Holonomy for Gerbes over Orbifolds

2-Categories, 4d state-sum models and gerbes

In this paper we give a short introduction to our results on the holonomy of gerbe-connections and explain our motivation coming from state-sum models.

Deligne Cohomology for Orbifolds, discrete torsion and B-fields

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.

A Stringy Product on Twisted Orbifold K-theory

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.

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…

Gerbes over posets and twisted C*-dynamical systems

A base $Δ$ generating the topology of a space $M$ becomes a partially ordered set (poset), when ordered under inclusion of open subsets. Given a precosheaf over $Δ$ of fixed-point spaces (typically C*-algebras) under the action of a group $G$, in general one cannot find a precosheaf of $G$-spaces having it as fixed-point precosheaf. Rather one gets a gerbe over $Δ$, that is, a "twisted precosheaf" whose twisting is encoded by a cocycle with coefficients in a suitable 2-group. We give a notion of holonomy for a gerbe, in terms of a non-abelian cocycle over the fundamental group $π_1(M)$. At the…

Twisted Equivariant Tate K-Theory

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.

Differential Characters on Orbifolds and String Connections I

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.

On cohomology of crystallographic groups with cyclic holonomy of split type

We disprove a conjecture stating that the integral cohomology of any crystallographic group Z^n \rtimes Z_m is given by the cohomology of Z_m with coefficients in the cohomology of the group Z^n, by providing a complete list of counterexamples up to dimension 5. We also find a counterexample with odd order holonomy, m=9, in dimension 8 and finish the computations of the cohomology of 6-dimensional crystallographic groups arising as orbifold fundamental groups of certain Calabi-Yau toroidal orbifolds.

Presentations of Noneffective Orbifolds

It is well-known that an effective orbifold M (one for which the local stabilizer groups act effectively) can be presented as a quotient of a smooth manifold P by a locally free action of a compact lie group K. We use the language of groupoids to provide a partial answer to the question of whether a noneffective orbifold can be so presented. We also note some connections to stacks and gerbes.

结点、边缘模式与 $G_2$-和乐轨形

Junctions, Edge Modes, and $G_2$-Holonomy Orbifolds

本文提出在奇异渐近锥形的 $G_2$-holonomy orbifolds 上的 M-theory 可实现带有 4D $\mathcal{N}=1$ edge modes 的强耦合 5D 体理论，并揭示体 generalized symmetries 在边界系统中的继承机制。

Compatible Actions and Cohomology of Crystallographic Groups

We compute the cohomology of crystallographic groups with holonomy of prime order. As an application we compute the group of gerbes associated to many six--dimensional toroidal orbifolds arising in string theory.

The Zigzag Hochschild Complex

In this paper, a new higher Hochschild Complex is defined with an Iterated Integral map to locally model differential forms on the space of bigons on $M$. In particular, given the local data for a gerbe with structure 2-group given by a crossed module of matrix-groups, there is an element in our curved zigzag Hochschild complex associated to the local 2-holonomy given by such a gerbe. This paper introduces an initial construction central to the author's PhD Thesis.

Parallel Transport of Higher Flat Gerbes as an Extended Homotopy Quantum Field Theory

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 …

Supersymmetric field theories from twisted vector bundles

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.

Cohomology of Toroidal Orbifold Quotients

Let $ϕ:\Z/p\to GL_{n}(\Z)$ denote an integral representation of the cyclic group of prime order $p$. This induces a $\Z/p$-action on the torus $X=\R^{n}/\Z^{n}$. The goal of this paper is to explicitly compute the cohomology groups $H^{*}(X/\Z/p;\Z)$ for any such representation. As a consequence we obtain an explicit calculation of the integral cohomology of the classifying space associated to the family of finite subgroups for any crystallographic group $Γ=\Z^n\rtimes\Z/p$ with prime holonomy.