is a Riemannian manifold, a Z/2 spinor on X is defined by a data set consisting of a closed set in X to be denoted by Z, a real line bundle over X-Z, and a nowhere zero section on X-Z of the tensor product of the real line bundle and a spinor bundle. The set Z and the spinor are jointly constrained by …
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.
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.
The aim of this paper is the construction of spinor bundles of Cartan type over certain non-orientable manifolds.
Constructs a fusion product on spinor bundle over loop space.
problem Describes a fusion product on the spinor bundle over loop space.
method Uses Connes fusion of von Neumann bimodules and string structures.
result Establishes a novel relation between string structures, loop fusion, and Connes fusion of Fock spaces.
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.
New -instantons found on 3-sphere's spinor bundle.
problem Classifying and constructing -instantons.
method Used -symmetries and asymptotically conical, co-homogeneity one -metric.
result Found new examples of -instantons with obstructed deformations.
Study torsion parallel spinors on Lorentzian 4-manifolds and their evolution flows.
problem Investigate torsion parallel spinors on Lorentzian four-manifolds.
method Geometric study via spinorial polyforms and supersymmetric NS-NS system.
result Globally hyperbolic evolution flow determined by supersymmetric solutions.
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…
Variant of Seiberg-Witten equations for multiple-spinors connects to stability of holomorphic bundles.
problem Detecting stability of holomorphic vector bundles using Seiberg-Witten equations.
method Abelian gauge-theoretic variant of Seiberg-Witten equations for multiple-spinors.
result Constructs a numerical invariant related to stability of holomorphic vector bundles.
We study an energy functional on the universal spinor bundle over a closed -dimensional spin manifold . The critical points of this functional, which is modelled on the total torsion functional of -structures in seven dimensions, are pairs of Ricci-flat metrics and real parallel spinor fields provided that $…
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.
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.
We establish, via geometric quantization of the supercotangent bundle sM of (M,g), a correspondence between its conformal geometry and those of the spinor bundle. In particular, the Kosmann Lie derivative of spinors is obtained by quantization of the comoment map, associated to the new Hamiltonian action of conf(M,g) o…
Parallel spinors help characterize G2* structures and isotropic forms.
problem Characterizing G2* structures and isotropic forms on pseudo-Riemannian manifolds.
method Using a correspondence between irreducible parallel spinors and solutions of a differential system for three-forms.
result Explicit description of isotropic irreducible spinors in signature (4,3) and characterization of G2* structures.
Kosmann-Lie derivatives in the bundle of Weyl spinors are considered. It is shown that the basic spin-tensorial fields of this bundle are constants with respect to these derivatives.
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…
In this paper connections between different gauge-theoretical problems in high and low dimensions are established. In particular it is shown that higher dimensional asd equations on total spaces of spinor bundles over low dimensional manifolds can be interpreted as Taubes-Pidstrygach's generalization of the Seiberg-Wit…
The paper extends Weierstrass representation to non-minimal conformal immersions.
problem Understanding conformal immersions of Riemann surfaces into 3D space.
method Generalizing Weierstrass representation to arbitrary conformal immersions and studying spin structures and spinors.
result Immersions give rise to spin structures and spinors, allowing for new ways to study the immersions.
Building on the universal covering group of the general linear group, we introduce the composite spinor bundle whose subbundles are Lorentz spin structures associated with different gravitational fields. General covariant transformations of this composite spinor bundle are canonically defined.
We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the Kähler-Atiyah bundle constructed by quantizing …
Develops complex spinorial forms for all dimensions and signatures, proving Brinkmann waves in supergravity.
problem Study of constrained parallelicity conditions for irreducible complex spinors.
method Differential theory of complex spinorial forms, reformulating conditions as equivalent differential systems.
result Every quasi-supersymmetric solution of Freedman's gauged supergravity belongs to a family of Brinkmann waves.
This paper constructs Seiberg-Witten monopoles from harmonic spinors on 3-manifolds.
problem Constructing Seiberg-Witten monopoles from -harmonic spinors on 3-manifolds.
method Gluing construction with alternating iteration to cancel infinite-dimensional obstruction bundle.
result A 1-parameter family of Seiberg-Witten monopoles converging to a -harmonic spinor.
Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are decomposed into irreducible modules including finding the hidden symmetry (Schur--Weyl--Howe type duality) given by a represent…
Calculates spinor heat flow using Gaussian-Grassmann integrals.
problem Computing the spinor heat flow symbol.
method Getzler calculus and Gaussian-Grassmann integrals.
result Computed the spinor heat flow symbol.
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.
Defines linear weightings for vector bundles and explores their applications.
problem Understanding and extending the concept of weightings in vector bundles.
method Constructs weighted normal bundles and deformation spaces; explains the relationship between weightings and differential operators.
result Captures the rescaled spinor bundle and related constructions.
Characterizes algebraic squares of irreducible complex spinors in various dimensions.
problem Understanding the relationship between spinors and exterior forms in different dimensions.
method Formalism using geometric product and algebraic relations.
result General correspondence between irreducible complex spinors and algebraically constrained exterior forms.
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.
New Fueter sections solve monopole equations for 3/2-spinors.
problem Existence of solutions to monopole equations for 3/2-spinors.
method Introduced 3/2-Fueter sections as solutions to an overdetermined non-linear elliptic differential equation.
result Non-compactness of moduli space of solutions is equivalent to existence of 3/2-Fueter sections.
Study of higher spin Killing spinors on 3D manifolds, proving rigidity and providing explicit expressions.
problem Understanding higher spin Killing spinors on 3D manifolds.
method Definition and detailed study of higher spin Killing spinors in arbitrary dimension, focusing on 3D manifolds. Rigidity result and explicit expressions for 3-sphere and 3-hyperbolic space.
result Proved a rigidity result for 3D manifolds admitting higher spin Killing spinors and provided explicit expressions for these spinors.
Theory of T-duality for transitive Courant algebroids developed.
problem Developing T-duality for transitive Courant algebroids.
method Introducing a map between sections of canonical spinor bundles and proving isomorphisms of invariant spinors and sections.
result Isomorphisms between spaces of invariant sections and spinors under T-duality.
A bundle gerbe is constructed from an oriented smooth vector bundle of even rank with a fiberwise inner product, over a compact connected orientable smooth manifold with Riemannian metric. From a trivialization of the bundle gerbe is constructed an irreducible Clifford module bundle, a spinor bundle over the smooth fre…
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 paper calculates determinants for Laplacians on spinor bundles over surfaces with flat metrics.
problem Calculating determinants for Laplacians on spinor bundles over surfaces with flat metrics.
method Explicit expressions for determinants of self-adjoint extensions of Laplacians using Bergman tau-function and theta-constants.
result An explicit expression for the determinant of the Szegö extension and comparison formulas for different extensions.
Study pseudo-laplacians and ζ(1) for spinor bundles over Riemann surfaces.
problem Analyzing self-adjoint extensions of Dolbeault Laplacians on Riemann surfaces.
method Defined -regularized determinants, introduced Robin mass, derived comparison formulas.
result Explicit expressions for Robin mass in spinor bundles and scalar cases.
Extended spinor connections associated with composite spin-tensorial bundles are considered. Commutation relationships for covariant and multivariate differentiations and corresponding curvature spin-tensors are derived.
Spinors prove rigidity for polyhedral spacetime data.
problem Rigidity of polyhedral spacetime data sets.
method Extending rigidity analysis from spacetime positive mass theorem.
result Dihedral rigidity connects mass theorem, trapped surfaces.
It is well known that spinors on oriented Riemannian manifolds cannot be defined as sections of a vector bundle associated with the frame bundle. For this reason spin and spin^c structures are often introduced. In this paper we prove that spin^c structures have a universal property among all other structures that enabl…
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 …