We introduce an equivariant version of cyclic cohomology for Hopf module algebras. For any H-module algebra A, where H is a Hopf algebra with S2 = idH we define the cocyclic module C ♮ H(A) and we find its relation with cyclic cohomology of crossed product algebra A ⋊ H. We define K 0 (A), the equivariant K-theory group of A, and its pairing with cyclic and periodic cyclic cohomology of C H(A).

In cohomological field theory we can obtain topological invariants as correlation functions of BRS cohomology classes. A proper understanding of BRS cohomology which gives non-trivial results requires the equivariant cohomology theory. Both topological Yang-Mills theory and topological string theory are typical examples of this fact. After reviewing the role of the equivariant cohomology in top...

Let G be a compact connected Lie group with maximal torus T , and H a closed subgroup containing T . We compute the equivariant cohomology ring and the equivariant characteristic numbers of the homogeneous space G/H under the natural action of the maximal torus T . The computation is based on the localization theorems of Borel and of Atiyah-Bott-Berline-Vergne. Let G be a compact connected Lie ...

A conjecture of D. Peterson, proved by W. Graham [Gr], states that the structure constants of the (T−)equivariant cohomology of a homogeneous space G/P satisfy a certain positivity property. In this paper we show that this positivity property holds in the more general situation of equivariant quantum cohomology.

This text is an introduction to equivariant cohomology, a classical tool for topological transformation groups, and to equivariant intersection theory, a much more recent topic initiated by D. Edidin and W. Graham. It is based on lectures given at Montréal is Summer 1997. Our main aim is to obtain explicit descriptions of cohomology or Chow rings of certain manifolds with group actions which ar...

The structure of equivariant cohomology in non-abelian localization formulas and topological field theories is discussed. Equivariance is formulated in terms of a nilpotent BRST symmetry, and another nilpotent operator which restricts the BRST cohomology onto the equivariant, or basic sector. A superfield formulation is presented and connections to reducible (BFV) quantization of topological Ya...

Let X be a rational homogeneous space and let QH(X)× loc be the group of invertible elements in the small quantum cohomology ring of X localised in the quantum parameters. We generalise results of [ChMaPe06] and realise explicitly the map π1(Aut(X)) → QH (X)× loc described in [Se97]. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring QH T (X)× ...

Recently it has been shown that D-branes in orientifolds are not always described by equivariant Real K–theory. In this paper we define a previously unstudied twisted version of equivariant Real K–theory which gives the Dbrane spectrum for such orientifolds. We find that equivariant Real K–theory can be twisted by elements of a generalised group cohomology. This cohomology classifies all orient...

We study the T−equivariant quantum cohomology of the Grassmannian. We prove the vanishing of a certain class of equivariant quantum Littlewood-Richardson coefficients, which implies an equivariant quantum Pieri rule. As in the equivariant case, this implies an algorithm to compute the equivariant quantum Littlewood-Richardson coefficients.

Using the language and terminology of relative homological algebra, in particular that of derived functors, we introduce equivariant cohomology over a general Lie-Rinehart algebra and equivariant de Rham cohomology over a locally trivial Lie groupoid in terms of suitably defined monads (also known as triples) and the associated standard constructions. This extends a characterization of equivari...