Colette Anné
We give a definition of the Maslov fibre bundle for a lagrangian submanifold of the cotangent bundle of a smooth manofold. This definition generelizes the definition given, in homotopic terms, by Arnol'd for lagrangian submanifolds of the cotangent bundle of the euclidean space and coincides with the one of Hörmander in his works about Fourier Inegral Operators.
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In this note it is shown that the Maslov Index for pairs of Lagrangian Paths as introduced by Cappell, Lee and Miller appears by parallel transporting elements of (a certain complex line-subbundle of) the symplectic spinorbundle over Euclidean space, when pulled back to an (embedded) Lagrangian submanifold $L$, along closed or non-closed paths therein. In especially, the CLM-Index mod 4 determines the holonomy group of this line bundle w.r.t. the Levi-Civita-connection on $L$, hence its vanishing mod 4 is equivalent to the existence of a trivializing parallel section. Moreover, it is shown tha…
The first purpose of this paper is to generalize the well-known Maslov indices of maps of open Riemann surfaces with boundary lying on Lagrangian submanifolds to maps with boundary lying on coisotropic submanifolds in symplectic manifolds. For this purpose, we first define the notion of {\it Maslov loops} of coisotropic Grassmanians and their indices. Then we introduce the notions of {\it transverse Maslov bundle} of coisotropic submanifolds, and {\it gradable coisotropic submanifolds}. We then define {\it graded coisotropic submanifolds} and the {\it coisotropic Maslov index} of the maps with…
Generalizing the construction of the Maslov class for a Lagrangian embedding in a symplectic vector space, we prove that it is possible to give a consistent definition of this class for any Lagrangian submanifold of a Calabi-Yau manifold. Moreover, we prove that this class can be represented by the contraction of the Kaehler form associated to the Calabi-Yau metric, with the mean curvature vector field of the Lagrangian embedding. Finally, we suggest a possible generalization of the Maslov class for Lagrangian submanifolds of any symplectic manifold, via the mean curvature representation.
In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifol…
We give a simplified definition of topological T-duality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for T-duals are shown to follow. We determine necessary and sufficient conditions for existence of a T-dual in the case of affine torus bundles. This is general enough to include all principal torus bundles as well as torus bundles with arbitrary monodromy representations. We show that isomorphisms in twisted cohomology, twisted K-theory and of Co…
We consider a {\em Hamiltonian setup} $\sextuple$, where $(\mathcal M,ω)$ is a symplectic manifold, $\mathfrak L$ is a distribution of Lagrangian subspaces in $\mathcal M$, $\mathcal P$ a Lagrangian submanifold of $ \mathcal M$, $H$ is a smooth time dependent Hamiltonian function on $\mathcal M$ and $Γ:[a,b]\to\mathcal M$ is an integral curve of the Hamiltonian flow $\Hf$ starting at $\mathcal P$. We do not require any convexity property of the Hamiltonian function $H$. Under the assumption that $Γ(b)$ is not $\mathcal P$-focal it is introduced the Maslov index $\maslov(Γ)$ of $Γ$ given in ter…
A pedagogical but concise overview of fiber bundles and their connections is provided, in the context of gauge theories in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations.…
Kock [Bull. Austral. Math. Soc., 25 (1982), 357-386] has considered differential forms with values in a group in a context where neighborhood relations are available. By doing so, he has made it clear where the so-called Maurer-Cartan formula should come from. In this paper, while we retain the classical definition of differential form with values in the Lie algebra of a group, we propose another definition of coboundary operator for the de Rham complex in a highly general microlinear context, in which neighborhood relations are no longer in view. Using this new definition of the coboundary op…
We provide a general, homotopy-theoretic definition of string group models within an $\infty$-category of smooth spaces, and we present new smooth models for the string group. Here, a smooth space is a presheaf of $\infty$-groupoids on the category of cartesian spaces. The key to our definition and construction of smooth string group models is a version of the singular complex functor, which assigns to a smooth space an underlying ordinary space. We provide new characterisations of principal $\infty$-bundles and group extensions in $\infty$-topoi, building on work of Nikolaus, Schreiber, and S…
We give a differential geometric description of the Cartan (or tractor) bundle and its canonical connection in CR geometry, thus offering a direct, alternative, definition to the usual abstract approach.
Metaplectic-c prequantizations and the Einstein--Brillouin--Keller quantizability condition
该研究证明每个辛流形存在辛形式，其复预量子化给出拉格朗日子流形可量化条件，当第一陈类消失时恢复经典爱因斯坦-布里渊-凯勒条件，并发展了相关障碍理论。
In this paper, we will provide an alternative definition for the singular Hermitian metric on a vector bundle. Moreover, we discuss the Griffiths and Nakano positivities under this circumstance and prove a generalised Griffiths' vanishing theorem.
We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. We focus on two definitions. The internal tangent space of a diffeological space is defined using smooth curves into the space, and the external tangent space is defined using smooth derivations on germs of smooth functions. We prove fundamental results about these tangent spaces, compute them in many examples, and observe that while they agree for smooth manifolds and many of the exam…
We present an alternate definition of the mod {\bf Z} component of the Atiyah-Patodi-Singer $η$ invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Patodi and Singer.
On a generalized complex manifold there is an associated definition of a generalized holomorphic bundle, introduced by Gualtieri. This notion in the case of an ordinary complex structure yields an object which we call a co-Higgs bundle and we consider the B-field action of a closed form of type (1,1), both local and global. The effect makes contact with both Nahm's equations and holomorphic gerbes.
Smooth one-dimensional topological field theories are vector bundles with connection
本文证明光滑一维拓扑场论与带联络的向量丛等价，通过定义编码切割律的光滑边界范畴并利用下降与配边假设实现。
Inspired by recent works of Zang Liu, Alan Weinstein and Ping Xu, we introduce the notions of CC algebroids and non asymmetric Courant algebroids and study these structures. It is shown that CC algebroids of rank greater than 3 are the same as Courant algebroids up to a constant factor, though the definition of CC algebroids is much simpler than that of Courant algebroids,requiring only 2 axioms instead of 5. The situation is similar to that of Lie algebroids, where in the usual definition used by all of he experts there is a redundant axiom, e.g.[GG,KO1,KO2,MK,PL]. Non asymmetric Courant alge…
We extend the notion of $r$-minimality of a submanifold in arbitrary codimension to $u$-minimality for a multi-index $u\in\mathbb{N}^q$, where $q$ is the codimension. This approach is based on the analysis on the frame bundle of orthonormal frames of the normal bundle to a submanifold and vector bundles associated with this bundle. The notion of $u$-minimality comes from the variation of $σ_u$-symmetric function obtained from the family of shape operators corresponding to all possible bases of the normal bundle. We obtain the variation field, which gives alternative definition of $u$--minimali…
We first apply the method and results in the previous paper to give a new proof of a result (hold in $ {\bf C}/{\bf Z}$) of Gilkey on the variation of h-invariants associated to non self-adjoint Dirac type operators. We then give an explicit local expression of certain h-invariant appearing in recent papers of Braverman-Kappeler on what they call refined analytic torsion, and propose an alternate formulation of their definition of the refined analytic torsion. A refinement in ${\bf C}$ of the above variation formula is also proposed.
In this largely expository paper we give a self-contained treatment of the Dirac operator. Emphasizing the algebraic point of view we first sketch the necessary prerequisites from Clifford algebras and their representations and then define (and characterize) spin structures and the corresponding Spin-Dirac operator purely in the spirit of irreducible representations. We also prove its equivalence with the usual approach via principal bundles. We conclude with other geometric Dirac operators and their properties using the afore-mentioned setting.
We study the conditions under which an almost Hermitian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ is K\" ahlerian. First, we obtain the algebraic conditions under which the manifold $(T^*M,G,J)$ is almost Hermitian. Next we get the integrability conditions for the almost complex structure $J$, then the conditions under which the associated 2-form is closed. The manifold $(T^*M,G,J)$ is K\" ahlerian iff it is almost Kahlerian and the almost complex structure $J$ is integrable. It follows that the family of Kahlerian structures…
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