A cohomological framework for homotopy moment maps

多重环面图的上同调提升

Cohomological lifting of multi-toric graphs

本文研究具有$T^{2}$-对称性的辛$SU(3)$-流形圆丛导出的$G_{2}$-流形，提出在多重哈密顿几何下$G_{2}$-结构多重矩图紧致部分可由基上同调提升获得，并在环面几何中说明该提升过程。

叶状结构的新上同调不变量

New cohomological invariants of foliations

本文引入闭流形叶状结构的反基本上同调，证明其维数不变性，并对黎曼叶状结构建立相应 Hodge 理论及 Betti 数的几何拓扑约束。

关于可微栈的上同调

On the cohomology of differentiable stacks

本文证明李群胚以同伦意义下的表示为系数的上同调是Morita不变量，从而成为可微栈的不变量，并附带证明高阶李群胚的上同调下降性质。

VB-groupoids and representation theory of Lie groupoids

A VB-groupoid is a Lie groupoid equipped with a compatible linear structure. In this paper, we describe a correspondence, up to isomorphism, between VB-groupoids and 2-term representations up to homotopy of Lie groupoids. Under this correspondence, the tangent bundle of a Lie groupoid G corresponds to the "adjoint representation" of G. The value of this point of view is that the tangent bundle is canonical, whereas the adjoint representation is not. We define a cochain complex that is canonically associated to any VB-groupoid. The cohomology of this complex is isomorphic to the groupoid cohomo…

Existence and unicity of co-moments in multisymplectic geometry

Given a multisymplectic manifold $(M,ω)$ and a Lie algebra $\frak{g}$ acting on it by infinitesimal symmetries, Fregier-Rogers-Zambon define a homotopy (co-)moment as an $L_{\infty}$-algebra-homomorphism from $\frak{g}$ to the observable algebra $L(M,ω)$ associated to $(M,ω)$, in analogy with and generalizing the notion of a co-moment map in symplectic geometry. We give a cohomological characterization of existence and unicity for homotopy co-moment maps and show its utility in multisymplectic geometry by applying it to special cases as exact multisymplectic manifolds and simple Lie groups and…

Lie 2-algebra moment maps in multisymplectic geometry

Consider a closed non-degenerate 3-form $ω$ with an infinitesimal action of a Lie algebra $\mathfrak{g}$. Motivated by the fact that the observables associated to $ω$ form a Lie 2-algebra, we introduce homotopy moment maps defined on a Lie 2-algebra rather than just on the Lie algebra $\mathfrak{g}$. We formulate existence criteria and provide a construction for such homotopy moment maps, by characterizing them in terms of cohomology.

Homotopy moment maps

Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L-infinity morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such m…

Fedosov dg manifolds associated with Lie pairs

Given any pair $(L,A)$ of Lie algebroids, we construct a differential graded manifold $(L[1]\oplus L/A,Q)$, which we call Fedosov dg manifold. We prove that the cohomological vector field $Q$ constructed on $L[1]\oplus L/A$ by the Fedosov iteration method arises as a byproduct of the Poincaré--Birkhoff--Witt map established in arXiv:1408.2903. Finally, using the homological perturbation lemma, we establish a quasi-isomorphism of Dolgushev--Fedosov type: the differential graded algebras of functions on the dg manifolds $(A[1],d_A)$ and $(L[1]\oplus L/A,Q)$ are homotopy equivalent.

On the extension problem for weak moment maps

We compare existence and equivariance phenomena for weak moment maps and homotopy moment maps in multisymplectic geometry.

Strongly Homotopy Lie Algebras from Multisymplectic Geometry

This Master Thesis is devoted to the study of $n$-plectic manifolds and the Strongly Homotopy Lie algebras, also called $L_{\infty}$-algebras, that can be associated to them. Since multisymplectic geometry and $L_{\infty}$-algebras are relevant in Theoretical Physics, and in particular in String Theory, we introduce the relevant background material in order to make the exposition accessible to non-experts, perhaps interested physicists. The background material includes graded and homological algebra theory, fibre bundles, basics of group actions on manifolds and symplectic geometry. We give an…

同伦和同调消失定理与随机流的稳定性

Homotopy and Homology Vanishing Theorems and the Stability of Stochastic Flows

我们将紧致流形$M$上的随机动力系统的稳定性（即矩指数）与$M$的同伦和积分同调群联系起来。在欧氏空间中$M$的等距浸入的梯度布朗系统的特殊情况下，这些矩指数可以根据浸入的第二基本形式来估计。这产生拓扑障碍等距浸入推广的结果劳森-西蒙斯和其他人。我们的工作也将这些作者的工作纳入Weitzenböck公式的一般框架。

多辛几何中的同伦共动量映射

Homotopy Comomentum Maps in Multisymplectic Geometry

本文研究多辛几何中群作用容许同伦共动量映射的条件，给出多辛球面上紧群作用的完整分类，显式构造泊松代数到扭李代数胚的高阶嵌入，并构造了体积保持微分同胚群在三维欧氏空间上的同伦共动量映射，揭示其与流体动力学和纽结理论的联系。