Atiyah sequences and connections on principal bundles over differentiable stacks

Lie Algebroids over Differentiable Stacks

We develop a theory of Lie algebroids over differentiable stacks that extends the standard theory of Lie algebroids over manifolds. In particular we show that Lie algebroids satisfy descent for submersions, define the category of Lie algebroids over a differentiable stack, construct a cohomology theory for these objects, and explain the relation to the theory of $\mathcal{LA}$-groupoids. We construct a number of examples.

Equivariant cohomology for differentiable stacks

We construct and analyse models of equivariant cohomology for differentiable stacks with Lie group actions extending classical results for smooth manifolds due to Borel, Cartan and Getzler. We also derive various spectral sequences for the equivariant cohomology of a differentiable stack generalising among others Bott's spectral sequence which converges to the cohomology of the classifying space of a Lie group.

On two notions of a Gerbe over a stack

Let $\mathcal{G}$ be a Lie groupoid. The category $B\mathcal{G}$ of principal $\mathcal{G}$-bundles defines a differentiable stack. On the other hand, given a differentiable stack $\mathcal{D}$, there exists a Lie groupoid $\mathcal{H}$ such that $B\mathcal{H}$ is isomorphic to $\mathcal{D}$. Define a gerbe over a stack as a morphism of stacks $F\colon \mathcal{D}\rightarrow \mathcal{C}$, such that $F$ and the diagonal map $Δ_F\colon \mathcal{D}\rightarrow \mathcal{D}\times_{\mathcal{C}}\mathcal{D}$ are epimorphisms. This paper explores the relationship between a gerbe defined above and a Mori…

可微叠上的向量场与导子

Vector fields and derivations on differentiable stacks

该研究引入并分析李群胚上乘法向量场dgla与函数分次代数的模结构，证明可微叠向量场在光滑函数上具Morita不变的分次Lie-Rinehart代数结构，且Van-Est型映射与模结构相容，还给出相关示例。

Measures on differentiable stacks

We introduce and study measures and densities (= geometric measures) on differentiable stacks, using a rather straightforward generalization of Haefliger's approach to leaf spaces and to transverse measures for foliations. In general we prove Morita invariance, a Stokes formula which provides reinterpretations in terms of (Ruelle-Sullivan type) algebroid currents, and a Van Est isomorphism. In the proper case we reduce the theory to classical (Radon) measures on the underlying space, we provide explicit (Weyl-type) formulas that shed light on Weinstein's notion of volumes of differentiable sta…

The smooth Hom-stack of an orbifold

For a compact manifold M and a differentiable stack \cX presented by a Lie groupoid X, we show the Hom-stack Hom(M,\cX) is presented by a Fréchet-Lie groupoid Map(M,X) and so is an infinite-dimensional differentiable stack. We further show that if \cX is an orbifold, presented by a proper étale Lie groupoid, then Map(M,X) is proper étale and so presents an infinite-dimensional orbifold.

Shifted Poisson structures on differentiable stacks

The purpose of this paper is to investigate shifted $(+1)$ Poisson structures in context of differential geometry. The relevant notion is shifted $(+1)$ Poisson structures on differentiable stacks. More precisely, we develop the notion of Morita equivalence of quasi-Poisson groupoids. Thus isomorphism classes of $(+1)$ Poisson stack correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following programs of independent interests: (1) We introduce a $\mathbb Z$-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its ho…

Lie Groupoids and Differentiable Stacks

This is a concise introduction to the theory of Lie groupoids, with emphasis in their role as models for stacks. After some preliminaries, we review the foundations on Lie groupoids, and we carefully study equivalences and proper groupoids. Differentiable stacks are geometric objects which have manifolds and orbifolds as special instances, and can be presented as the transverse geometry of a Lie groupoid. Two Lie groupoids are equivalent if they are presenting the same stack, and proper groupoids are presentations of separated stacks, which by the linearization theorem are locally modeled by l…

S^1-bundles and gerbes over differentiable stacks

We study $S^1$-bundles and $S^1$-gerbes over differentiable stacks in terms of Lie groupoids, and construct Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvature.

Riemannian metrics on differentiable stacks

We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve similar results for classic geometries. Then we establish the Morita invariance for our metrics, introduce a notion for metrics on stacks, and use them to construct stacky tubular neighborhoods and to prove a stacky Ehresmann theorem.

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。

String principal bundles and Courant algebroids

Just like Atiyah Lie algebroids encode the infinitesimal symmetries of principal bundles, exact Courant algebroids are believed to encode the infinitesimal symmetries of $S^1$-gerbes. At the same time, transitive Courant algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and the non-commutative analogue of exact Courant algebroids. In this article, we explore what the "principal bundle" behind transitive Courant algebroids are, and they turn out to be principal 2-bundles of string groups. First, we construct the stack of principal 2-bundles of string groups with connecti…

Lie n-groupoids and stacky Lie groupoids

We discuss two sorts of generalization of Lie groupoids. One is Lie $n$-groupoids defined as simplicial manifolds with trivial $π_{k\geq n+1}$. The other is the stacky Lie groupoid $\cG\rra M$ with $\cG$ a differentiable stack. We build 1-1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to a certain Morita equivalence. Equivalences of these higher groupoids are described.

Discrete dynamics and differentiable stacks

In this paper we relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold encodes the dynamics up to conjugation and inversion. We also prove a generalization of this result for arbitrary discrete groups and non-simply connected manifolds, and relate it to the covering theory of stacks. As applications, we obtain a geometric version of Rieffel's theorem on irrational rotations of the circle, we compute the stack-theoretic fundame…

Strictification of etale stacky Lie groups

We define stacky Lie groups to be group objects in the 2-category of differentiable stacks. We show that every connected and etale stacky Lie group is equivalent to a crossed module of the form (H,G) where H is the fundamental group of the given stacky Lie group and G is the connected and simply connected Lie group integrating the Lie algebra of the stacky group. Our result is closely related to a strictification result of Baez and Lauda.

Differentiable Stacks and Gerbes

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study $S^1$-bundles and $S^1$-gerbes over differentiable stacks. In particular, we establish the relationship between $S^1$-gerbes and groupoid $S^1$-central extensions. We define connections and curvings for groupoid $S^1$-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for $S^1$-gerbes over manifolds. We develop a Chern-Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier-Douady classes in terms …

The volume of a differentiable stack

We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself. Since there is no natural measure in the smooth case like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a natural line bundle over the groupoid. Sections of a square root of this line bundle …

可微栈上的移位接触结构

Shifted Contact Structures on Differentiable Stacks

本文在可微栈上定义0-移位和+1-移位接触结构，奠定移位接触几何基础，并证明李群胚上乘法1-形式的核总可作为可微栈存在，通过辛到接触字典建立与移位辛结构的联系。

可微栈的移位余迷向结构

Shifted coisotropic structures for differentiable stacks

本文在1-移位辛李群胚上利用扭曲狄拉克结构定义余迷向结构，证明其交为0-移位泊松结构，并通过森田等价传递到可微栈，从而推广了 Marsden-Weinstein-Meyer 辛约化。

Twisted K-theory of differentiable stacks

In this paper, we develop twisted $K$-theory for stacks, where the twisted class is given by an $S^1$-gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure $K^i_α\otimes K^j_β\to K^{i+j}_{α+β}$ are derived. Our approach provides a uniform framework for studying various twisted $K$-theories including the usual twisted $K$-theory of topological spaces, twisted equivariant $K$-theory, and the twisted $K$-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted $K$-groups can be ex…

Diffeological Coarse Moduli Spaces of Stacks over Manifolds

In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the "underlying" diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left adjoint: the functor sending any stack to its diffeological coarse moduli space. As an application, we restrict our attention to differentiable stacks and examine the geometry behind the coarse moduli space construction in terms of Lie groupoids and their principal bundles. With…

n-群胚与堆群胚

n-Groupoids and Stacky Groupoids

本文提出李n-群胚与堆李群胚两种推广，证明李2-群胚与堆李群胚在Morita等价下的一一对应，并推广到微分与拓扑范畴。

Differential characters as stacks and prequantization

We generalize geometric prequantization of symplectic manifolds to differentiable stacks. Our approach is atlas-independent and provides a bijection between isomorphism classes of principal circle bundles (with or without connections) and second cohomology groups of certain chain complexes.

博士论文：可微层上的平移接触结构

PhD Thesis: Shifted Contact Structures on Differentiable Stacks

本论文提出李群胚上0和+1平移接触结构，利用线丛值1-形式及同伦核概念，受辛-接触词典启发，涵盖轨道流形及前量子化等例子。

Non Abelian Differentiable Gerbes

We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids. More precisely, we develop the theory of connections on Lie groupoid $G$-extensions, which we call "connections on gerbes", and study the induced connections on various associated bundles. We also prove analogues of the Bianchi identities. In particular, we develop a cohomology theory which measures the existence of connections and curvings for $G$-gerbes over stacks. We also introduce $G$-central extensions of groupoids, generalizing the standard groupoid $S^1$-central extensions. As an example, we apply…

G-gerbes, principal 2-group bundles and characteristic classes

Let $G$ be a Lie group and $G\to\Aut(G)$ be the canonical group homomorphism induced by the adjoint action of a group on itself. We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group $[G\to\Aut(G)]$-bundles over Lie groupoids and, on the other hand, $G$-extensions of Lie groupoids (i.e.\ between principal $[G\to\Aut(G)]$-bundles over differentiable stacks and $G$-gerbes over differentiable stacks). This approach also allows us to identify $G$-bound gerbes and $[Z(G)\to 1]$-group bundles over differentiable stacks, wher…

Parallel Transport on Principal Bundles over Stacks

In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldof from manifolds to stacks. In the process of proving our main result we simplify Schreiber and Waldorf's definition of a transport functor for principal bundles with connections over manifolds and provide a more direct proof of the correspondence between principal bundles with conn…

可微栈上的同调向量场

Homological vector fields over differentiable stacks

本文利用超几何将 Lie 和 Courant algebroid 视为分级 groupoid 上同伦同调的向量场，从代数与范畴论双视角统一描述，为其提供了 Morita 不变定义及统一研究框架。

Integrating Lie algebroids via stacks

Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie algebroid can be integrated into a Weinstein groupoid.

On the de Rham Cohomology of Differential and Algebraic Stacks

We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids. Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack …

可微群胚对象及其抽象李代数胚

Differentiable groupoid objects and their abstract Lie algebroids

本文将李群胚到李代数胚的微分过程推广到具有抽象切结构和环对象标量乘法的任意范畴中的群胚对象，证明不变向量场在李括号下封闭且满足莱布尼茨法则，从而公理化地定义了抽象李代数胚，并给出了多个几何与物理中的实例。

李群胚上的Morse理论

Morse theory on Lie groupoids

本文引入了Morita不变的Morse李群胚态射概念，证明了群胚版Morse引理以描述临界子群胚层的拓扑行为，并构造了Morse双复形以同构于Bott-Shulman-Stasheff上同调。

Yang-Mills Connections On Orientable and Nonorientable Surfaces

In math.SG/0605587, we studied Yang-Mills functional on the space of connections on a principal G_R-bundle over a closed, connected, nonorientable surface, where G_R is any compact connected Lie group. In this sequel, we generalize the discussion in "The Yang-Mills equations over Riemann surfaces" by Atiyah and Bott, and math.SG/0605587. We obtain explicit descriptions (as representation varieties) of Morse strata of Yang-Mills functional on orientable and nonorientable surfaces for non-unitary classical groups SO(n) and Sp(n). It turns out to be quite different from the unitary case. we use L…

On the Lie 2-algebra of sections of an LA-groupoid

In this work we introduce the category of multiplicative sections of an $\la$-groupoid. We prove that this category carries natural strict Lie 2-algebra structures, which are Morita invariant. As applications, we study the algebraic structure underlying multiplicative vector fields on a Lie groupoid and in particular vector fields on differentiable stacks. We also introduce the notion of geometric vector field on the quotient stack of a Lie groupoid, showing that the space of such vector fields is a Lie algebra. We describe the Lie algebra of geometric vector fields in several cases, including…

Orbispaces as differentiable stratified spaces

We present some features of the smooth structure, and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures - it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper Lie groupoid. The canonical smooth structure on an orbispace is studied mainly via Spallek's framework of differentiable spaces, and two alternative frameworks are then presented. For the canonical str…

李群胚上的联络与 Chern-Weil 理论

Connections on Lie groupoids and Chern-Weil theory

本文在带可积联络分布的李群胚上建立 de Rham 上同调，研究主丛联络的 Atiyah 序列与可微栈构造，并发展 Chern-Weil 理论以刻画示性类。

On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces

We describe various equivalent ways of associating to an orbifold, or more generally a higher étale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we show that for a differentiable stack X arising from a Lie groupoid G, the weak homotopy type of X agrees with that of BG. Using this machinery, we are able to find new presentations for the weak homotopy type of certain classifying spaces. In particular, we give a new presentation for the Borel construction of an almost free action of a Lie group G on a smooth …

通过栈积分李代数胚及其在雅可比流形上的应用

Integrating Lie algebroids via stacks and applications to Jacobi manifolds

针对李代数胚无法总能积分成李群胚的问题，本文引入“温斯坦群胚”（类群胚公理的可微栈）解决其积分问题，并将结果应用于雅可比流形构造接触群胚，还拓展到预量子化等领域。

Deformations of vector bundles in the categories of Lie algebroids and groupoids

This thesis deals with deformations of VB-algebroids and VB-groupoids. They can be considered as vector bundles in the categories of Lie algebroids and groupoids and encompass several classical objects, including Lie algebra and Lie group representations, 2-vector spaces and the tangent and the cotangent algebroid (groupoid) to a Lie algebroid (groupoid). Moreover, they are geometric models for some kind of representations of Lie algebroids (groupoids), namely 2-term representations up to homotopy. Finally, it is well known that Lie groupoids are "concrete" incarnations of differentiable stack…

关于李群胚的Bott谱序列的若干观点

Perspectives on the Bott spectral sequence for Lie groupoids

本文关联统一两类改进可微栈上同调计算的方法：Arias Abad将Bott谱序列推广至李群胚，作者则提出正则真李群胚对应可微栈的上同调模型。

Sheaf Theory for Étale Geometric Stacks

We generalize the notion of a small sheaf of sets over a topological space or manifold to define the notion of a small stack of groupoids over an étale topological or differentiable stack. We then provide a construction analogous to the étalé space construction in this context, establishing an equivalence of 2-categories between small stacks over an étale stack and local homeomorphisms over it. We go on to characterize small sheaves and gerbes. We show that ineffective data of étale stacks is completely described by the theory of small gerbes. Furthermore, it is shown that étale stacks (and in…

导出微分几何中的对称性与高形式连接

Symmetries and Higher-Form Connections in Derived Differential Geometry

本文提出导出微分几何中主∞-丛的高形式连接定义，并通过L∞-代数与更高对称性关系，揭示了其在更高规范理论和超引力中的应用。

Modular Classes of Lie Groupoid Representations up to Homotopy

We describe a construction of the modular class associated to a representation up to homotopy of a Lie groupoid. In the case of the adjoint representation up to homotopy, this class is the obstruction to the existence of a volume form, in the sense of Weinstein's "The volume of a differentiable stack".

The van Est Map on Geometric Stacks

We generalize the van Est map and isomorphism theorem in three ways, and we discuss conjectured connections with homotopy theory, including a proposal of a category which unifies differentiable stacks, Lie algebroids and homotopy theory. In Part 2 of this thesis we generalize the van Est map from a comparison map between Lie groupoid cohomology and Lie algebroid cohomology to a (more conceptual) comparison map between the cohomology of a stack $\mathcal{G}$ and the cohomology of a simple foliation $\mathcal{H}\to\mathcal{G}$. In Part 1 we generalize the functions that we can take cohomology of…

Morita equivalences of vector bundles

We study vector bundles over Lie groupoids, known as VB-groupoids, and their induced geometric objects over differentiable stacks. We establish a fundamental theorem that characterizes VB-Morita maps in terms of fiber and basic data, and use it to prove the Morita invariance of VB-cohomology, with implications to deformation cohomology of Lie groupoids and of classic geometries. We discuss applications of our theory to Poisson geometry, providing a new insight over Marsden-Weinstein reduction and the integration of Dirac structures. We conclude by proving that the derived category of VB-groupo…

0-移位余辛结构在微分堆栈上的哈密顿作用

Hamiltonian actions on 0-shifted cosymplectic groupoids

本文研究了微分堆栈上的0-移位余辛结构，发展了余辛作用的动量映射理论，并给出了Morse-Bott李群状映射的例子。

李群胚上主2-丛中的规范理论与平行传输研究

On gauge theory and parallel transport in principal 2-bundles over Lie groupoids

本研究引入李群胚上的主李2-群丛概念，构建其Atiyah序列并定义严格与半严格联络，揭示半严格联络的扩展对称性，建立Haefliger路径上的平行传输理论，沟通了规范理论与纤维化范畴的核心概念。

Resolutions of proper Riemannian Lie groupoids

In this paper we prove that every proper Lie groupoid admits a desingularization to a regular proper Lie groupoid. When equipped with a Riemannian metric, we show that it admits a desingularization to a regular Riemannian proper Lie groupoid, arbitrarily close to the original one in the Gromov-Hausdorff distance between the quotient spaces. We construct the desingularization via a successive blow-up construction on a proper Lie groupoid. We also prove that our construction of the desingularization is invariant under Morita equivalence of groupoids, showing that it is a desingularization of the…

Deformations of Linear Lie Brackets

A VB-algebroid is a vector bundle object in the category of Lie algebroids. We attach to every VB-algebroid a differential graded Lie algebra and we show that it controls deformations of the VB-algebroid structure. Several examples and applications are discussed. This is the first in a series of papers devoted to deformations of vector bundles and related structures over differentiable stacks.

Equivariant Dixmier-Douady Classes

An equivariant bundle gerbe à la Meinrenken over a $G$-manifold $M$ is known to be a special type of $S^1$-gerbe over the differentiable stack $[M/G]$. We prove that the natural morphism relating the Cartan and simplicial models of equivariant cohomology in degree 3 maps the Dixmier-Douady class of an equivariant bundle gerbe à la Meinrenken to the Behrend-Xu-Dixmier-Douady class of the corresponding $S^1$-gerbe.

The Yang-Mills equations over Klein surfaces

Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients, and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projective curve. From the algebraic point of view, these Lagrangian quotients are connected sets of real points inside a complex moduli variety endowed with a real structure; when the rank and the degree are coprime, they are in fact the connected components of the fixed-point set of th…

有界群胚卷积的函子性

Functoriality of bornological groupoid convolution

本文证明李群胚的完全有界卷积代数和群胚双束的卷积双模定义了从可微栈的2-范畴到完全有界代数的Morita 2-范畴的幺半函子，并展示了卷积代数的自诱导性和拟单位性。

李群胚上李 2-群丛沿 Haefliger 路径的平行移动

Parallel transport on a Lie 2-group bundle over a Lie groupoid along Haefliger paths

本文证明了李 2-群挠子版本的纤维化范畴与伪函子对应关系，借此在李群胚上主 2-丛框架中沿 Haefliger 路径推导出光滑的平行移动，并应用于 VB-群胚。

Higher Groupoid Bundles, Higher Spaces, and Self-Dual Tensor Field Equations

We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable differentiable stacks, as e.g. orbifolds. We start off with a self-contained review on simplicial sets as models of $(\infty,1)$-categories. We then discuss principal bundles in terms of simplicial maps and their homotopies. We explain in detail a differentiation procedure, suggested by Severa, that maps higher gro…