The Higher Spin Dirac Operators on 3-Dimensional Manifolds
Yasushi Homma
研究了三维流形上的高自旋Dirac算子，证明了对于每个配丛，存在两个Laplace型算子。进一步给出了这些Laplace型算子第一特征值的下界估计。
We study the higher spin Dirac operators on 3-dimensional manifolds and show that there exist two Laplace type operators for each associated bundle. Furthermore, we give lower bound estimations for the first eigenvalues of these Laplace type operators.
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We define two categories of Dirac manifolds, i.e. manifolds with complex Dirac structures. The first notion of maps I call \emph{Dirac maps}, and the category of Dirac manifolds is seen to contain the categories of Poisson and complex manifolds as full subcategories. The second notion, \emph{dual-Dirac maps}, defines a \emph{dual-Dirac category} which contains presymplectic and complex manifolds as full subcategories. The dual-Dirac maps are stable under B-transformations. In particular we get two structures of a category on Hitchin'sgeneralized complex manifolds, i.e., two reasonable notions …
A new method of singular reduction is extended from Poisson to Dirac manifolds. Then it is shown that the Dirac structures on the strata of the quotient coincide with those of the only other known singular Dirac reduction method.
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 $n = 7$.
Solutions of the Einstein-Dirac Equation on Riemannian 3-Manifolds with Constant Scalar Curvature
本文包含了一个分类的所有3维流形的常数数量曲率$S \not= 0$进行非平凡的解决方案的Einstein-Dirac方程。
The first eigenvalue of the Dirac operator on locally reducible Riemannian manifolds
证明了具有正数量曲率的紧致局部可约黎曼自旋流形上Dirac算子第一特征值的一个较低估计。我们还确定的流形上的最小可能的特征值达到的通用覆盖。
In this paper, we obtain two Lichnerowicz type formulas for the Dirac-Witten operators. And we give the proof of Kastler-Kalau-Walze type theorems for the Dirac-Witten operators on 4-dimensional and 6- dimensional compact manifolds with (resp.without) boundary
The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtained short time existence and the existence of a global weak solution was established by Jost, Liu, and Zhu. We prove short time existence of the heat flow for Dirac-harmonic maps on closed manifolds.
We show that for generic Riemannian metrics on a closed spin manifold of dimension three the Dirac operator has only simple eigenvalues.
Complex Dirac structures with constant real index on flag manifolds
本文通过李代数的根刻画极大旗流形上具有常实指标的不变复Dirac结构，并在$B$-变换下给出完全分类。
Lower bounds of the Dirac eigenvalues on compact Riemannian spin manifolds with locally product structure
研究了几乎积黎曼结构和几乎厄米特结构之间的一些相似性。受相似性的启发，我们证明了狄拉克算子在具有局部乘积结构的紧致黎曼自旋流形上的下特征值估计。我们还提供了一些例子（极限流形）的极限情况下的估计。
In this paper, we define lower dimensional volumes of compact Riemannian manifolds with boundary. For five dimensional spin manifolds with boundary, we prove a Kastler-Kalau-Walze type theorem associated with one-form perturbations of Dirac operators in this case.
In this paper, we extend the Hijazi inequality, involving the Energy-Momentum tensor, for the eigenvalues of the Dirac operator on $Spin^c$ manifolds without boundary. The limiting case is then studied and an example is given.
In this paper, we get estimates on the higher eigenvalues of the Dirac operator on locally reducible Riemannian manifolds, in terms of the eigenvalues of the Laplace-Beltrami operator and the scalar curvature. These estimates are sharp, in the sense that, for the first eigenvalue, they reduce to the result of Alexandrov.
In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.
Geometric connections and geometric Dirac operators on contact manifolds
我们在一般度量切触流形上构造了几个对几何背景更敏感的自然联络和Dirac型算子。在CR流形的特殊情况下，这些连接也与CR结构兼容，其中包括韦伯斯特连接。我们还描述了几个Weitzenbock型公式。我们的方法是基于工作的P. Gauduchon适用于两个殆复流形自然相关的一个给定的度量接触流形。
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.
In Kähler-Einstein case of positive scalar curvature and even complex dimension, an improved lower bound for the first eigenvalue of the Dirac operator is given. It is shown by a general construction that there are manifolds for which this new lower bound itself is the first eigenvalue.
We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds $(M,g)$ which are traceless cyclic with respect to some quotient expression $M=G/K$ and reductive decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{m}$. Using transversally symmetric fibrations of noncompact type, we give a list of them.
An integer valued topological index of a Dirac operator is introduced for a pair of a 4n+2 dimensional open Spin^c manifold and a section of the determinant line bundle satisfying some property. We show a relation between the index and an index of a Dirac operator of its characteristic submanifold, by a localization of K-class.
The Dirac spectrum on manifolds with gradient conformal vector fields
我们证明了自旋流形上的Dirac算子不允许$L^2$本征旋量，只要该度规具有一定的渐近行为，并且在无穷远处是一个翘曲积。这些条件的度量是满足特别是如果流形是完整的，并进行了非完整的向量场以外的紧集是梯度共形和非零。
The spectrum of twisted Dirac operators on compact flat manifolds
设$M$是具有自旋结构的可定向紧致平坦黎曼流形.本文确定了作用在$M$的扭旋量丛光滑截面上的Dirac算子的谱，并导出了相应的η级数公式。在具有和乐群$\Z_2^k$的流形的情况下，我们给出了特征值重数的一个非常简单的表达式，该表达式允许根据Riemann-Hurwitz zeta函数的值显式地计算$η$ -级数和$η$ -不变量。给出了调和旋量空间的维数，刻画了所有具有非对称Dirac谱的$\Z_2^k$ -流形.此外，我们展示了许多不满足其他类型的等谱性的$\Z_2^k$ -流形的Dirac等谱对的例子。在一个主要的例子中，我们构造了一个大的Dirac等谱紧致平坦$n$ -流形族，两两非同胚。
Eigenvalue estimates of the Kohn-Dirac operator on CR manifolds
本文在紧致CR流形上建立Kohn-Dirac算子的新特征值估计，刻画等式情形为CR扭量旋量的存在，并分类此类流形，构造例子。
Conformal perturbations of dirac operators and general Kastler-Kalau-Walze type theorems for even dimensional manifolds with boundary
本文证明了带边与无边偶维紧流形上Dirac算子共形扰动的一般Kastler-Kalau-Walze型定理。
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 these operators.
The Rescaled Dirac operator fDh and the noncommutative residue for 6-dimensional manifolds
本文计算了6维紧流形上重标度Dirac算子fDh的非交换留数，证明了有边界情形的Kastler-Kalau-Walze型定理，并给出若干重要特例。
Along the lines of the classic Hodge-De Rham theory a general decomposition theorem for sections of a Dirac bundle over a compact Riemannian manifold is proved by extending concepts as exterior derivative and coderivative as well as as elliptic absolute and relative boundary conditions for both Dirac and Dirac Laplacian operators. Dirac sections are shown to be a direct sum of harmonic, exact and coexact spinors satisfying alternatively absolute and relative boundary conditions. Cheeger's estimation technique for spectral lower bounds of the Laplacian on differential forms is generalized to th…
For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the metric is extended over a handle of codimension at least two attached at the boundary. Applications of this result include the construction of non-isotopic metrics with invertible Dirac operator, and a concordance existence and classification theorem.
The Hodge-Dirac operator and Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary
本文将定向偶数维 Riemannian 流形上 Hodge-Dirac 算子的谱 Einstein 双线性泛函结果推广至带边界的 4 维情形，并证明了带边流形上相关的 Dabrowski-Sitarz-Zalecki 型定理。
The Einstein-Dirac Equation on Riemannian Spin Manifolds
我们构造了Einstein-Dirac方程的精确解，该方程通过能量动量张量将引力场与Dirac算子的本征旋量耦合起来。为此，我们引入一个新的场方程推广的概念杀死旋。这个旋量场方程的解称为弱Killing旋量（WK-旋量）。它们是Einstein-Dirac方程的特解，在n=3维中两个方程基本重合。证明了任何具有以某种特殊方式与度量张量以及切触结构相关的Ricci张量的Sasakian流形都存在WK-旋量。这一结果是一个后果的调查特殊旋量场方程的Sasakian流形（Sasakian准Killing旋量）。总之，在奇数维中，接触几何产生爱因斯坦-狄拉克方程的解。此外，我们还证明了在n > 8的所有维数下，Einstein-Dirac方程的解的存在性，这些解不是WK-旋量。
A note on coupled Dirac-harmonic maps
本文提出一种从 $S^1\ imes M$ 型流形构造耦合狄拉克-调和映射的方法，其中 $M$ 为带满足范数条件的调和旋量对的闭自旋流形。
Spectrum of the Dirac operator on Compact Riemannian Manifolds
本文研究紧黎曼流形上Dirac算子的特征值问题，通过外在估计推导出任意连续n个特征值之和的不等式，并应用于欧氏空间、单位球面和射影空间中的紧子流形，同时给出曲率条件下的普适界及Atiyah-Singer Laplacian特征值和的上界估计。
The Dirac Operator and Conformal Compactification
本文给出了与具有非空边界的黎曼流形内部共形等价的完备黎曼流形上Dirac算子的L^2核和谱的结果.
It is well-known that the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator on a closed Riemannian $\text{spin}^{\mathbb{C}}$ manifold $M^{2k}$ of dimension $2k$ for $k \in \mathbb{N}$ is symmetric. In this article, we prove that over an odd-dimensional Riemannian product $M_{1}^{2p} \times M_{2}^{2q+1}$ with a product $\text{spin}^{\mathbb{C}}$ structure for $p \geq 1, q \geq 0$, the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator given by a product connection is symmetric if and only if either the $\text{spin}^{\mathbb{C}}$ Dirac spectrum of $M_{2}^{2q+1}$ is symmetric or $(e^{…
In this paper, we prove a Kastler-Kalau-Walze type theorem for 4-dimensional and 6-dimensional spin manifolds with boundary associated with the conformal Robertson-Walker metric. And we give two kinds of operator theoretic explanations of the gravitational action for boundary in the case of 4-dimensional manifolds with flat boundary. In particular, for 6-dimensional spin manifolds with boundary with the conformal Robertson-Walker metric, we obtain the noncommutative residue of the composition of $π^+D^{-1}$ and $π^+D^{-3}$ is proportional to the Einstein-Hilbert action for manifolds with bound…
The J-twist D_J of the Dirac operator and the Kastler-Kalau-Walze type theorem for six-dimensional manifolds with boundary
本文将已有三维、四维带边近积黎曼 spin 流形上关于 Dirac 算子 J-扭变 D_J 的 Kastler-Kalau-Walze 型定理推广到六维情形。
We study the spectrum of the Dirac operator $D$ on pseudo-Riemannian spin manifolds of signature $(p,q)$, considered as an unbounded operator in the Hilbert space $L^2_ξ(S)$. The definition of $L^2_ξ(S)$ involves the choice of a $p$-dimensional time-like subbundle $ξ\subset TM$. We establish a sufficient criterion for the spectra of $D$ induced by two maximal time-like subbundles $ξ_1,ξ_2\subset TM$ to be equal. If the base manifold $M$ is compact, the spectrum does not depend on $ξ$ at all. We then proceed by explicitely computing the full spectrum of $D$ for $\mathbb R^{p,q}$, the flat torus…
The Dirac-Dolbeault Operator Approach to the Hodge Conjecture
该研究利用Dirac-Dolbeault算子相关的Green函数表示与Nash-Moser广义反函数定理，在特定假设下证明复射影流形满足条件的复子流形存在性，进而证明复射影流形的Hodge猜想。
The well known conformal covariance of the Dirac operator acting on spinor fields over a semi Riemannian spin manifold does not extend to powers thereof in general. For odd powers one has to add lower order curvature correction terms in order to obtain conformal covariance. We derive an algorithmic construction in terms of associated tractor bundles to compute these correction terms. Depending on the signature of the semi Riemannian manifold in question, the obtained conformal powers of the Dirac operator turn out to be formally self-adjoint with respect to the $L^2-$scalar product, or formall…
We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact Kähler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of Kählerian Killing ${\rm spin}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a Kählerian Killing ${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case the result of A. Moroianu stating that, on a compact Kähler-Einstein man…
In this study, we obtain a spinorial Gauss formula for a lightlike hypersurface in Lorentzian manifold with 4-dimension. Then, we take into account the changes caused by degenerate metric on hypersurface and investigate Dirac operator for lightlike hypersurface. Later, we establish the relation between Dirac operators and Riemannian curvatures of manifold and hypersurface.
We prove that on a compact $n$-dimensional spin manifold admitting a non-trivial harmonic 1-form of constant length, every eigenvalue $λ$ of the Dirac operator satisfies the inequality $λ^2 \geq \frac{n-1}{4(n-2)}\inf_M Scal$. In the limiting case the universal cover of the manifold is isometric to $R\times N$ where $N$ is a manifold admitting Killing spinors.
A Reilly-Type Inequality for CR Dirac Operators on CR Manifolds and Applications
本文推导CR流形上CR Dirac算子的Reilly型不等式，并应用于p-质量下界估计及有界域上CR Dirac方程的求解。
Holomorphic spinors and the Dirac equation
一个具有正数量曲率且狄拉克算子的第一特征值最小的闭自旋Kähler流形被全纯旋量刻画。证明了在任意自旋Kähler-Einstein流形上，每个全纯旋量都是Dirac算子平方的本征旋量的有限和。证明了全纯旋量的消灭定理。
Spectral Bounds for Dirac Operators on Open Manifolds
我们将紧流形上Dirac算子的几个经典特征值估计推广到非紧甚至不完备流形上。这包括弗里德里希的估计流形与积极的数量曲率以及作者的估计表面。
In this paper we study the modular classes of Dirac manifolds and of Dirac maps, and we discuss their basic properties. We apply these results to explain the relationship between the modular classes of the various structures involved in the reduction of a Poisson manifold under the action by of a Poisson Lie group.
Dirac submanifolds of Jacobi manifolds
Poisson流形的Dirac子流形的概念由Xu（arXiv：math.SG/0110326）研究。我们给出了一个解释徐的定义焊接到一个规范化的子流形张量场的一般概念。然后，这个解释被用来定义狄拉克子流形的雅可比流形。讨论了几个性质和例子。
Reduction of Jacobi manifolds via Dirac structures theory
我们首先回顾了Jacobi流形，广义李双代数胚，广义Courant代数胚和Dirac结构的一些基本定义和事实。建立了Jacobi流形$(M,Λ,E)$（其中1是容许函数）的广义李双代数胚的可约Dirac结构与$M$的Jacobi商流形之间的一一对应关系.本文从Dirac结构理论的角度研究了Jacobi约化，并给出了一些例子和应用。
Eigenvalue Estimates for the Dirac Operator on Quaternionic Kaehler Manifolds
我们考虑紧四元数Kaehler流形上的Dirac算子，并证明了其谱的一个下界。这个估计是尖锐的，因为它是狄拉克算子在四元数射影空间上的第一本征值。
Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds
本文研究非紧致流形上Dirac-Schrödinger算子的相对eta不变量，推导非紧致流形上的谱流公式，并推广Atiyah-Patodi-Singer索引定理至非紧边界情况。
In this paper, we define the spectral Einstein functional associated with the Dirac operator for manifolds with boundary. And we give the proof of Kastler-Kalau-Walze type theorem for the spectral Einstein functional associated with the Dirac operator on 4-dimensional manifolds with boundary.
How to Construct a Dirac Operator in Infinite Dimensions
我们定义的概念，一个合作黎曼结构，并显示它如何可以用来定义狄拉克算子在一个适当的无限维流形。特别是，这种方法适用于所谓的弦流形的光滑闭路空间。
Eigenvalue estimates of the Dirac operator depending on the Ricci tensor
利用精化Weitzenböck技巧证明了紧致黎曼自旋流形上Dirac算子第一特征值的一个新的下界.它适用于具有调和曲率张量的流形，并依赖于Ricci张量。示例显示了它与其他已知边界相比的行为。
Eigenvalue estimates for the Dirac operator depending on the Weyl curvature tensor
我们证明了紧流形上Dirac算子第一特征值的新下界，其Weyl张量或曲率张量分别是发散自由的。在爱因斯坦流形的特殊情况下，我们得到的估计依赖于外尔张量。
We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.
We study the transversely metaplectic structure and the transversely symplectic Dirac operator on a transversely symplectic foliation. Moreover, we give the Weitzenbock type formula for transversely symplectic Dirac operators and we estimate the lower bound of the eigenvalues of the transversely symplectic Dirac operator on special spinors space on transverse Kahler foliations.
For a Hamiltonian, proper and free action of a Lie group $G$ on a Dirac manifold $(M,L)$, with a regular moment map $μ:M\to \mathfrak{g}^*$, the manifolds $M/G$, $μ^{-1}(0)$ and $μ^{-1}(0)/G$ all have natural induced Dirac structures. If $(M,L)$ is an integrable Dirac structure, we show that $M/G$ is always integrable, but $μ^{-1}(0)$ and $μ^{-1}(0)/G$ may fail to be integrable, and we describe the obstructions to their integrability.
On the local structure of Dirac manifolds
我们给出了Dirac结构的一个局部标准形。作为结果，我们证明了狄拉克流形的前辛叶的维数具有相同的奇偶性。我们还表明，给定一个点$m$的狄拉克流形$M$，有一个明确的横向泊松结构的前辛叶$P$通过$m$。最后，我们描述了一个前辛叶的几何数据的邻域。这一描述与Vorobjev对Poisson情形给出的描述一致
Twisted Jacobi manifolds, twisted Dirac-Jacobi structures and quasi-Jacobi bialgebroids
我们研究扭曲的雅可比流形，一个概念，我们已经介绍了在以前的说明。扭雅可比流形可以用扭Dirac-雅可比来刻画，扭Dirac-雅可比是Courant-Jacobi代数胚的子丛。我们证明了每个扭曲的Jacobi流形都有一个相关联的李代数胚，它具有一个1-上圈。我们引入了拟Jacobi双代数胚的概念，并证明了每个扭曲Jacobi流形都有一个正则相关的拟Jacobi双代数胚。此外，拟Jacobi双代数胚的二重化是Courant-Jacobi代数胚。给出了扭Jacobi流形和扭Dirac-Jacobi结构的几个例子。
In this note, we prove lower and upper bounds for Dirac operators of submanifolds in certain ambient manifolds in terms of conformal and extrinsic quantities.
The spectral torsion for the one form rescaled Dirac operator
受谱挠率与1-形式缩放Dirac算子启发，本文提出了Dirac算子谱挠率的新扩展，并计算了无边偶数维自旋流形上1-形式缩放Dirac算子的谱挠率。
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