Equivariant cohomology for differentiable stacks

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Abstract

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.

Introduction

An important principle in geometry and physics is to exploit symmetry whenever possible. A common manifestation of such symmetry is for example given by an action of a Lie group G on a smooth manifold X. Equivariant cohomology is a way of exploiting this symmetry and provides an important algebraic invariant to study Lie group actions on smooth manifolds. A first approach when looking for a notion of cohomology in this framework is to use the singular cohomology of the quotient space XG, however even though singular cohomology is well defined, the quotient and its singular cohomology may lose a lot of geometric information. In order to overcome this obstruction, one can construct alternative and better models for equivariant cohomology using a weaker, but more adequate notion of a quotient for Lie group actions on smooth manifolds by employing homotopy theory or using appropriate equivariant versions of the de Rham complex of differential forms. There are basically three important models for equivariant cohomology. Firstly, the Borel model provides a good topological model for equivariant cohomology by considering the homotopy quotient space instead, also called Borel construction, EG×GX of X (see [9], [21]). Secondly, the Cartan model employs the notion of equivariant differential forms on X, but restricted to compact Lie group actions (see [12]). Both models provide the same equivariant cohomology groups (see [8]). And finally, the Getzler model (see [19]), which also employs an appropriate complex of equivariant differential forms, provides a generalisation and an alternative to the Cartan model in the more general case of arbitrary, not necessarily compact Lie group actions on smooth manifolds. This model is also essential for many important applications in global analysis, differential geometry and mathematical physics.

In this article we provide an alternative and extension to the classical constructions of equivariant cohomology using the more general framework of Lie group actions on differentiable stacks. This approach relies on the construction of a good quotient stack or stacky quotient MG for a general action of a Lie group G on a given differentiable stack M. In the special situation of an action of a Lie group G on a smooth manifold X we recover the quotient stack [XG]. We will show that this stacky quotient MG is again a differentiable stack and has a homotopy type given by a good homotopy quotient constructed via the fat geometric realisation of the nerve of the Lie groupoid associated to the differentiable stack MG. In more detail, we can understand this homotopy type of MG as being constructed from the bisimplicial smooth manifold given by G×X, where Gp is the p-fold cartesian product of the Lie group G and Xn describes the nth component of the simplicial smooth manifold given by the nerve associated to the Lie groupoid constructed from a given G-atlas XM of the original differentiable G-stack M. These constructions are in fact independent of a particular choice of an atlas and are therefore stacky by nature. In the special situation of an action of a Lie group G on a smooth manifold X we obtain the classical quotient stack [XG] and its homotopy type simply recovers the homotopy quotient EG×GX. Consequently, we obtain a Borel model for equivariant cohomology of general differentiable G-stacks which extends and generalises the classical Borel model for smooth G-manifolds. Furthermore using simplicial techniques, we extend the Cartan and Getzler models for equivariant cohomology based on particular complexes of equivariant differential forms for differentiable stacks with compact or non-compact Lie group actions. For example, the stacky Cartan model is induced from constructions of adequate complexes of differential forms for simplicial smooth manifolds based on Meinreken’s work (see [30], [37], [38]) while for the stacky Getzler model we also use simplicial techniques from [27] applied to the simplicial smooth manifold build from the nerve of the associated Lie groupoid of the given differentiable G-stack. Group actions on stacks were first studied systematically by Romagny [33] in the context of algebraic geometry where general actions of flat group schemes on algebraic stacks were considered. More recently general actions of topological groups on topological stacks were also studied by Ginot–Noohi [20] in their general approach to equivariant string topology [6]. Since differentiable stacks generalise orbifolds (see [28], [35], [39]), the models of equivariant cohomology for Lie group actions on differentiable stacks provided here also establish good models for equivariant orbifold cohomology which we aim to explore in future work. Equivariant Chen-Ruan orbifold cohomology for example has many important applications in symplectic geometry, in particular when considering Hamiltonian torus actions on orbifolds (compare for example [25], [29]).

This article is structured as follows. In the first section we recall the definition and basic properties of general Lie group actions on differentiable stacks. We start by defining the notion of an action of a Lie group G on a differentiable stack M and then describe the associated 2-category GSt of G-stacks. This is very much in the flavour of [33] and [20], but in the context of smooth manifolds and Lie groups. The second section features the construction of the quotient stack MG for a differentiable stack M with an action of a Lie group G. We then introduce the notion of a differentiable G-stack M using an appropriate version of a G-atlas given as a smooth manifold with G-action and relate it to the simplicial G-manifold constructed from the associated Lie groupoid of M. Finally we show that the quotient stack MG is in fact a differentiable stack and we explicitly describe its homotopy type given by the fat geometric realisation of the simplicial nerve of the Lie groupoid associated to MG. In the third section we recall and discuss several cohomology theories for differentiable stacks, namely de Rham cohomology, sheaf cohomology and hypercohomology, following in parts the expositions in [4], [5] and [22]. In the fourth section we introduce the concepts of equivariant cohomology for differentiable G-stacks and derive and describe the Borel, Cartan and Getzler models in this general context. We analyse several fundamental properties of equivariant cohomology, in particular concerning the effect of restricting the acting Lie group. Finally in the fifth and last section we derive several spectral sequences that all converge to the equivariant cohomology of a differentiable G-stack. They relate the cohomology of the simplicial nerve of the Lie groupoid associated to a differentiable G-stack M with the equivariant cohomology of M. We will then analyse these in particular situations and discuss their homological properties. As special cases we obtain generalisations of spectral sequences previously constructed for equivariant cohomology of smooth G-manifolds, including the celebrated Bott spectral sequence converging to the cohomology of the classifying space of a Lie group (see [16], [36] and [10]).

Section snippets

Group actions on differentiable stacks

The general notion of a group action on a stack was first developed and applied by Rogmany [33] in the context of group scheme actions on algebraic stacks. More recently, Ginot and Noohi [20] studied topological group actions on topological stacks in their approach to equivariant string topology. In this section we will recall and analyse the main definitions and constructions within the framework of differentiable stacks. For us here, a stack will always mean a pseudo-functor M:DiffopGrpds,

Quotient stacks

The concept of quotient stacks for group actions on stacks was first developed and studied by Romagny [33] for algebraic stacks and by Ginot–Noohi [20] for topological stacks. This generalises the classical notions of quotient stacks arising from group actions on schemes, manifolds or topological spaces. Here again we will be working entirely in the differentiable setting of differentiable stacks with Lie group actions.

Definition 2.1

Let G be a Lie group acting on a differentiable stack M. Consider the

Cohomology of differentiable stacks

We will now discuss several cohomology theories for differentiable stacks, namely de Rham cohomology, sheaf cohomology and hypercohomology. As general references for cohomology of stacks, in particular de Rham and sheaf cohomology, we refer to [4], [7] and [22].

Models for equivariant cohomology of differentiable stacks

In this section we will now describe the Borel, Cartan and Getzler models for equivariant cohomology of differentiable stacks with Lie group actions and study some of their main properties.

Spectral sequences for equivariant cohomology of differentiable G-stacks

In this final section we will derive some auxiliary results about spectral sequences for the equivariant cohomology of a differentiable G-stack. These will be derived in the frameworks of sheaf cohomology, hypercohomology and continuous cohomology. We also recover in special cases some spectral sequences previously constructed in the context of smooth manifolds with Lie group actions, namely by Felder–Henriques–Rossi–Zhu [16], Stasheff [36] and Bott [10].

Acknowledgments

The first author was supported by the Colciencias Conv. 646 scholarship of the Colombian government. Both authors like to thank the referee for helpful suggestions, comments and corrections to improve this article.

References (41)

  • Arias AbadC. et al.

    On the equivariant de Rham cohomology for non-compact Lie groups

    Differential Geom. Appl.

    (2015)
  • Barbosa-TorresL.A.

    Differentiable Stacks and Equivariant Cohomology

    (2019)
  • BehrendK.

    Cohomology of stacks

  • BehrendK. et al.

    String topology for stacks

    Asterisque

    (2012)
  • BehrendK. et al.

    Differentiable stacks and gerbes

    J. Symplectic Geom.

    (2011)
  • BerlineN. et al.

    Heat kernels and Dirac operators

  • BorelA.

    Seminar on transformation groups, With contributions by G. Bredon, E.E. Floyd, D. Montgomery, R. Palais

  • BottR. et al.

    Differential forms in algebraic topology

  • CartanH.

    La transgression dans un groupe de Lie et dans un espace fibré principal

  • DeligneP.

    Théorie de Hodge III

    Inst. Hautes Études Sci. Publ. Math.

    (1974)
  • View full text