S^1-bundles and gerbes over differentiable stacks

Differentiable Categories, gerbes and G-structures

The theories of strings and $D$-branes have motivated the development of non Abelian cohomology techniques in differential geometry, on the purpose to find a geometric interpretation of characteristic classes. The spaces studied here, like orbifolds are not often smooth. In classical differential geometry, non smooth spaces appear also naturally, for example in the theory of foliations, the space of leaves can be an orbifold with singularities. The scheme to study these structures is identical: classical tools used in differential geometry, like connections, curvature are adapted. The purpose …

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…

Bundle Gerbes and Surface Holonomy

Hermitian bundle gerbes with connection are geometric objects for which a notion of surface holonomy can be defined for closed oriented surfaces. We systematically introduce bundle gerbes by closing the pre-stack of trivial bundle gerbes under descent. Inspired by structures arising in a representation theoretic approach to rational conformal field theories, we introduce geometric structure that is appropriate to define surface holonomy in more general situations: Jandl gerbes for unoriented surfaces, D-branes for surfaces with boundaries, and bi-branes for surfaces with defect lines.

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.

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…

Four Equivalent Versions of Non-Abelian Gerbes

We recall and partially improve four versions of smooth, non-abelian gerbes: Cech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We prove that all these four versions are equivalent, and so establish new relations between interesting recent developments. Prominent partial results we prove are a bijection between continuous and smooth non-abelian cohomology, and an explicit equivalence between bundle gerbes and principal 2-bundles as 2-stacks.

A quantum Leray-Hirsch theorem for banded gerbes

For a gerbe $\Y$ over a smooth proper Deligne-Mumford stack $\B$ banded by a finite group $G$, we prove a structure result on the Gromov-Witten theory of $\Y$, expressing Gromov-Witten invariants of $\Y$ in terms of Gromov-Witten invariants of $\B$ twisted by various flat $U(1)$-gerbes on $\B$. This is interpreted as a Leray-Hirsch type of result for Gromov-Witten theory of gerbes.

Non-abelian gerbes and some applications in string theory

We review a systematic construction of the 2-stack of bundle gerbes via descent, and extend it to non-abelian gerbes. We review the role of non-abelian gerbes in orientifold sigma models, for the anomaly cancellation in supersymmetric sigma models, and in a geometric description of so-called non-geometric T-duals.

Differential Cocycles and Dixmier-Douady Bundles

This paper exhibits equivalences of 2-stacks between certain models of $\mathbb{S}^1$-gerbes and differential 3-cocycles. We focus primarily on the model of Dixmier-Douady bundles, and provide an equivalence between the 2-stack of Dixmier-Douady bundles and the 2-stack of differential 3-cocycles of height 1, where the 'height' is related to the presence of connective structure. Differential 3-cocycles of height 2 (resp. height 3) are shown to be equivalent to $\mathbb{S}^1$-bundle gerbes with connection (resp. with connection and curving). These equivalences extend to the equivariant setting o…

A classification of equivariant gerbe connections

Let G be a compact Lie group acting on a smooth manifold M. In this paper, we consider Meinrenken's G-equivariant bundle gerbe connections on M as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to M, and isomorphism classes of G-equivariant gerbe connections are classified by degree three differential equivariant cohomology. Finally, we consider the existence and uniqueness of conjugation-equivariant gerbe connections on compact semisimple Lie groups.

Geometric structures on Lie groupoids and differentiable stacks

In this PhD thesis, we have studied certain geometric structures over Lie groupoids and differentiable stacks. This thesis is based on the work [arXiv:2103.04560, arXiv:2012.08447, arXiv:2012.08442, arXiv:1907.00375]. In [arXiv:1907.00375], we have discussed the correspondence between two notions of a gerbe over a stack. In [arXiv:2012.08447], we have developed the notion of Chern-Weil map for principal bundles over a Lie groupoid when the Lie groupoid is equipped with a connection. In [arXiv:2012.08442], we discuss the notion of connection on the principal bundle over a Deligne Mumford stack …

拓扑群胚的扩张以及 Serre、Hurewicz 态射

Extension of topological groupoids and Serre, Hurewicz morphisms

该研究提出拓扑群胚扩张概念，将其关联于拓扑栈上的 gerbe，并考察 Serre、Hurewicz 栈上 gerbe 的性质。

Atiyah sequences, connections and Chern-Weil theory for algebraic and differentiable stacks

We construct connections and characteristic forms for principal bundles over groupoids and stacks in the differentiable, holomorphic and algebraic category using Atiyah sequences associated to transversal tangential distributions.

Higher Structures: gerbes and Nijenhuis forms

In the first chapter, we give a precise and general description of gerbes valued in arbitrary crossed module and over an arbitrary differential stack. We do it using only Lie groupoids, hence ordinary differential geometry, by considering differential stacks as being Lie groupoids up to Morita equivalence. We prove the coincidence with the existing notions by comparing our construction with non-Abelian cohomology. More precisely, we introduce the key notion of extension of Lie groupoids valued in a crossed-module. We relate it with Dedecker's non-Abelian 1-cocycles, and we then show that Morit…

The Lusternik-Schnirelmann category for a differentiable stack

We introduce the notion of Lusternik-Schnirelmann category for differentiable stacks and establish its relation with the groupoid Lusternik-Schnirelmann category for Lie groupoids.

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…

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。

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.

Atiyah sequences and connections on principal bundles over differentiable stacks

We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.

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.

规范理论中的类格结构构造

A gerbe-like construction in gauge theory

本文证明同伦K3曲面族中沿纤维切丛与自对调和2形式丛的直和具有典范自旋结构，通过族Seiberg–Witten方程构造O(1)-格提升。

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…

Equivariance In Higher Geometry

We study (pre-)sheaves in bicategories on geometric categories: smooth manifolds, manifolds with a Lie group action and Lie groupoids. We present three main results: we describe equivariant descent, we generalize the plus construction to our setting and show that the plus construction yields a 2-stackification for 2-prestacks. Finally we show that, for a 2-stack, the pullback functor along a Morita-equivalence of Lie groupoids is an equivalence of bicategories. Our results have direct applications to gerbes and 2-vector bundles. For instance, they allow to construct equivariant gerbes from loc…

Vector Fields and Flows on Differentiable Stacks

This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell. This extends the usual result on the existence and uniqueness of flows on a manifold as well as the author's existing results for orbifolds. It sets the scene for a discussion of Morse Theory on a general proper stack and also paves the way for the categorification of other key aspects of differential geometry such as the tangent bundle and the Lie a…

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…

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…

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…

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.

Principal actions of stacky Lie groupoids

Stacky Lie groupoids are generalizations of Lie groupoids in which the "space of arrows" of the groupoid is a differentiable stack. In this paper, we consider actions of stacky Lie groupoids on differentiable stacks and their associated quotients. We provide a characterization of principal actions of stacky Lie groupoids, i.e., actions whose quotients are again differentiable stacks in such a way that the projection onto the quotient is a principal bundle. As an application, we extend the notion of Morita equivalence of Lie groupoids to the realm of stacky Lie groupoids, providing examples tha…

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。

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 …

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。

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。

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。

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 …

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…

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.

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.

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.

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.

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…

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。

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。

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…

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…

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…

Extended higher cup-product Chern-Simons theories

The proper action functional of (4k+3)-dimensional U(1)-Chern-Simons theory including the instanton sectors has a well known description: it is given on the moduli space of fields by the fiber integration of the cup product square of classes in degree-(2k+2) differential cohomology. We first refine this statement from the moduli space to the full higher smooth moduli stack of fields, to which the higher order-ghost BRST complex is the infinitesimal approximation. Then we generalize the refined formulation to cup product Chern-Simons theories of nonabelian and higher nonabelian gauge fields, su…

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。