Frank Klinker
In this note we compare the spinor bundle of a Riemannian manifold $(M=M_1\times...\times M_N,g)$ with the spinor bundles of the Riemannian factors $(M_i,g_i)$. We show, that - without any holonomy conditions - the spinor bundle of $(M,g)$ for a special class of metrics is isomorphic to a bundle obtained by tensoring the spinor bundles of $(M_i,g_i)$ in an appropriate way.
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We describe, by their holonomy groups, all simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalise the Riemannian case and the pseudo-Riemannian one.
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 immersed into Lorentzian manifolds and on surfaces conformally immersed into Minkowski spacetime is defined.
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.
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.
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) on sM. We study then the conf(M,g)-module structures induced on the space of differential operators acting on spinor densities and on its spaces of symbols (functions on sM). In the conformally flat case, we classify their conformal invariants, including the conformally odd powers…
The decomposition of the spinor bundle of the spin Grassmann manifolds $G_{m,n}=SO(m+n)/SO(m)\times SO(n)$ into irreducible representations of $\mathfrak{so}(m)\oplus\mathfrak{so}(n)$ is presented. A universal construction is developed and the general statement is proven for $G_{2k+1,3}$, $G_{2k,4}$, and $G_{2k+1,5}$ for all $k$. The decomposition is used to discuss properties of the spectrum and the eigenspaces of the Dirac operator.
Physics and Geometry from a Lagrangian: Dirac Spinors on a Generalised Frame Bundle
本文澄清了Hélein和Vey在无结构10-流形上提出的Einstein-Cartan引力变分原理结构，引入广义标架丛并耦合Dirac旋量，证明了所得变分方程在标准标架丛下蕴含通常的Einstein-Cartan-Dirac场方程。
Inspired by the recent work of Physicists Hertog-Horowitz-Maeda, we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admits nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. In fact, we show that the Lichnerowicz Laplacian, which governs the second variation, is the square of a twisted Dirac operator. Our second result, which is a local version of the first one, shows that any metrics of posi…
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.
We give examples illustrating the fact that the different space/time splittings of the tangent bundle of a semi-Riemannian spin manifold give rise to non-equivalent norms on the space of compactly supported sections of the spinor bundle, and as a result, to different completions. We give a necessary and sufficient condition for two space/time splittings to define equivalent norms in terms of a generalized Doppler shift between maximal negative definite subspaces. We explore some consequences for the Noncommutative Geometry program.
We describe, by their holonomy groups, all complete simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalize the Riemannian SpinC case and the pseudo-Riemannian Spin one.
Complex spinorial forms, Brinkmann four-manifolds, and self-dual bundle gerbes
本文研究复旋量形式的微分理论，通过将约束平行性条件转化为外形式的微分系统，探讨了复旋量在不同维度和签名下的性质，并应用于超引力中的超对称条件，证明了某些超对称解属于特定的Brinkmann波族。
In this paper, we extend the study of generalized Killing spinors on Riemannian Spin$^c$ manifolds started by Moroianu and Herzlich to complex Killing functions. We prove that such spinor fields are always real Spin$^c$ Killing spinors or imaginary generalized Spin$^c$ Killing spinors, providing that the dimension of the manifold is greater or equal to 4. Moreover, we classify Riemannian Spin$^c$ manifolds carrying imaginary and imaginary generalized Killing spinors.
We describe the possible holonomy groups of simply connected irreducible non-locally symmetric pseudo-Riemannian spin manifolds which admit parallel spinors.
We study a Killing spinor type equation on spin Riemannian flows. We prove integrability conditions and partially classify those Riemannian flows $M$ carrying non-trivial solutions to that equation in case $M$ is a local Riemannian product, a Sasakian manifold or 3-dimensional.
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.
We introduce a notion of twisted pure spinor in order to characterize, in a unified way, all the special Riemannian holonomy groups just as a classical pure spinor characterizes the special Kähler holonomy. Motivated by certain curvature identities satisfied by manifolds admitting parallel twisted pure spinors, we also introduce the Clifford monopole equations as a natural geometric generalization of the Seiberg-Witten equations. We show that they restrict to the Seiberg-Witten equations in 4 dimensions, and that they admit non-trivial solutions on manifolds with special Riemannian holonomy.
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 represented by the Pauli matrices. In order to uniquely prolong the Killing equation from the tangent bundle onto the spinor bundle over the space-time manifold, it is necessary to pass to the complexification of the manifold and the corresponding bundles, and then to pass to the rea…
Semi-Riemannian $\text{spin}^c$ manifolds carrying generalized Killing spinors and the classification of Riemannian $\text{spin}^c$ manifolds admitting a type I imaginary generalized Killing spinor
本文分类了携带第一类虚广义Killing旋量的黎曼spin^c流形，通过构造平行旋量，并研究半黎曼情形及修正Dirac流，同时提供第二类实例。
From Clifford Bundles to Spinor Classification: Algebraic and Geometric Approaches, and New Spinor Fields in Flux Compactifications
本论文以Kähler-Atiyah丛为框架，结合代数与几何方法研究Clifford结构与旋量场，并通过双线性协变量和Fierz恒等式对旋量进行分类，进而在AdS$_3 \ imes M_8$翘曲通量紧化中识别出新的旋量场类。
On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions can be extended to a Lorentzian manifold with a parallel null vector field. Similarly, every parallel null spinor on a Lorentzian manifold induces an imaginary generalised Killing spinor on a space-like hypersurface. Then, based on the fact that a parallel spinor field induces a…
The article consists of the Russian and English variants of Ph.D. Thesis in which the answers is given on the following questions: 1. how to construct the spinor formalism for n=6; 2. how to construct the spinor formalism for n=8; 3. how to prolong the Riemannian connection from the tangent bundle into the spinor one with the base: a complex analytical 6-dimensional Riemannian space; 4. how to construct the real and complex representations of this bundles; 5. how to construct the curvature spinors and to investigate its properties; 6. how to obtain the canonical form of a bilinear form for the…
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 spinor structures, generalising some earlier results. I will explain how principal fibre bundles and covering spaces provide the key ingredients to the proof. A different type of classification can also be attempted, in terms of the underlying principal fibre bundle, and this allows u…
The aim of this paper is the construction of spinor bundles of Cartan type over certain non-orientable manifolds.
The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold $(M,g)$ is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of $(M,g)$. We characterize the following simply connected pseudo-Riemannian manifolds admitting such subbundles in terms of their holonomy algebras: Riemannian manifolds; Lorentzian manifolds; pseudo-Riemannian manifolds with irreducible holonomy algebras; pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions.
I discuss geometry and normal forms for pseudo-Riemannian metrics with parallel spinor fields in some interesting dimensions. I also discuss the interaction of these conditions for parallel spinor fields with the condition that the Ricci tensor vanish (which, for pseudo-Riemannian manifolds, is not an automatic consequence of the existence of a nontrivial parallel spinor field).
We prove that a smooth Riemannian manifold admitting an imaginary generalized Killing spinor whose Dirac current satisfies an additional algebraic constraint condition can be embedded as spacelike Cauchy hypersurface in a smooth Lorentzian manifold on which the given spinor extends to a null parallel spinor. This is in contrast to a corresponding Cauchy problem for real generalized Killing spinors into Riemannian manifolds. The construction is based on first order symmetric hyperbolic PDE-methods. In fact, the coupled evolution equations for metric and spinor as considered here extend and gene…
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 spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two L…
The purpose of this Note is to prove that each of the following conditions is equivalent to that of the foliation ${\cal F}$ is riemannian: 1) the lifted foliation ${\cal F}^{r}$ on the bundle of $r$-transverse jets is riemannian for an $r\geq 1$; 2) the foliation ${\cal F}_{0}^{r}$ on the slashed ${\cal J}_{0}^{r}$ is riemannian and vertically exact for an $r\geq 1 $; 3) there is a positively admissible transverse lagrangian on ${\cal J}%_{0}^{r}E$, the $r$-transverse slashed jet bundle of a foliated bundle $% E\rightarrow M$, for an $r\geq 1$.
Uncoupled Dirac-Yang-Mills Pairs on Closed Riemannian Spin Manifolds
本文研究闭合黎曼流形上解耦狄拉克-杨-米尔斯方程的解，通过扰动理论和指标定理分析其存在性和结构，提出构造显式解的方法。
We consider Dirac operators on odd-dimensional compact spin manifolds which are twisted by a product bundle. We show that the space of connections on the twisting bundle which yield an invertible operator has infinitely many connected components if the untwisted Dirac operator is invertible and the dimension of the twisting bundle is sufficiently large.
The purpose of this paper is to prove that each of the following conditions is equivalent to that the foliation ${\cal F}$ is riemannian: 1) the lifted foliation ${\cal F}^{r}$ on the $r$-transverse bundle $ν^{r}{\cal F}$ is riemannian for an $r\geq 1$; 2) the foliation ${\cal F}_{0}^{r}$ on a slashed $ν_{\ast}^{r}{\cal F}$ is riemannian and vertically exact for an $r\geq 1$; 3) there is a positively admissible transverse lagrangian on a $ν_{\ast}^{r}{\cal F}$, for an $r\geq 1$. Analogous results have been proved previously for normal jet vector bundles.
A classical result of Gelfand shows that the topologized spectrum of characters on commutative Banach algebra is homeomorphic to the underlying space. This fact is used in solving the Calderón problem in dimension 2 via the boundary control (BC) method. To apply the BC method in dimension 3, the algebra of complex holomorphic functions can be replaced by the space of harmonic quaternion fields, but this space is no longer an algebra and is not commutative. Nonetheless, it has been shown that a suitable notion of a spectrum exists and in the case when the underlying space is convex, the spectru…
Riemannian manifolds with non-zero Killing spinors are Einstein manifolds. Klaus Kröncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in \cite{Kro15}. In this paper, we obtain a new proof for this stability result by using a Bochner type formula in \cite{DWW05} and \cite{Wan91}. Moreover, existence of real Killing spinors is closely related to the Sasaki-Einstein structure. A regular Sasaki-Einstein manifold is essentially the total space of a certain principal $S^{1}$-bundle over a Kähler-Einstein manifold. We prove that if the ba…
Connes fusion of spinors on loop space
本文通过将环路空间上的旋量丛实现为von Neumann代数双模丛，利用Connes融合逐纤维构造了覆盖环路融合的积，建立了string结构、环路融合与Fock空间Connes融合的新关系。
We study an energy functional on the universal spinor bundle over a closed $n$-dimensional spin manifold $M$. The critical points of this functional, which is modelled on the total torsion functional of $G_2$-structures in seven dimensions, are pairs of Ricci-flat metrics and real parallel spinor fields provided that $n$ equals $3$ or $7$. We then modify the functional to obtain the analogue in arbitrary dimensions. Finally we apply the universal spinor bundle approach to solve some ODEs problems concerning $G_2$-structures.
Generalized Killing spinors as Lagrangian sections of principal spin bundles
本文建立了三维伪黎曼旋量流形上ε-单位广义Killing旋量与其主旋量丛上Lagrangian截面的一一对应关系。
We determine the holonomy of generalized Killing spinor covariant derivatives of the form $D= \nabla + Ω$ on pseudo-Riemannian reductive homogeneous spaces in a purely algebraic and algorithmic way, where $Ω: TM \rightarrow Λ^*(TM)$ is a left-invariant homomorphism. This is essentially an application of the theory of invariant principal bundle connections defined over homogeneous principal bundles. Moreover, the algorithm allows a computation of the associated Killing superalgebra in certain cases. The procedure is demonstrated by determining the supersymmetries of certain homogeneous M2 duals…
Riemannian Metrics and Harmonic Sections of Spinor Bundles
研究度量变化下 Dirac 算子最低非负特征值的聚集现象，并证明维数大于等于四的闭 Spin 流形存在允许非平凡调和旋量的 Riemannian 度量。
We show that in the analytic category, given a Riemannian metric $g$ on a hypersurface $M\subset \Z$ and a symmetric tensor $W$ on $M$, the metric $g$ can be locally extended to a Riemannian Einstein metric on $Z$ with second fundamental form $W$, provided that $g$ and $W$ satisfy the constraints on $M$ imposed by the contracted Codazzi equations. We use this fact to study the Cauchy problem for metrics with parallel spinors in the real analytic category and give an affirmative answer to a question raised in Bär, Gauduchon, Moroianu (2005). We also answer negatively the corresponding questions…
We revisit and generalize a recent result of Cederbaum [C2, C3] concerning the rigidity of the Schwarzschild manifold for spin manifolds. This includes the classical black hole uniqueness theorems [BM, GIS, Hw] as well as the more recent uniqueness theorems for pho-ton spheres [C1, CG1, CG2].
We prove the existence of harmonic spinor fields in axisymmetric Riemannian 3-manifolds having nonnegative scalar curvature and asymptotic to the usual constant time hypersurface of Melvin's magnetic universe. Such a spinor can be used in the proof of the uniqueness of the magnetized Schwarzschild solution.
The spinor bundle on loop space
受Stolz和Teichner启发，本文构造了string流形环路空间的旋量丛及融合积，并证明流形上带融合积的旋量丛存在当且仅当该流形具有string结构。
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 the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein-Dirac-Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory.
We prove the instability of some families of Riemannian manifolds with non-trivial real Killing spinors. These include the invariant Einstein metrics on the Aloff-Wallach spaces $N_{k, l}={\rm SU}(3)/i_{k, l}(S^{1})$ (which are all nearly ${\rm G}_2$ except $N_{1,0}$), and Sasaki Einstein circle bundles over certain irreducible Hermitian symmetric spaces. We also prove the instability of most of the simply connected non-symmetric compact homogeneous Einstein spaces of dimensions $5, 6, $ and $7$, including the strict nearly Kähler ones (except ${\rm G}_2/{\rm SU}(3)$).
Let $M$ be a pseudo-Riemannian spin manifold of dimension $n$ and signature $s$ and denote by $N$ the rank of the real spinor bundle. We prove that $M$ is locally homogeneous if it admits more than ${3/4}N$ independent Killing spinors with the same Killing number, unless $n\equiv 1 \pmod 4$ and $s\equiv 3 \pmod 4$. We also prove that $M$ is locally homogeneous if it admits $k_+$ independent Killing spinors with Killing number $λ$ and $k_-$ independent Killing spinors with Killing number $-λ$ such that $k_++k_->{3/2}N$, unless $n\equiv s\equiv 3\pmod 4$. Similarly, a pseudo-Riemannian manifold …
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 bundle, is a bigraded Chern-Dirac bundle with spaces of harmonic spinors isomorphic to the full Dolbeault cohomology class. A similar construction establishes isomorphisms between other types of harmonic spinors and Bott-Chern, Aeppli and twisted cohomology.
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 free loop space of the manifold. First, a Clifford algebra bundle over the loop space is constructed from the vector bundle. A polarization class bundle is constructed, choosing continuously over each point of the loop space a polarization class of Lagrangian subspaces of the comple…
We show that given a conformal structure whose holonomy representation fixes a totally lightlike subspace of arbitrary dimension, there is always a local metric in the conformal class off a singular set which is Ricci-isotropic and gives rise to a parallel, totally lightlike distribution on the tangent bundle. This naturally applies to parallel spin tractors resp. twistor spinors on conformal spin manifolds and clarifies which twistor spinors are locally equivalent to parallel spinors. Moreover, we study the zero set of a twistor spinor using the curved orbit decomposition for parabolic geomet…
We prove that the Cauchy problem for parallel null vector fields on smooth Lorentzian manifolds is well posed. The proof is based on the derivation and analysis of suitable hyperbolic evolution equations given in terms of the Ricci tensor and other geometric objects. Moreover, we classify Riemannian manifolds satisfying the constraint conditions for this Cauchy problem. It is then possible to characterise certain holonomy reductions of globally hyperbolic manifolds with parallel null vector in terms of flow equations for Riemannian special holonomy metrics. For exceptional holonomy groups thes…
In this paper we complete the classification of spin manifolds admitting parallel spinors, in terms of the Riemannian holonomy groups. More precisely, we show that on a given n-dimensional Riemannian manifold, spin structures with parallel spinors are in one to one correspondence with lifts to Spin_n of the Riemannian holonomy group, with fixed points on the spin representation space. In particular, we obtain the first examples of compact manifolds with two different spin structures carrying parallel spinors.
Torsion parallel spinors on Lorentzian four-manifolds and supersymmetric evolution flows on bundle gerbes
本文利用旋量多重形式理论研究洛伦兹四流形上的挠率平行旋量，并探讨由四维超对称NS-NS系统整体双曲解导出的演化流，将斜对称挠率解释为阿贝尔丛格上联络的曲率。
A supermanifold M is canonically associated to any pseudo Riemannian spin manifold (M_0,g_0). Extending the metric g_0 to a field g of bilinear forms g(p) on T_p M, p\in M_0, the pseudo Riemannian supergeometry of (M,g) is formulated as G-structure on M, where G is a supergroup with even part G_0\cong Spin(k,l); (k,l) the signature of (M_0,g_0). Killing vector fields on (M,g) are, by definition, infinitesimal automorphisms of this G-structure. For every spinor field s there exists a corresponding odd vector field X_s on M. Our main result is that X_s is a Killing vector field on (M,g) if and o…
The Standard Model - the Commutative Case: Spinors, Dirac Operator and de Rham Algebra
本文以模论方法综述经典场论的数学基础，涵盖旋量丛、狄拉克算子及 Connes 微分代数，并给出外代数丛与 Connes 微分代数同构的新证明。
Based on the observation that Cacic [10]'s characterization of almost commutative spectral triples as Clifford module bundles can be pushed to endomorphim algebras of Dirac bundles, with the geometric Dirac operator related to the Dirac operator of the spectral triple by a perturbation, the question of complete positivity of the heat semigroups generated by connection laplacian and Dirac and Kostant's cubic Dirac laplacians is approached using spin geometry and C *-Dirichlet forms. The geometric heat semigroups for on endomorphosm algebras of spinor bundles are shown to be quantum dynamical se…
We show how to lift a Riemannian metric and almost symplectic form on a manifold to a Riemannian structure on a canonically associated supermanifold known as the antitangent or shifted tangent bundle. We view this construction as a generalisation of Sasaki's construction of a Riemannian metric on the tangent bundle of a Riemannian manifold.
On a Kähler spin manifold Kählerian twistor spinors are a natural analogue of twistor spinors on Riemannian spin manifolds. They are defined as sections in the kernel of a first order differential operator adapted to the Kähler structure, called Kählerian twistor (Penrose) operator. We study Kählerian twistor spinors and give a complete description of compact Kähler manifolds of constant scalar curvature admitting such spinors. As in the Riemannian case, the existence of Kählerian twistor spinors is related to the lower bound of the spectrum of the Dirac operator.
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 called weak Killing spinors (WK-spinors). They are special solutions of the Einstein-Dirac equation and in dimension n=3 the two equations essentially coincide. It turns out that any Sasakian manifold with Ricci tensor related in some special way to the metric tensor as well as to…
We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincaré-Einstein metrics $(X,g_+)$. Our approach is based on the spectral theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor valued meromorphic family of distributions with residues given by the residue family operators $\slashed{D}_N^{res}(h;λ)$ on spinors. We develop basic aspects and properties of $\slashed{D}_N^{res}(h;λ)$ including conformal covariance, factorization properties by conformally covariant operators…
Strichartz's conjecture for the spinor bundle over the real hyperbolic space
本文研究实双曲空间旋量丛上的 Poisson 变换像，并据此证明广义谱投影的 $L^2$ 一致估计，将 Strichartz 猜想推广至旋量情形。
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