Equivariant smooth Deligne cohomology

Differential Borel equivariant cohomology via connections

For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the Cartan-Weil equivariant forms and to Borel's equivariant integral cohomology. We show the Chern-Weil homomorphism for equivariant vector bundles with connection naturally factors through differential equivariant cohomology.

Higher-twisted periodic smooth Deligne cohomology

Degree one twisting of Deligne cohomology, as a differential refinement of integral cohomology, was established in previous work. Here we consider higher degree twists. The Rham complex, hence de Rham cohomology, admits twists of any odd degree. However, in order to consider twists of integral cohomology we need a periodic version. Combining the periodic versions of both ingredients leads us to introduce a periodic form of Deligne cohomology. We demonstrate that this theory indeed admits a twist by a gerbe of any odd degree. We present the main properties of the new theory and illustrate its u…

Projective unitary representations of smooth Deligne cohomology groups

We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle. We also classify such representations under a certain condition. The number of the equivalence classes of irreducible representations is finite, and is determined by the cohomology of the manifold.

Chern-Weil homomorphism in twisted equivariant cohomology

We describe the Cartan and Weil models of twisted equivariant cohomology together with the Cartan homomorphism among the two, and we extend the Chern-Weil homomorphism to the twisted equivariant cohomology. We clarify that in order to have a cohomology theory, the coefficients of the twisted equivariant cohomology must be taken in the completed polynomial algebra over the dual Lie algebra of $G$. We recall the relation between the equivariant cohomology of exact Courant algebroids and the twisted equivariant cohomology, and we show how to endow with a generalized complex structure the finite d…

Equivariant de Rham Cohomology: Theory and Applications

This is a survey on the equivariant cohomology of Lie group actions on manifolds, from the point of view of de Rham theory. Emphasis is put on the notion of equivariant formality, as well as on applications to ordinary cohomology and to fixed points.

Hermitian-holomorphic Deligne cohomology, Deligne pairing for singular metrics, and hyperbolic metrics

We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup-product in cohomology is shown to correspond to a generalization of the Deligne pairing to line bundles with "good" hermitian metrics in the sense of Mumford and others. A particular case is that of the tangent bundle of the curve twisted by the negative of the singularity divisor of a hyperbolic metric: its cup square (corrected by the total area) is shown to b…

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.

Equivariant cohomology of quaternionic flag manifolds

The main result of the paper is a Borel type description of the $Sp(1)^n$-equivariant cohomology ring of the manifold $Fl_n(\mathbb{H})$ of all complete flags in $\mathbb{H}^n$. To prove this, we obtain a Goresky-Kottwitz-MacPherson type description of that ring.

The equivariant cohomology ring of a cohomogeneity-one action

We compute the rational Borel equivariant cohomology ring of a cohomogeneity-one action of a compact Lie group.

S^1-equivariant Morse cohomology

We construct a deformed Morse complex computing the equivariant cohomology of a manifold M endowed with a smooth S^1-action. The deformation of the coboundary operator is given by counting gradient flow lines of a Morse function f that are allowed to "jump" along orbits of the S^1-action for finite time intervalls.

Equivariant Differential Cohomology

The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: The definition of equivariant characteristic forms in the Cartan model due to Nicole Berline and Michèle Vergne motivates a refinement of equivariant integral cohomology by all Cartan cocycles. In view of this, we will also review previous definitions critically, in particular the one given by Kiyonori Gomi.

Extended manifolds and extended equivariant cohomology

We define the category of manifolds with extended tangent bundles, we study their symmetries and we consider the analogue of equivariant cohomology for actions of Lie groups in this category. We show that when the action preserves the splitting of the extended tangent bundle, our definition of extended equivariant cohomology agrees with the twisted equivariant de Rham model of Cartan, and for this case we show that there is localization at the fixed point set, à la Atiyah-Bott.

Delocalized equivariant cohomology and resolution

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type was established by the authors in the context of manifolds with corners; the canonical construction induces fibrations on the boundary faces of the resolution resulting in an `equivariant resolution structure'. Here, equivariant K-theory and the Cartan model for equivariant cohomology are tracked under the resolution procedure as is the delocalized equivariant cohomology of Baum, Brylinski and MacPherson. This leads to resolved models…

Aeppli-Bott-Chern cohomology and Deligne cohomology from a viewpoint of Harvey-Lawson's spark complex

By comparing Deligne complex and Aeppli-Bott-Chern complex, we construct a differential cohomology $\widehat{H}^*(X, *, *)$ that plays the role of Harvey-Lawson spark group $\widehat{H}^*(X, *)$, and a cohomology $H^*_{ABC}(X; \Z(*, *))$ that plays the role of Deligne cohomology $H^*_{\mathcal{D}}(X; \Z(*))$ for every complex manifold $X$. They fit in the short exact sequence $$ 0\rightarrow H^{k+1}_{ABC}(X; \Z(p, q)) \rightarrow \widehat{H}^k(X, p, q) \overset{δ_1}{\rightarrow} Z^{k+1}_I(X, p, q) \rightarrow 0$$ and $\widehat{H}^{\bullet}(X, \bullet, \bullet)$ possess ring structure and refin…

Index theory, eta forms, and Deligne cohomology

The Chern classes of a K-theory class which is represented by a vector bundle with connection admit refinements to Cheeger-Simons classes in Deligne cohomology. In the present paper we consider similar refinements in the case where the classes in K-theory are represented by geometric families of Dirac operators. In low dimensions these refinements correspond to the exponentiated eta-invariant, the determinant line bundle with Quillen metric and Bismut-Freed connection, and Lott's index gerbe with connection and curving. We give a unified treatement of these cases as well as their higher genera…

Massey products in differential cohomology via stacks

We extend Massey products from cohomology to differential cohomology via stacks, organizing and generalizing existing constructions in Deligne cohomology. We study the properties and show how they are related to more classical Massey products in de Rham, singular, and Deligne cohomology. The setting and the algebraic machinery via stacks allow for computations and make the construction well-suited for applications. We illustrate with several examples from differential geometry and mathematical physics.

Equivariant Cohomology and Wall Crossing Formulas in Seiberg-Witten Theory

We use localization formulas in the theory of equivariant cohomology to rederive the wall crossing formulas of Li-Liu and Okonek-Teleman for Seiberg-Witten invariants.

The Cartan Model for Equivariant Cohomology

In this article, we will discuss a new operator $d_{C}$ on $W(\mathfrak{g})\otimesΩ^{*}(M)$ and to construct a new Cartan model for equivariant cohomology. We use the new Cartan model to construct the corresponding BRST model and Weil model, and discuss the relations between them.

Formal Frobenius manifold structure on equivariant cohomology

For a closed Kähler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model.

李群胚的等变基本上同调

Equivariant basic cohomology of Lie groupoids

本文发展了带有李群弱作用的李群胚的等变基本上同调理论，通过Kan纤维化编码弱作用，并与Bott-Shulman-Stasheff上同调比较，应用于orbifold的等变积分和定位，以及惯性群胚上的Chen-Ruan上同调环的等变细化。

On the localization formula in equivariant cohomology

We give two generalizations of the Atiyah-Bott-Berline-Vergne localization theorem for the equivariant cohomology of a torus action: 1) replacing the torus action by a compact connected Lie group action, 2) replacing the manifold having a torus action by an equivariant map. This provides a systematic method for calculating the Gysin homomorphism in ordinary cohomology of an equivariant map. As an example, we recover a formula of Akyildiz-Carrell for the Gysin homomorphism of flag manifolds.

On the equivariant cohomology of hyperpolar actions on symmetric spaces

We show that the equivariant cohomology of any hyperpolar action of a compact and connected Lie group on a symmetric space of compact type is a Cohen-Macaulay ring. This generalizes some results previously obtained by the authors.

Chiral Equivariant Cohomology III

This is the third of a series of papers on a new equivariant cohomology that takes values in a vertex algebra, and contains and generalizes the classical equivariant cohomology of a manifold with a Lie group action a la H. Cartan. In this paper, we compute this cohomology for spheres and show that for any simple connected group G, there is a sphere with infinitely many actions of G which have distinct chiral equivariant cohomology, but identical classical equivariant cohomology. Unlike the classical case, the description of the chiral equivariant cohomology of spheres requires a substantial am…

Singular equivariant asymptotics and the momentum map. Residue formulae in equivariant cohomology

Let $M$ be a smooth manifold and $G$ a compact connected Lie group acting on $M$ by isometries. In this paper, we study the equivariant cohomology of ${\bf X}=T^\ast M$, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae. In case that $\bf X$ is a compact symplectic manifold with a Hamiltonian $G$-action, similar residue formulae were derived by Jeffrey, Kirwan et al.

通过稳定化研究等价不变的莫尔斯-博特上同调

Equivariant Morse-Bott cohomology through stabilization

本文通过稳定化技术研究等价不变的莫尔斯-博特上同调，证明小扰动产生稳定不变函数，并利用通用度量实现横截性和定向性假设。

On the localization theorem in equivariant cohomology

We present a simple proof of a precise version of the localization theorem in equivariant cohomology. As an application, we describe the cohomology algebra of any compact symplectic variety with a multiplicity-free action of a compact Lie group. This applies in particular to smooth, projective spherical varieties.

Equivariant Poisson Cohomology and a Spectral Sequence Associated with a Moment Map

We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The spectral sequence converges to the Poisson cohomology of the manifold and has the $E_2$-term equal to the tensor product of the cohomology of the Lie algebra and the equivariant Poisson cohomology of the manifold. The latter is defined as the equivariant cohomology of the multi-vector fields made into a G-differential complex by means of the momentum mapping.…

Chiral Equivariant Cohomology I

We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, with the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai-Qu…

The quantum equivariant cohomology of toric manifolds through mirror symmetry

Using mirror symmetry as described by Hori and Vafa, we compute the quantum equivariant cohomology ring of toric manifolds. This ring arises naturally in topological gauged sigma-models and is related to the Hamiltonian Gromov-Witten invariants of the target manifold.

Bouquets revisited and equivariant elliptic cohomology

Let M be a spin manifold with a circular action. Given an elliptic curve E, we introduce, as in Grojnowski, elliptic bouquets of germs of holomorphic equivariant cohomology classes on M. Following Bott-Taubes and Rosu, we show that integration of an elliptic bouquet is well defined. In particular, this imply Witten's rigidity theorem. We emphasize the similarity between integration in K-theory and in elliptic cohomology.

On localization in holomorphic equivariant cohomology

We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu's localization formulas.