Cartan Spinor Bundles on Manifolds

A non-normal Fefferman-type construction of split-signature conformal structures admitting twistor spinors

We treat a non-normal Fefferman-type construction based on an inclusion $\SL(n+1)\embed\Spin(n+1,n+1)$. The construction associates a split signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. For $n\geq 3$ the induced conformal Cartan connection is shown to be normal if and only if it is flat. The main technical work of this article consists in showing that in the non-flat case the normalised conformal Cartan connection still allows a parallel (pure) spin-tractor and thus a corresponding (pure) twistor spinor on the conformal space. The Fefferman-type constru…

A vertical Liouville subfoliation on the cotangent bundle of a Cartan space and some related structures

In this paper we study some problems related to a vertical Liouville distribution (called vertical Liouville-Hamilton distribution) on the cotangent bundle of a Cartan space. We study the existence of some linear connections of Vrănceanu type on Cartan spaces related to some foliated structures. Also, we identify a certain $(n,2n-1)$--codimensional subfoliation $(\mathcal{F}_V,\mathcal{F}_{C^*})$ on $T^*M_0$ given by vertical foliation $\mathcal{F}_V$ and the line foliation $\mathcal{F}_{C^*}$ spanned by the vertical Liouville-Hamilton vector field $C^*$ and we give a triplet of basic connecti…

Two-spinor tetrad and Lie derivatives of Einstein-Cartan-Dirac fields

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 geometrically constructed from the unique assumption of a complex vector field $S\to M$ with 2-dimensional fibers. The Lie derivatives of objects of all considered types, with respect to a vector field $X:M\to TM$, are well-defined without making any special assumption about $X$, and fulfil…

Chern-Dirac bundles on non-Kähler Hermitian manifolds

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.

以Lorentz-Minkowski时空未来光锥为模型的Cartan几何

Cartan geometries with model the future lightlike cone of Lorentz-Minkowski spacetime

本文建立以Lorentz-Minkowski时空未来光锥为模型的光like Cartan几何理论，通过构造标准tractor丛给出其向量丛数据刻画，并揭示Cartan联络中超出度量与根数据的额外几何信息。

Universal Cartan-Lie algebroid of an anchored bundle with connection and compatible geometries

Consider an anchored bundle $(E,ρ)$, i.e. a vector bundle $E\to M$ equipped with a bundle map $ρ\colon E \to TM$ covering the identity. M.~Kapranov showed in the context of Lie-Rinehard algebras that there exists an extension of this anchored bundle to an infinite rank universal free Lie algebroid $FR(E)\supset E$. We adapt his construction to the case of an anchored bundle equipped with an arbitrary connection, $(E,\nabla)$, and show that it gives rise to a unique connection $\tilde \nabla$ on $FR(E)$ which is compatible with its Lie algebroid structure, thus turning $(FR(E), \tilde \nabla)$ …

Algebraic constructions in the category of vector bundles

The category of generalized Lie algebroids is presented. We obtain an exterior differential calculus for generalized Lie algebroids. In particular, we obtain similar results with the classical and modern results for Lie algebroids. So, a new result of Maurer-Cartan type is presented. Supposing that any vector subbundle of the pullback vector bundle of a generalized Lie algebroid is called interior differential system (IDS) for that generalized Lie algebroid, a theorem of Cartan type is obtained. Extending the classical notion of exterior differential system (EDS) to generalized Lie algebroids,…

Cartan Spaces and Natural Foliations on the Cotangent Bundle

In this paper, the natural foliations in cotangent bundle T*M of Cartan space (M,K) are studied. It is shown that the geometry of these foliations is closely related to the geometry of the Cartan space (M,K) itself. This approach is used to obtain new characterizations of Cartan spaces with negative constant curvature.

Cones and Cartan geometry

We show that the extended principal bundle of a Cartan geometry of type $(A(m,\mathbb{R}),GL(m,\mathbb{R}))$, endowed with its extended connection $\hatω$, is isomorphic to the principal $A(m,\mathbb{R})$-bundle of affine frames endowed with the affine connection as defined in classical Kobayashi-Nomizu volume I. Then we classify the local holonomy groups of the Cartan geometry canonically associated to a Riemannian manifold. It follows that if the holonomy group of the Cartan geometry canonically associated to a Riemannian manifold is compact then the Riemannian manifold is locally a product …

Spin spaces, Lipschitz groups, and spinor bundles

It is shown that every bundle $\varSigma\to M$ of complex spinor modules over the Clifford bundle $\Cl(g)$ of a Riemannian space $(M,g)$ with local model $(V,h)$ is associated with an lpin ("Lipschitz") structure on $M$, this being a reduction of the ${\Ort}(h)$-bundle of all orthonormal frames on M to the Lipschitz group $\Lpin(h)$ of all automorphisms of a suitably defined spin space. An explicit construction is given of the total space of the $\Lpin(h)$-bundle defining such a structure. If the dimension m of M is even, then the Lipschitz group coincides with the complex Clifford group and t…

The zero loci of Z/2 harmonic spinors in dimension 2, 3 and 4

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 the following requirement: The norm of the spinor must extend across Z as a continuous function vanishing on Z. In particular, the vanishing locus of the norm of the spinor is the complement of the set where the real line bundle is defined, and hence where the spinor is defined. …

The spinor and Weierstrass representations of surfaces in space

In this paper, following Sullivan, Kusner, and Schmitt, we study conformal immersions of Riemann surfaces into the three-dimensional Euclidean space. Regarding such immersions as special bundle maps from the tangent bundle of the surface to the cotangent bundle of the 2-dimensional sphere, we generalize the classical Weierstrass representation of minimal surfaces to the case of arbitrary conformal immersions. We study how such an immersion gives rise to a spin structure on the surface together with a pair of spinors and how the immersion itself can be studied by means of these spinors.

On the Cartan Model of the Canonical Vector Bundles over Grassmannians

We give a representation of canonical vector bundles over Grassmannian manifolds as non-compact affine symmetric spaces as well as their Cartan model in the group of the Euclidean motions.

黎曼几何中对应空间导出的嘉当几何

A Cartan geometry from a correspondence space in Riemannian geometry

该研究探讨黎曼几何中对应空间导出的嘉当几何，证明(Euc(n),O(n-1))型嘉当几何与底流形的近接触度量结构等数据一一对应，且n≥3时存在正则嘉当联络。

(1,1)-forms acting on Spinors on Kähler Surfaces

It is known that, for Dirac operators on Riemann surfaces twisted by line bundles with Hermitian-Einstein connections, it is possible to obtain estimates for the first eigenvalue in terms of the topology of the twisting bundle \cite{JL2}. Attempts to generalize topological estimates for higher rank bundles or higher dimensional manifolds have been so far unsuccessful. In this work we construct a class of examples which indicates one problem that arises on such attempts to derive topological estimates.

The spinor bundle of Riemannian products

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.

3-旋量丛上的$G_2$-瞬子

$G_2$-instantons on the spinor bundle of the 3-sphere

本研究对具有$SU(2)^3$对称性的$G_2$-瞬子分类，在带Bryant-Salamon型渐近锥形$G_2$度量的3-球旋量丛上构造新实例，并证明非$SU(2)^3$不变的同渐近行为$G_2$-瞬子必有受阻形变。

Remarks on non-maximal integral elements of the Cartan plane in jet spaces

There is a natural filtration on the space of degree-$k$ homogeneous polynomials in $n$ independent variables with coefficients in the algebra of smooth functions on the Grassmannian $\mathrm{Gr}(n,s)$, determined by the tautological bundle. In this paper we show that the space of $s$-dimensional integral elements of a Cartan plane on $J^{k-1}(E,n)$, with $\mathrm{dim}\, E=n+m$, has an affine bundle structure modeled by the the so-obtained bundles over $\mathrm{Gr}(n,s)$, and we study a natural distribution associated with it. As an example, we show that a third-order nonlinear PDE of Monge-Am…

On canonical Cartan connections associated to filtered G-structures

A filtered manifold is a smooth manifold $M$ together with a filtration of the tangent bundle by smooth subbundles which is compatible with the Lie bracket of vector fields in a certain sense. The Lie bracket of vector fields then induces a bilinear operation on the associated graded of each tangent space of $M$ making it into a nilpotent graded Lie algebra. Assuming that these symbol algebras are the same for all points, one obtains a natural frame bundle for the associated graded to the tangent bundle, and filtered G--structures are defined as reductions of structure group of this bundle. Ge…

On the Szegö kernel of Cartan-Hartogs domains

Inspired by the work of Z. Lu and G. Tian [21] in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan-Hartogs domain based on a bounded symmetric domain. The main ingredients in our analysis are the fact that every Cartan-Hartogs domain can be viewed as an "iterated" disk bundle over its base and the the ideas given in [4] for the computation of the Szegö kernel of the disk bundle over an Herimitian symmetric space of compact type.

从拉格朗日量推导的物理与几何：广义标架丛上的狄拉克旋量

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。

On operator fields in the bundle of Dirac spinors

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.

A bundle gerbe construction of a spinor bundle from the smooth free loop of a vector bundle

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…

旋量丛的黎曼度量与调和截面

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。

环路空间上的旋量丛

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。

Homogeneous Cartan Geometries

We describe invariant principal and Cartan connections on homogeneous principal bundles and show how to calculate the curvature and the holonomy; in the case of an invariant Cartan connection we give a formula for the infinitesimal automorphisms. The main result of this paper is that the above calculations are purely algorithmic. As an example of an homogeneous parabolic geometry we treat a conformal structure on the product of two spheres.

洛伦兹四流形上的挠率平行旋量及丛格上的超对称演化流

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 note on metric connections for chiral and Dirac spinors

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.

Poisson fiber bundles and coupling Dirac structures

We give sufficient conditions for the existence of a Dirac structure on the total space of a Poisson fiber bundle endowed with a compatible connection. We also show that Cartan and Cartan-Hannay-Berry connections give rise to coupling Dirac structures.

标架丛的局部推广

A local generalisation of frame bundles

本文引入广义标架丛，通过余标架与弱化Maurer-Cartan方程刻画局部结构，并研究李群作用及底流形构造，发现其底层流形可为奇异流形。

A Local Existence Theorem for the Einstein-Dirac Equation

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 $M^n$ admitting real Killing spinors (resp. parallel spinors), there exist warped product metrics $\barη$ on $M^n \times {\mathbb R}$ such that $(M^n \times {\mathbb R}, \barη)$ admit Einstein spinors (resp. weak Killing spinors). To prove the result we split the Einstein-Dirac equation into evolution equations and constraints, by means of Cartan's frame formalism, and apply the local preservation property of constraints.

Comparison of two formulas for metric connections in the bundle of Dirac spinors

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.

实双曲空间上旋量丛的 Strichartz 猜想

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。

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We…

Conformal geometry of the supercotangent and spinor bundles

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…

Quantum diffusion on almost commutative spectral triples and spinor bundles

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…

On the spinor structure of the homogeneous and isotropic universe in closed model

The closed homogeneous and isotropic universe is considered. The bundles of Weyl and Dirac spinors for this universe are explicitly described. Some explicit formulas for the basic fields and for the connection components in stereographic and in spherical coordinates are presented.

Spinors as automorphisms of the tangent bundle

We show that, on a 4-manifold M endowed with a spin^c structure induced by an almost-complex structure, a self-dual (= positive) spinor field ϕ\in Γ(W^+) is the same as a bundle morphism ϕ: TM \to TM acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of ϕon tangent vectors, and that the squaring map σ: W^+ \to Λ^+ acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kahler and symplectic structures.

从Clifford丛到旋量分类：代数与几何方法，以及通量紧化中的新旋量场

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。

A note on real Killing spinors in Weyl geometry

This text is dedicated to the real Killing equation on 3-dimensional Weyl manifolds. Any manifold admitting a real Killing spinor of weight 0 satisfies the conditions of a Gauduchon-Tod geometry. Conversely, any simply connected Gauduchon-Tod geometry has a 2-dimensional space of solutions of the real Killing equation on the spinor bundle of weight 0.

The decomposition of the spinor bundle of Grassmann manifolds

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.

A Universal Property of the Groups Spin^c and Mp^c

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 enable the construction of spinor bundles. We proceed to prove a similar result for metaplectic^c structures on symplectic manifolds.

Dirac operators on real spinor bundles of complex type

Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. We compute the obstruction for a vector bundle $S$ over $(M,g)$ to admit a Dirac operator whose principal symbol induces on $S$ the structure of a bundle of irreducible real Clifford modules of complex type, that is, a real spinor bundle of irreducible complex type. In order to do this, we use the theory of Lipschitz structures in signature $p-q\equiv_8 3,7$ to reformulate the problem as the obstruction problem for $(M,g)$ to admit a $\mathrm{Spin}^{o}_α$ structure with $α= -1$ if $p-q \equiv_{8} 3$ or $α= +1$ if $ p-q \equiv_{8…

A note on Kosmann-Lie derivatives of Weyl spinors

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.

Non-relativistic twistor theory and Newton--Cartan geometry

We develop a non-relativistic twistor theory, in which Newton--Cartan structures of Newtonian gravity correspond to complex three-manifolds with a four-parameter family of rational curves with normal bundle ${\mathcal O}\oplus{\mathcal O}(2)$. We show that the Newton--Cartan space-times are unstable under the general Kodaira deformation of the twistor complex structure. The Newton--Cartan connections can nevertheless be reconstructed from Merkulov's generalisation of the Kodaira map augmented by a choice of a holomorphic line bundle over the twistor space trivial on twistor lines. The Coriolis…

复旋量形式、Brinkmann四维流形与自对偶束丛

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。

Gauge theory, calibrated geometry and harmonic spinors

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-Witten equations. By collapsing each fibre of the spinor bundle to a point, solutions of the Taubes-Pidstrygach equations are related to generalized harmonic spinors. This approach is also generalized for arbitrary fibrations (without singular fibres) compatible with an appropriate …

On the spinor formalism for even n

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…

Courant Algebroids in Parabolic Geometry

Let $p$ be a Lie subalgebra of a semisimple Lie algebra $g$ and $(G,P)$ be the corresponding pair of connected Lie groups. A Cartan geometry of type $(G,P)$ associates to a smooth manifold $M$ a principal $P$-bundle and a Cartan connection, and a parabolic geometry is a Cartan geometry where $P$ is parabolic. We show that if $P$ is parabolic, the adjoint tractor bundle of a Cartan geometry, which is isomorphic to the Atiyah algebroid of the principal $P$-bundle, admits the structure of a (pre-)Courant algebroid, and we identify the topological obstruction to the bracket being a Courant bracket…

具有平凡对数切丛的复流形上的对数 Cartan 几何

Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle

本文研究了具有平凡对数切丛的紧复流形上的全纯 Cartan 几何，证明了其同构于自同构群作用拉回的充要条件是主丛存在保持几何结构的对数连接。

Classification of 1st order symplectic spinor operators over contact projective geometries

We give a classification of $1^{st}$ order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so called higher symplectic, harmonic or generalized Kostant spinor modules. Higher symplectic spinor modules are arising from the Segal-Shale-Weil representation of the metaplectic group by tensoring it by finite dimensional modules. We show that for all pairs of the considered bundles, there is at most one $1^{st}$…

On the space of connections having non-trivial twisted harmonic spinors

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.

旋量丛上的拉普拉斯算子：自共轭扩展和正则化行列式

Laplacians in spinor bundles over translation surfaces: self-adjoint extentions and regularized determinants

本文研究了平坦共形度量下自旋丛上自共轭扩展的正则化行列式，并提出新的正则化方法。

Field Theories found Geometrically from Embeddings in Flat Frame Bundles

We present two families of exterior differential systems (EDS) for non-isometric embeddings of orthonormal frame bundles over Riemannian spaces of dimension q = 2, 3, 4, 5.... into orthonormal frame bundles over flat spaces of sufficiently higher dimension. We have calculated Cartan characters showing that these EDS satisfy Cartan's test and are well-posed dynamical field theories. The first family includes a constant-coefficient (cc) EDS for classical Einstein vacuum relativity (q = 4). The second family is generated only by cc 2-forms, so these are integrable (but nonlinear) systems of parti…