Differentiable Stacks Research Articles

Shifted Contact Structures on Differentiable Stacks

Shifted Contact Structures on Differentiable Stacks

Connections on Lie groupoids and Chern–Weil theory

Let [Formula: see text] be a Lie groupoid equipped with a connection, given by a smooth distribution [Formula: see text] transversal to the fibers of the source map. Under the assumption that the distribution [Formula: see text] is integrable, we define a version of de Rham cohomology for the pair [Formula: see text], and we study connections on principal [Formula: see text]-bundles over [Formula: see text] in terms of the associated Atiyah sequence of vector bundles. We also discuss associated constructions for differentiable stacks. Finally, we develop the corresponding Chern–Weil theory and describe characteristic classes of principal [Formula: see text]-bundles over a pair [Formula: see text].

Atiyah sequences and connections on principal bundles over Lie groupoids and differentiable stacks

We construct and study general and integrable connections on Lie groupoids and differentiable stacks, as well as on principal bundles over them using an Atiyah exact sequence of vector bundles associated to transversal tangential distributions.

Shifted Poisson Structures on Differentiable Stacks

Abstract The purpose of this paper is to investigate $(+1)$-shifted Poisson structures in the context of differential geometry. The relevant notion is that of $(+1)$-shifted Poisson structures on differentiable stacks. More precisely, we develop the notion of the Morita equivalence of quasi-Poisson groupoids. Thus, isomorphism classes of $(+1)$-shifted Poisson stacks correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following program, which is of independent interest: (1) We introduce a ${\mathbb{Z}}$-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under the Morita equivalence of Lie groupoids, and thus they can be considered to be polyvector fields on the corresponding differentiable stack ${\mathfrak{X}}$. It turns out that $(+1)$-shifted Poisson structures on ${\mathfrak{X}}$ correspond exactly to elements of the Maurer–Cartan moduli set of the corresponding dgla. (2) We introduce the notion of the tangent complex $T_{\mathfrak{X}}$ and the cotangent complex $L_{\mathfrak{X}}$ of a differentiable stack ${\mathfrak{X}}$ in terms of any Lie groupoid $\Gamma{\rightrightarrows } M$ representing ${\mathfrak{X}}$. They correspond to a homotopy class of 2-term homotopy $\Gamma$-modules $A[1]\rightarrow TM$ and $T^{\vee } M\rightarrow A^{\vee }[-1]$, respectively. Relying on the tools of theory of VB-groupoids including homotopy and Morita equivalence of VB-groupoids, we prove that a $(+1)$-shifted Poisson structure on a differentiable stack ${\mathfrak{X}}$ defines a morphism ${L_{\mathfrak{X}}}[1]\to{T_{\mathfrak{X}}}$.

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.

GEODESICS ON RIEMANNIAN STACKS

Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on Riemannian stacks, measuring their length using stacky metrics, and introducing stacky geodesics. Our main results show that the length of stacky curves measure distances on the orbit space, characterize stacky geodesics as locally minimizing curves, and establish a stacky version of Hopf-Rinow Theorem. We include a concise overview that bypasses nonessential technicalities, and we lay stress on the examples of orbit spaces of isometric actions and leaf spaces of Riemannian foliations.

On two notions of a gerbe over a stack

Let G be a Lie groupoid. The category BG of principal G-bundles defines a differentiable stack. On the other hand, given a differentiable stack D, there exists a Lie groupoid H such that BH is isomorphic to D. Define a gerbe over a stack as a morphism of stacks F:D→C, such that F and the diagonal map ΔF:D→D×CD are epimorphisms. This paper explores the relationship between a gerbe defined above and a Morita equivalence class of a Lie groupoid extension.

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 fundamental group of hyperbolic toral automorphisms, and we revisit the classification of lens spaces.

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 stacks; in particular, in the symplectic case, we prove the conjecture left open in Weinstein’s “The volume of a differentiable stack” [33].We also revisit the notion of Haar systems (and the existence of cut-off functions). Our original motivation comes from the study of Poisson manifolds of compact types [8–10], which provide two important examples of such measures: the affine and the Duistermaat–Heckman measures.

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.

On the Lie 2-algebra of sections of an [formula omitted]-groupoid

In this work we introduce the category of multiplicative sections of an LA-groupoid. We prove that this category carries a natural strict Lie 2-algebra structure, which is 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 classifying stacks, quotient stacks of regular Lie groupoids and in particular orbifolds, and foliation groupoids.

Étale stacks as prolongations

We give a complete and categorical characterization of étale stacks (generalized orbifolds) in various geometric contexts, including differentiable stacks and topological stacks. This characterization states that étale stacks are precisely those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms, and makes no mention of an atlas or groupoid presentation. Moreover, we show that the bicategory of étale differentiable stacks and local diffeomorphisms is equivalent to the bicategory of stacks on the site of smooth manifolds and local diffeomorphisms. An analogous statement holds for other flavors of manifolds (topological, Ck, complex, super...), and topological spaces locally homeomorphic to a given space X. A slight modification of this result also holds in an even more general context, including all étale topological stacks, and Zariski étale stacks, which are analogues of Deligne-Mumford stacks for the Zariski topology, and we also sketch a proof of an analogous characterization of classical Deligne-Mumford algebraic stacks (which we prove in full in [7] by different methods). We go on to characterize effective étale stacks (e.g. reduced orbifolds) as precisely those stacks arising as the prolongations of sheaves of sets (rather than stacks of groupoids). It follows that étale stacks (and in particular orbifolds) induce a small gerbe over their effective part, and all gerbes over effective étale stacks arise in this way. We show that well known Lie groupoids from foliation theory literature can be reproduced from this framework as natural presentations for moduli stacks of certain geometric structures. For example, there exists a classifying stack for Riemannian metrics, presented by Haefliger's groupoid RΓ [13] and submersions into this stack classify Riemannian foliations, and similarly for symplectic structures, with the role of RΓ replaced with ΓSp. We also prove some unexpected results, for example: the category of smooth n-manifolds and local diffeomorphisms has binary products.

Resolutions of Proper Riemannian Lie Groupoids

Abstract In this paper we prove that every proper Lie groupoid admits a regularization to a regular proper Lie groupoid. When equipped with a Riemannian metric, we show that it admits regularization 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 regularization via a successive blow-up construction on a proper Lie groupoid. We also prove that our construction of the regularization is invariant under Morita equivalence of groupoids, showing that it is a desingularization of the underlying differentiable stack.

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

Abstract 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, that is, 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 that naturally arise from non-integrable Lie algebroids.

Morita Equivalences of Vector Bundles

Abstract 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-groupoids is a Morita invariant, which leads to a notion of VB-stacks, and solves (an instance of) an open question on representations up to homotopy.

A current-reuse biomedical amplifier with a NEF < 1

Noise Efficiency Factor (NEF) is the most employed figure of merit to compare different low-noise biomedical signal amplifiers, taking into account current consumption, noise, or bandwidth trade-offs. A small NEF means a more efficient amplifier, and was assumed to be always NEF > 1 (an ideally efficient single BJT amplifier). In this work current-reuse technique will be utilized to exceed this limit in a very efficient CMOS amplifier. A micro-power, ultra-low-noise amplifier, aimed at electro-neuro-graph signal recording in a specific single-channel implantable medical device, is presented. The circuit is powered with a standard medical grade 3.6 V(nom) secondary battery. The amplifier input stage stacks twelve differential pairs to maximize current-reuse. The differential pair stacking technique is very efficient: allows most of the energy to be dissipated in the input transistors that amplify and not in mirror or bias transistors, and allows also the input transistors to operate with a reduced VDS just above saturation. The amplifier was implemented in a 0.6 μm technology, it has a total gain of almost 80 dB, with a 4 kHz bandwidth. The measured input referred noise is 4.5 nV/Hz1/2@1 kHz, and 330 nVrms in the band of interest, with a total current consumption of only 16.5 μA from the battery (including all the 4 stages and the auxiliary circuits). The measured NEF is only 0.84, below the classic NEF = 1 limit.

Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields

We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth [Theory Appl. Categ. 22 (2009), 542-587] in his study of vector fields on differentiable stacks. Here we argue in favor of the usefulness of replacing an equivariant vector field by an isomorphic one to study nonlinear stability of relative equilibria. In particular, we use this idea to obtain a criterion for nonlinear stability. As an application, we offer an alternative proof of Montaldi and Rodr\'iguez-Olmos's criterion [arXiv:1509.04961] for stability of Hamiltonian relative equilibria.

On symplectic vortex equations over a compact orbifold Riemann surface

Making use of theory of differentiable stacks, we study symplectic vortex equations over a compact orbifold Riemann surface. We discuss the category of representable morphisms from a compact orbifold Riemann surface to a quotient stack. After that we define symplectic vortex equations over a compact orbifold Riemann surface. We also discuss the moduli space of solutions to the equations for linear actions of the circle group on the complex plane.

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 (∞, 1)‐categories. We then discuss principal bundles in terms of simplicial maps and their homotopies. We explain in detail a differentiation procedure, suggested by Ševera, that maps higher groupoids to ‐algebroids. Generalising this procedure, we define connections for higher groupoid bundles. As an application, we obtain six‐dimensional superconformal field theories via a Penrose–Ward transform of higher groupoid bundles over a twistor space. This construction reduces the search for non‐Abelian self‐dual tensor field equations in six dimensions to a search for the appropriate (higher) gauge structure. The treatment aims to be accessible to theoretical physicists.