Ernesto Lupercio, Bernardo Uribe
The purpose of this paper is to introduce the notion of loop groupoid associated to a groupoid. After studying the general properties of the loop groupoid, we show how this notion provides a very natural geometric interpretation for the twisted sectors of an orbifold, and for the inner local systems introduced by Ruan by means of a natural generalization of the concept holonomy of a gerbe.
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In this paper we construct an explicit geometric model for the group of gerbes over an orbifold $X$. We show how from its curvature we can obtain its characteristic class in $H^3(X)$ via Chern-Weil theory. For an arbitrary gerbe $\LL$, a twisting $\Korb{\LL}(X)$ of the orbifold $K$-theory of $X$ is constructed, and shown to generalize previous twisting by Rosenberg, Witten, Atiyah-Segal and Bowknegt et. al. in the smooth case and by Adem-Ruan for discrete torsion on an orbifold.
Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact étale Lie groupoid $M$, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on $M$. This result can be seen as an equivariant Serre-Swan theorem for twisted vector bundles.
The twisted Drinfeld double (or quasi-quantum double) of a finite group with a 3-cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3-cocycle. The twisted representation theory of finite groupoids is developed and used to derive properties of the Drinfeld double, such as representations being classified by their characters. This is all motivated by gerbes and 3-dimensional topological quantum field theory. In particular the representation …
We consider descent data in cosimplicial crossed groupoids. This is a combinatorial abstraction of the descent data for gerbes in algebraic geometry. The main result is this: a weak equivalence between cosimplicial crossed groupoids induces a bijection on gauge equivalence classes of descent data.
We compute the integral cohomology of certain semi-direct products arising from a linear G-action on the n-torus, where G is a finite group. The main application is the complete calculation of torsion gerbes for certain six dimensional examples arising from string theory.
Given a double cover $π: \mathcal{G} \rightarrow \hat{\mathcal{G}}$ of finite groupoids, we explicitly construct twisted loop transgression maps, $τ_π$ and $τ_π^{ref}$, thereby associating to a Jandl $n$-gerbe $\hatλ$ on $\hat{\mathcal{G}}$ a Jandl $(n-1)$-gerbe $τ_π(\hatλ)$ on the quotient loop groupoid of $\mathcal{G}$ and an ordinary $(n-1)$-gerbe $τ^{ref}_π(\hatλ)$ on the unoriented quotient loop groupoid of $\mathcal{G}$. For $n =1,2$, we interpret the character theory (resp. centre) of the category of Real $\hatλ$-twisted $n$-vector bundles over $\hat{\mathcal{G}}$ in terms of flat secti…
In this paper we compute explicit formulas for the holonomy map for a gerbe with connection over an orbifold. We show that the holonomy descends to a transgression map in Deligne cohomology. We prove that this recovers both the inner local systems in Ruan's theory of twisted orbifold cohomology and the local system of Freed-Hopkins-Teleman in their work in twisted K-theory. In the case in which the orbifold is simply a manifold we recover previous results of Gawedzki and Brylinski.
Gerbes are locally connected presheaves of groupoids. They are classified up to local weak equivalence by path components in a 2-cocycle category taking values in all sheaves of groups, their isomorphisms and homotopies. If F is a full presheaf of sheaves of groups, isomorphisms and homotopies, then [*,BF] is isomorphic to equivalence classes of gerbes locally equivalent to groups appearing in F. Giraud's non-abelian cohomology object of equivalence classes of gerbes with band L is isomorphic to morphisms in the homotopy category from the point * to the homotopy fibre over L for a map defined …
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.
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。
We construct a 2-category version of tom Dieck's equivariant fundamental groupoid for representable orbifolds and show that the discrete fundamental groupoid is Morita invariant; hence an orbifold invariant for representable orbifolds.
We define a 2-category structure (Pre-Orb) on the category of reduced complex orbifold atlases. We construct a 2-functor F from (Pre-Orb) to the 2-category (Grp) of proper étale effective groupoid objects over the complex manifolds. Both on (Pre-Orb) and (Grp) there are natural equivalence relations on objects: (a natural extension of) equivalence of orbifold atlases in (Pre-Orb) and Morita equivalences in (Grp). We prove that F induces a bijection between the equivalence classes of its source and target.
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…
This is a survey paper based on my talk at the Workshop on Orbifolds and String Theory, the goal of which was to explain the role of groupoids and their classifying spaces as a foundation for the theory of orbifolds.
We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps. We use this result to define an orbifold version of Bredon cohomology.
For a finitely presented discrete group $Γ$, we introduce two generalizations of the orbifold Euler characteristic and $Γ$-orbifold Euler characteristic to a class of proper topological groupoids large enough to include all cocompact proper Lie groupoids. The $Γ$-Euler characteristic is defined as an integral with respect to the Euler characteristic over the orbit space of the groupoid, and the $Γ$-inertia Euler characteristic is the usual Euler characteristic of the $Γ$-inertia space associated to the groupoid. A key ingredient is the application of o-minimal structures to study orbit spaces …
In this paper we introduce the concept of Deligne cohomology of an orbifold. We prove that the third Deligne cohomology group of a smooth étale groupoid classify gerbes with connection over the groupoid. We argue that the $B$-field and the discrete torsion in type II superstring theories are special kinds of gerbes with connection, and finally, for each one of them, using Deligne cohomology we construct a flat line bundle over the inertia groupoid, namely a Ruan inner local in the case of an orbifold.
We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the notion of orbifold 1-homotopy type induced by a 1-homotopy between presentations of the orbifold maps.
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…
We propose a new homotopy invariant for Lie groupoids which generalizes the classical Lusternik-Schnirelmann category for topological spaces. We use a bicategorical approach to develop a notion of contraction in this context. We propose a notion of homotopy between generalized maps given by the 2-arrows in a certain bicategory of fractions. This notion is invariant under Morita equivalence. Thus, when the groupoid defines an orbifold, we have a well defined LS-category for orbifolds. We prove an orbifold version of the classical Lusternik-Schnirelmann theorem for critical points.
Configuration Lie groupoids and orbifold braid groups
本文提出配置李群胚的两种定义，证明一类李群胚的Fadell-Neuwirth型纤维化定理，并利用纯轨道折叠辫群的多重几乎自由结构，推导出相应Artin群及轨道折叠辫群的Farrell-Jones猜想。
In this paper we introduce the Cheeger-Simons cohomology of a global quotient orbifold. We prove that the Cheeger-Simons cohomology of the orbifold is isomorphic to its Beilinson-Deligne cohomology. Furthermore we construct a string connection (à la Segal) from a global gerbe with connection over the loop orbifold, refining the corresponding differential character.
Lie groupoids generalize transformation groups, and so provide a natural language for studying orbifolds and other noncommutative geometries. In this paper, we investigate a connection between orbifolds and equivariant stable homotopy theory using such groupoids. A different sort of twisted sector, along with a classical theorem of tom Dieck, allows for a natural definition of stable orbifold homotopy groups, and motivates defining extended unstable orbifold homotopy groups generalizing previous definitions.
In this paper we define an associative stringy product for the twisted orbifold K-theory of a compact, almost complex orbifold X. This product is defined on the twisted K-theory of the inertia orbifold of X, where the twisting gerbe is assumed to be in the image of the inverse transgression map.
We prove that the parallel transport of a flat $n-1$-gerbe on any given target space gives rise to an $n$-dimensional extended homotopy quantum field theory. In case the target space is the classifying space of a finite group, we provide explicit formulae for this homotopy quantum field theory in terms of transgression. Moreover, we use the geometric theory of orbifolds to give a dimension-independent version of twisted and equivariant Dijkgraaf-Witten models. Finally, we introduce twisted equivariant Dijkgraaf-Witten theories giving us in the 3-2-1-dimensional case a new class of equivariant …
Starting with a $\mathbb{C}^*$-valued cocycle on the global quotient orbifold $X // G$, we apply transgression techniques for 2-gerbes, as developed by Lupercio and Uribe, to construct a gerbe on the orbifold loop space $\mathcal{L}(X//G)$. This gives a loop based definition of twisted equivariant Tate K-theory for finite groups that conforms to the definition that Luecke provides for compact, connected Lie groups. We relate our construction to Ganter's work on Moonshine and Huan's quasi-elliptic cohomology.
On aspherical configuration Lie groupoids
本文研究了2维orbifold超平面补空间的非球面性，证明了一类2维非球面orbifold的情形，并在李群胚范畴中推广了该问题。
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…
Novikov type inequalities for orbifolds
本文将 Novikov 数推广到有限生成型 proper Lie groupoid 上闭基本 1-形式的基本上同调类，并据此证明紧 orbifold 的 Novikov 型不等式。
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 …
We give a description of the delocalized twisted cohomology of an orbifold and the Chern character of a twisted vector bundle in terms of supersymmetric Euclidean field theories. This includes the construction of a twist functor for $1|1$-dimensional EFTs from the data of a gerbe with connection.
For a finitely generated discrete group $Γ$, the $Γ$-sectors of an orbifold $Q$ are a disjoint union of orbifolds corresponding to homomorphisms from $Γ$ into a groupoid presenting $Q$. Here, we show that the inertia orbifold and $k$-multi-sectors are special cases of the $Γ$-sectors, and that the $Γ$-sectors are orbifold covers of Leida's fixed-point sectors. In the case of a global quotient, we show that the $Γ$-sectors correspond to orbifolds considered by other authors for global quotient orbifolds as well as their direct generalization to the case of an orbifold given by a quotient by a L…
The universal Euler characteristic and Burnside group for definable groupoids
本文为可定义群胚引入万有欧拉示性数，证明加性乘性不变量均可由环同态统一表示，并构造推广经典伯恩赛德环的伯恩赛德群。
On a new geometric homology theory and an application in categorical Gromov-Witten theory
本文证明带角光滑orbifold可三角剖分，并据此提出几何同调理论GHT以克服奇异同调的组合刚性，同时展示其在范畴Gromov-Witten理论中的应用。
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