Tomas Salac
The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the theory of exterior differential systems to the setting of weighted differential operators.
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We apply the Cartan-Kahler theorem for the k-Dirac operator studied in Clifford analysis and to the parabolic version of this operator. We show that for k = 2 the tableaux of the first prolongations of these two operators are involutive. This gives us a new characterization of the set of initial conditions for the 2-Dirac operator.
We introduce linear Dirac and generalized complex structures on Cartan geometries and give criteria for Dirac subalgebras of $\frkg\ltimes\frkg^*$ representing Dirac structures on a Cartan geometry. We prove that there is a bijection between the linear generalized structures on a torsion free Cartan geometry and the equivariant generalized structures on its corresponding homogeneous space.
We establish an upper estimate for the small eigenvalues of the twisted Dirac operator on Kahler submanifolds in Kahler manifolds carrying Kahlerian Killing spinors. We then compute the spectrum of the twisted Dirac operator of the canonical embedding \CP^d \rightarrow \CP^n in order to test the sharpness of the upper bounds
The Cartan-Kähler theorem for exterior differential systems on transitive Lie algebroids
将外微分系统推广到李代数胚，在锚映射满射条件下给出两个Cartan-Kähler定理版本，并应用于变分学的不变逆问题。
In this paper, we give an optimal lower bound for the eigenvalues of the basic Dirac operator on a quaternion-Kahler foliation. The limiting case is characterized by the existence of quaternion-Kahler Killing spinors. We end this paper by giving some examples.
In this paper, we prove Kirchberg inequalities for any kahler spin foliations. Their limiting cases are then characterized as being transversal minimal Einstein foliations. The key point is to introduce the transversal kahlerian twistor operators.
It is shown that on a compact spin symmetric space with a Kähler or Quaternion-Kähler structure, the first eigenvalue of the Dirac operator is linked to a ''{lowest}'' action of the holonomy, given by the fiberwise action on spinors of the canonical forms characterized by this holonomy. The result is also verified for the symmetric space $\mathrm{F}_4/\mathrm{Spin}_9$, proving that it is valid for all the ''{possible}'' holonomies in the Berger's list occurring in that context. The proof requires a characterization of the first eigenvalue of the Dirac operator which was given in Commun. Math. …
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…
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.
The famous theorems of Cartan, related to the axiom of $r$-planes, and Leung-Nomizu about the axiom of $r$-spheres were extended to Kähler geometry by several authors. In this paper we replace the strong notions of totally geodesic submanifolds ($r$-planes) and extrinsic spheres ($r$-spheres) by a wider class of special isometric immersions such that theorems of type "axioms of $r$-special submanifolds" could hold. We verify also that there are plenty of special submanifolds in real and complex space forms and, in the codimension one case, in Einstein manifolds.
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 theorems for holomorphic spinors are proved.
We calculate relations on characteristic classes which are obstructions preventing closed Kähler manifolds from carrying holomorphic Cartan geometries. We apply these relations to give global constraints on the phase spaces of complex analytic determined and underdetermined systems of differential equations.
We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit branched flat holomorphic projective structure. We also give an example of a non-flat branched holomorphic normal projective structure on a compact complex surface. It is known that no compact complex surface admits such a structure…
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.
In this paper we introduce the Dirac and spin-Dirac operators associated to a connection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac operators associated with a Levi-Civita connection on a Riemannian (Lorentzian) space(time) and calculate the square of these operators, which play an important role in several topics of modern Mathematics, in particular in the study of the geometry of moduli spaces of a class of black holes, the geometry of NS-5 brane solutions of type II supergravity theories and BPS solitons in some string theories. We obtain a generalized Lichnerowicz form…
This paper addresses the problem of existence of generalized Landsberg structures on surfaces using the Cartan-Kähler Theorem and a Path Geometry approach.
We use Dirac operator techniques to establish a sharp lower bound for the first eigenvalue of the twisted Dolbeault Laplacian on holomorphic line bundles over compact Kähler manifolds.
Characteristic forms of complex Cartan geometries II
本文研究复Cartan几何的特征形式，通过允许非紧致和非Kähler流形，推导了向量丛的上同调不变量，并首次给出了Cartan几何的Chern-Simons不变量结果。
On the Bergman metric of Cartan-Hartogs domains
本文在 Cartan-Hartogs 域上引入 Bergman 对偶，证明其双全纯等价于单位球、Bergman 度量是 Kahler-Ricci 孤子及对偶有限射影诱导三者等价，并与其他典范度量比较，提出开问题。
We use Dirac operator techniques to a establish sharp lower bound for the first eigenvalue of the Dolbeault Laplacian twisted by Hermitian-Einstein connections on vector bundles of negative degree over compact Kähler manifolds.
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…
In the present paper, the $(HM',S,T)$-Cartan connections on pseudo-Finsler manifolds, introduced by A. Bejancu and H.R. Farran, are obtained by the natural almost complex structure arising from the nonlinear connection $HM'$. We prove that the natural almost complex linear connection associated to a $(HM',S,T)$-Cartan connection is a metric linear connection with respect to the Sasaki metric $G$. Finally we give some conditions for $(M', J, G)$ to be a Kähler manifold.
In this paper, we define a differential operator as a modified Dirac operator. Using the operator, we introduce a quaternionic $k$-vector field on a quaternionic Kähler manifold and show that any quaternionic $k$-vector field corresponds to a holomorphic $k$-vector field on the twistor space. We calculate the dimension of the space of quaternionic $k$-vector fields on $\mathbb{H}P^n$.
In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the differential calculus associated to a split Lie 2-algebroid and establish the Manin triple theory for split Lie 2-algebroids. More precisely, we give the notion of a strict Dirac structure and define a Manin triple for split Lie 2-algebroids to be a CLWX 2-algebroid with two transversal strict Dirac structures. We show that there is a one-to-one correspondence between Manin triples for split Lie 2-alg…
We partially describe equivariant Dirac and generalized complex structures on a homogeneous space $G/K$ by giving equivalent data involving only the Lie algebra. We consider real semisimple adjoint orbits in any semisimple Lie algebra over $\mathbb R$ and real nilpotent orbits in $sl_n (\mathbb R)$. We give a complete classification for Riemannian symmetric spaces and for a compact group modulo a closed, connected subgroup containing a Cartan subgroup.
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.
The generality of closed $G_2$ solitons
本文通过构造自然的外微分系统并应用Cartan-Kahler理论，证明了闭$G_2$结构拉普拉斯流孤子空间在微分同胚意义下依赖于6变量的16个函数，并指出梯度孤子情形的非对合性使其普遍性仍待解。
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.
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。
We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge-de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.
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.
Cartan Geometry and Infinite-Dimensional Kempf-Ness Theory
本文率先开发了Kempf-Ness定理的严格无穷维框架，提出基于Cartan丛的新方法，利用Maurer-Cartan形式定义轨道复模型与Kempf-Ness函数并证明其凸性，可应用于Kähler几何、形变量化与规范理论。
For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class $\mathcal C$, whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class $\m…
In this paper, we consider the symmetries of the Dirac operator derived from a connection with skew-symmetric torsion. We find that the generalized conformal Killing-Yano tensors give rise to symmetry operators of the massless Dirac equation, provided an explicitly given anomaly vanishes. We show that this gives rise to symmetries of the Dirac operator in the case of strong Kahler with torsion (KT) and strong hyper-Kahler with torsion (HKT) manifolds.
Dolbeault-Dirac operators on compact Kähler manifolds in Banach noncommutative geometry
本文研究了紧Kähler流形上带有Hermite全纯向量丛系数的Dolbeault-Dirac算子的L^p理论，建立了闭L^p实化算子的双侧性，证明了其具有有界H^∞函数分析，获得了Gaffney型估计和L^p-Hodge分解，并证明了该算子生成紧Banach谱三元组，其索引与全纯欧拉特征数一致且与p无关。
We present a clear-cut example of the importance of the functorial approach of gauge-natural bundles and the general theory of Lie derivatives for classical field theory, where the sole correct geometrical formulation of Einstein (-Cartan) gravity coupled with Dirac fields gives rise to an unexpected indeterminacy in the concept of conserved quantities.
The notion of a Kähler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite Kähler structure has a canonically associated triple satisfying, up to the compact resolvent condition, Connes' axioms for a spectral triple. In this paper we begin the development of a robust framework in which to investigate the compact resolvent condition, and moreover, the general spectral behaviour of covariant Kähler structures. Th…
The reduced Dirac structure of General Relativity on manifolds with corners
本文推导四维Palatini-Cartan引力的角泊松结构，利用边界约束代数得到角场空间的预狄拉克结构及约化过程，将约化角理论呈现为类BF理论并导出BF²V形式，为引力的体、边界和角结构提供统一框架基础。
We describe both the Hodge - de Rham and the spin manifold Dirac operator on the spheres ${\rm S}^3$ and ${\rm S}^2$, following the formalism introduced by Kähler, and exhibit a complete spectral resolution for them in terms of suitably globally defined eigenspinors.
Compact Quantum Homogeneous Kähler Spaces
本文研究紧量子齐性凯勒空间，探讨其与Woronowicz量子群理论的兼容性，分析Dolbeault-Dirac算子的分析行为及扭后的Fredholm性与欧拉特征数的关系。
This note elaborates on Th. Voronov's construction [math/0304038,math/0412202] of $L_\infty$-structures via higher derived brackets with a Maurer-Cartan element. It is shown that gauge equivalent Maurer-Cartan elements induce $L_\infty$-isomorphic structures. Applications in symplectic, Poisson and Dirac geometry are discussed.
We show that tensoring the Laplace and Dolbeault-Dirac operators of a Kähler structure (with closed integral) by a negative Hermitian holomorphic module, produces operators with spectral gaps around zero. The proof is based on the recently established Akizuki-Nakano identity of a noncommutative Kähler structure. This general framework is then applied to the Heckenberger-Kolb calculi of the irreducible quantum flag manifolds, and it is shown that twisting their Dirac and Laplace operators by negative line bundles produces a spectral gap, for q sufficiently close to 1. The main technical challen…
For a Kähler Manifold $M$, the "symplectic Dolbeault operators" are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar\partial$ and $\bar\partial^*$, arise from Dirac operators on the canonical complex spinors on $M$. We give special attention to two special classes of Kähler manifolds: Riemann surfaces and flag manifolds ($G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). In the case of flag manifolds, we work with the Hermitian structure induced by the Killing form and a cho…
Generalized Kahler geometry is the natural analogue of Kahler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We explore the fundamental aspects of this geometry, including its equivalence with the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2,2) supersymmetry, as well as the relation to ho…
We characterize the Lie derivative of spinor fields from a variational point of view by resorting to the theory of the Lie derivative of sections of gauge-natural bundles. Noether identities from the gauge-natural invariance of the first variational derivative of the Einstein(--Cartan)--Dirac Lagrangian provide restrictions on the Lie derivative of fields.
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