Connective structure for principal gerbes

Connections and curvings on lifting bundle gerbes

We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection and with the curving coincides with the obstruction class of the lifting problem with these structures.

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…

Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry

We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module there is a simplicial group NC, the nerve of the 1-category defined by the crossed module and its geometric realization |NC|. Equivalence classes of principal bundles with structure group |NC| are shown to be one-to-one with stable equivalence classes of what we call crossed module gerbes bundle gerbes. We can also associate to a crossed module a 2-category C'. Then there are two equivalent ways how to view classifying spaces of NC-bundles and hence of |NC|-bundles and…

Hermitian-holomorphic (2)-Gerbes and tame symbols

We observe that the line bundle associated to the tame symbol of two invertible holomorphic functions also carries a fairly canonical hermitian metric, hence it represents a class in a Hermitian holomorphic Deligne cohomology group. We put forward an alternative definition of hermitian holomorphic structure on a gerbe which is closer to the familiar one for line bundles and does not rely on an explicit ``reduction of the structure group.'' Analogously to the case of holomorphic line bundles, a uniqueness property for the connective structure compatible with the hermitian-holomorphic structure …

A Loop Space Formulation for Geometric Lifting Problems

We review and then combine two aspects of the theory of bundle gerbes. The first concerns lifting bundle gerbes and connections on those, developed by Murray and Gomi. Lifting gerbes represent obstructions against extending the structure group of a principal bundle. The second is the transgression of gerbes to loop spaces, initiated by Brylinski and McLaughlin and with recent contributions of the author. Combining these two aspects, we obtain a new formulation of lifting problems in terms of geometry on the loop space. Most prominently, our formulation explains the relation between (complex) s…

Chern character in twisted K-theory: equivariant and holomorphic cases

It has been argued by Witten and others that in the presence of a nontrivial B-field, D-brane charges in type IIB string theories are measured by twisted K-theory. In joint work with Bouwknegt, Carey and Murray it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are ordinary vector bundles on a principal projective unitary bundle, with an action of the bundle gerbe determined by the principal projective unitary bundle. The principal projective unitary bundle is in turn determined by the twist. In this paper, we study in more detail the Chern-W…

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…

Equivariant bundle gerbes

We develop the theory of simplicial extensions for bundle gerbes and their characteristic classes with a view towards studying descent problems and equivariance for bundle gerbes. Equivariant bundle gerbes are important in the study of orbifold sigma models. We consider in detail two examples: the basic bundle gerbe on a unitary group and a string structure for a principal bundle. We show that the basic bundle gerbe is equivariant for the conjugation action and calculate its characteristic class; we show also that a string structure gives rise to a bundle gerbe which is equivariant for a natur…

Lifting Problems and Transgression for Non-Abelian Gerbes

We discuss various lifting and reduction problems for bundles and gerbes in the context of a strict Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen's long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in non-abelian cohomology. This transgression map relates the degree one non-abelian cohomology of a smooth manifold (represented by non-abelian gerbes) with the degree zero non-abelian cohomology of the free loop space (represented by principal bundles). We prove several properties…

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 …

双李群胚的态射：一种单纯方法

Morphisms of double Lie groupoids: a simplicial approach

本文从单纯观点研究双李群胚的态射，扩展余对角函子至主双纤维丛和方形态射，并应用于非阿贝尔gerbes模型等价及传递Courant algebroid中Manin三元组的积分细化，证明积分在辛Morita等价意义下良定义。

Semistrict Higher Gauge Theory

We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2-bundles with connective structures. Principal 2-bundles are obtained in terms of weak 2-functors from the Cech groupoid to weak Lie 2-groups. As is demonstrated, some of these Lie 2-groups can be differentiated to semistrict Lie 2-algebras by a method due to Severa. We further derive the full description of connective structures on semistrict principal 2-bundles including the non-linear gauge transformations. As an application, we use…

高阶主丛的Chern对应关系

Chern correspondence for higher principal bundles

本文研究高阶 gauge 理论中的 Chern 对应关系，通过定义乘法上同调丛的连接结构，提出 2-群的复化概念，建立新的高阶连接概念，统一了文献中两种高阶连接定义。

Bigerbes

The bigerbes introduced here give a refinement of the notion of 2-gerbes, representing degree four integral cohomology classes of a space. Defined in terms of bisimplicial line bundles, bigerbes have a symmetry with respect to which they form 'bundle 2-gerbes' in two ways; this structure replaces higher associativity conditions. We provide natural examples, including a Brylinski-McLaughlin bigerbe associated to a principal G-bundle for a simply connected simple Lie group. This represents the first Pontryagin class of the bundle, and is the obstruction to the lifting problem on the associated p…

群oids的规范作用、影子中的弦以及涌现的Poisson σ-模型

Gaugings of Groupoids, Strings in Shadows, and Emergent Poisson $σ$-Models

本文提出了一种规范原理，用于刚性李群oids对称性G→M的Polyakov-Alvarez-Gawędzki二维非线性σ-模型，通过CS微分字符从阿贝尔gerbe产生的WZ项，并探讨了该原理在Poisson σ-模型中的应用。

关于模空间的微分 $K$-理论

On the differential $K$-theory of moduli stacks

本文计算了主 $G$-丛带联络模空间的连通微分 $K$-理论和微分上同调，结果以不变多项式和表示环表述，并利用空间与谱预层的同伦理论证明。

Lie范畴基础与乘法Ehresmann联络的Yang-Mills理论

Fundamentals of Lie categories and Yang-Mills theory for multiplicative Ehresmann connections

本文研究Lie范畴的基本性质，并将Yang-Mills理论从主丛推广到一般Lie群胚与代数胚，通过乘法Ehresmann联络、障碍类和变分原理，得到不可积非传递情形下的一对规范不变方程，并以$S^1$-丛gerbes为例加以说明。