Equivariant Cohomology Groups Research Articles

Motivic Atiyah-Segal Completion Theorem

Abstract Let $T$ be a torus, $X$ a smooth separated scheme of finite type equipped with a $T$-action, and $[X/T]$ the associated quotient stack. Given any localizing ${\mathbb {A}}^{1}$-homotopy invariant of dg categories $E$ (homotopy $K$-theory, algebraic $K$-theory with coefficients, étale $K$-theory with coefficients, $l$-adic algebraic $K$-theory, $l$-adic étale $K$-theory, semi-topological $K$-theory, topological $K$-theory, periodic cyclic homology, etc), we prove that the derived completion of $E([X/T])$ at the augmentation ideal $I$ of the representation ring $R(T)$ of $T$ agrees with the classical Borel construction associated to the $T$-action on $X$. Moreover, for certain localizing ${\mathbb {A}}^{1}$-homotopy invariants, we extend this result to the case of a linearly reductive group scheme $G$. As a first application, we obtain an alternative proof of Krishna’s completion theorem in algebraic $K$-theory, of Thomason’s completion theorem in étale $K$-theory with coefficients, and also of Atiyah-Segal’s completion theorem in topological $K$-theory (for those topological $M$-spaces $X^{\textrm {an}}$ arising from analytification; $M$ is a(ny) maximal compact Lie subgroup of $G^{\textrm {an}}$). These alternative proofs lead to a spectral enrichment of the corresponding completion theorems and also to the following improvements: in the case of Thomason’s completion theorem the base field $k$ no longer needs to be separably closed, and in the case of Atiyah-Segal’s completion theorem the topological $M$-space $X^{\textrm {an}}$ no longer needs to be compact and the $M$-equivariant topological $K$-theory groups of $X^{\textrm {an}}$ no longer need to be finitely generated over the representation ring $R(M)$. As a second application, we obtain new completion theorems in $l$-adic étale $K$-theory, in semi-topological $K$-theory and also in periodic cyclic homology. As a third application, we obtain a description of the different equivariant cohomology groups in the literature (motivic, $l$-adic, morphic, Betti, de Rham, etc) in terms of derived completion. Finally, in two appendixes of independent interest, we extend a result of Weibel on homotopy $K$-theory from the realm of schemes to the broad setting of quotient stacks and establish some useful properties of semi-topological $K$-theory.

G∞–ring spectra and Moore spectra for β–rings

In this paper, we introduce the notion of $G_\infty$-ring spectra. These are globally equivariant homotopy types with a structured multiplication, giving rise to power operations on their equivariant homotopy and cohomology groups. We illustrate this structure by analysing when a Moore spectrum can be endowed with a $G_\infty$-ring structure. Such $G_\infty$-structures correspond to power operations on the underlying ring, indexed by the Burnside ring. We exhibit a close relation between these globally equivariant power operations and the structure of a $\beta$-ring, thus providing a new perspective on the theory of $\beta$-rings.

Cup-Product for Equivariant Leibniz Cohomology and Zinbiel Algebras

We study finite group actions on Leibniz algebras, and define equivariant cohomology groups associated to such actions. We show that there exists a cup-product operation on this graded cohomology, which makes it a graded Zinbiel algebra.

T-duality of singular spacetime compactifications in an H-flux

We begin by presenting a symmetric version of the circle equivariant T-duality result in a joint work of the second author with Siye Wu, thereby generalising the results there. We then initiate the study of twisted equivariant Courant algebroids and equivariant generalised geometry and apply it to our context. As before, T-duality exchanges type II A and type II B string theories. In our theory, both spacetime and the T-dual spacetime can be singular spaces when the fixed point set is non-empty; the singularities correspond to Kaluza-Klein monopoles. We propose that the Ramond-Ramond charges of type II string theories on the singular spaces are classified by twisted equivariant cohomology groups, consistent with the previous work of Mathai and Wu, and prove that they are naturally isomorphic. We also establish the corresponding isomorphism of twisted equivariant Courant algebroids.

On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms

Let $$\mathrm {R}$$ be a real closed field. We prove that for any fixed d, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $$\mathrm {R}^k$$ defined by polynomials of degrees bounded by d vanishes in dimensions d and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of $$d^{O(d)} s^d k^{\lfloor d/2 \rfloor -1} $$ on the equivariant Betti numbers of closed symmetric semi-algebraic subsets of $$\mathrm {R}^k$$ defined by quantifier-free formulas involving s symmetric polynomials of degrees bounded by d, where $$1 < d \ll s,k$$ . This bound is tight up to a factor depending only on d. These results significantly improve upon those obtained previously in Basu and Riener (Adv Math 305:803–855, 2017) which were proved using different techniques. Our new methods are quite general, and also yield bounds on the equivariant Betti numbers of certain special classes of symmetric definable sets (definable sets symmetrized by pulling back under symmetric polynomial maps of fixed degree) in arbitrary o-minimal structures over $$\mathrm {R}$$ . Finally, we utilize our new approach to obtain an algorithm with polynomially bounded complexity for computing these equivariant Betti numbers. In contrast, the problem of computing the ordinary Betti numbers of (not necessarily symmetric) semi-algebraic sets is considered to be an intractable problem, and all known algorithms for this problem have doubly exponential complexity.

AGT relations for abelian quiver gauge theories on ALE spaces

We construct level one dominant representations of the affine Kac–Moody algebra g l ̂ k on the equivariant cohomology groups of moduli spaces of rank one framed sheaves on the orbifold compactification of the minimal resolution X k of the A k − 1 toric singularity C 2 / Z k . We show that the direct sum of the fundamental classes of these moduli spaces is a Whittaker vector for g l ̂ k , which proves the AGT correspondence for pure N = 2 U ( 1 ) gauge theory on X k . We consider Carlsson–Okounkov type Ext-bundles over products of the moduli spaces and use their Euler classes to define vertex operators. Under the decomposition g l ̂ k ≃ h ⊕ s l ̂ k , these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra h and primary fields of s l ̂ k . We use these operators to prove the AGT correspondence for N = 2 superconformal abelian quiver gauge theories on X k .

EQUIVARIANT ANDERSON DUALITY AND MACKEY FUNCTOR DUALITY

AbstractWe show that the$\mathbb{Z}$/2-equivariantnth integral MoravaK-theory with reality is self-dual with respect to equivariant Anderson duality. In particular, there is a universal coefficients exact sequence in integral Morava K-theory with reality, and we recover the self-duality of the spectrumKOas a corollary. The study of$\mathbb{Z}$/2-equivariant Anderson duality made in this paper gives a nice interpretation of some symmetries ofRO($\mathbb{Z}$/2)-graded (i.e. bigraded) equivariant cohomology groups in terms of Mackey functor duality.

Conjugation spaces and equivariant Chern classes

Let $\eta$ be a Real bundle, in the sense of Atiyah, over a space $X$. This is a complex vector bundle together with an involution which is compatible with complex conjugation. We use the fact that $BU$ has a canonical structure of a conjugation space, as defined by Hausmann, Holm, and Puppe, to construct equivariant Chern classes in certain equivariant cohomology groups of $X$ with twisted integer coefficients. We show that these classes determine the (non-equivariant) Chern classes of $\eta$, forgetting the involution on $X$, and the Stiefel-Whitney classes of the real bundle of fixed points.

Topology and Flux of T-Dual Manifolds with Circle Actions

We present an explicit formula for the topology and H-flux of the T-dual of a general type II compactification, significantly generalizing earlier results. Our results apply to T-dualities with respect to any circle action on spacetime. As before, T-duality exchanges type IIA and type IIB string theories. A new consequence is that the T-dual spacetime is a singular space when the fixed point set is non-empty; the singularities correspond to Kaluza-Klein monopoles. We propose that the Ramond-Ramond charges of type II string theories on the singular dual are classified by twisted equivariant cohomology groups. We also include the K-theory approach.

Vertex operators and quasimodularity of Chern numbers on the Hilbert scheme

In Carlsson and Okounkov (preprint) [7], Okounkov and the author defined a family of vertex operators on the equivariant cohomology groups of the Hilbert scheme of points on a smooth quasi-projective surface as a characteristic class of certain canonical bundles on Hilb⁎S×Hilb⁎S. We then proved a bosonization formula in terms of Nakajimaʼs Heisenberg operators (Nakajima, 1997 [23]). In this paper we apply this operator in the special case when S=C2 with a particular action of a torus, and prove that the generating functions of equivariant Chern numbers on Hilbn over n, are quasimodular forms in the generating variable. This property determines the answer up to a finite-dimensional vector space of functions of the generating variable, q. These generating functions can be thought of as the analogous correlation functions to Nekrasovʼs partition function in rank 1.We present a different proof of the bosonization formula which is based on the proof of a more general formula in K-theory given in an upcoming paper by Nekrasov, Okounkov and the author (in preparation) [6], but specialized to the surface of interest. By altering a certain bundle that appears in Carlsson et al. (in preparation) [6], and specializing the surface, the proof actually reduces to a much simpler self-contained application of the infinite wedge representation. This picture is consistent with both the original introduction of this operator in Nekrasov and Okounkov (2006) [26] and with Haimanʼs character theory of the Bridgeland, King and Reid isomorphism (Haiman, 2003 [11]).

Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology

In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Λ on a compact Riemannian manifold M with boundary ∂ M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field X M on M, Witten defines an inhomogeneous coboundary operator d X M = d + ι X M on invariant forms on M. The main purpose is to adapt Belishev–Sharafutdinovʼs boundary data to invariant forms in terms of the operator d X M in order to investigate to what extent the equivariant topology of a manifold is determined by the corresponding variant of the DN map. We define an operator Λ X M on invariant forms on the boundary which we call the X M -DN map and using this we recover the X M -cohomology groups from the generalized boundary data ( ∂ M , Λ X M ) . This shows that for a Zariski-open subset of the Lie algebra, Λ X M determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the ring structure of X M -cohomology groups from Λ X M . These results explain to what extent the equivariant topology of the manifold in question is determined by Λ X M .

Vertex Operators, Grassmannians, and Hilbert Schemes

We describe a well-known collection of vertex operators on the infinite wedge representation as a limit of geometric correspondences on the equivariant cohomology groups of a finite-dimensional approximation of the Sato grassmannian, by cutoffs in high and low degrees. We prove that locality, the boson-fermion correspondence, and intertwining relations with the Virasoro algebra are limits of the localization expression for the composition of these operators. We then show that these operators are, almost by definition, the Hilbert scheme vertex operators defined by Okounkov and the author in \cite{CO} when the surface is $\mathbb{C}^2$ with the torus action $z\cdot (x,y) = (zx,z^{-1}y)$.

A new description of equivariant cohomology for totally disconnected groups

AbstractWe consider smooth actions of totally disconnected groups on simplicial complexes and compare different equivariant cohomology groups associated to such actions. Our main result is that the bivariant equivariant cohomology theory introduced by Baum and Schneider can be described using equivariant periodic cyclic homology. This provides a new approach to the construction of Baum and Schneider as well as a computation of equivariant periodic cyclic homology for a natural class of examples. In addition we discuss the relation between cosheaf homology and equivariant Bredon homology. Since the theory of Baum and Schneider generalizes cosheaf homology we finally see that all these approaches to equivariant cohomology for totally disconnected groups are closely related.

Yang-Mills connections on nonorientable surfaces

Let G be a compact, connected Lie group. In [1], Atiyah and Bott identified the affine space A of connections on a principal G-bundle P over a Riemann surface with the affine space C of holomorphic structures on P = P ×G G, where G is the complexification of G. The identification A ∼= C is an isomorphism of affine spaces, thus a diffeomorphism. It was conjectured in [1] that under this identification the Morse stratification of the Yang–Mills functional on A exists and coincides with the stratification of C from algebraic geometry [14, 25]. The conjecture was proved by Daskalopoulos in [6] (see also [24] by Rade). The top stratum Css of C consists of semi-stable holomorphic structures on P. Atiyah and Bott showed that the stratification of C is G-perfect, where G = Aut(P). It has strong implications on the topology of the moduli space M(P ) of S-equivalence classes of semi-stable holomorphic structures on P. When M(P ) is smooth, Atiyah and Bott found a complete set of generators of the cohomology groups H∗(M(P );Q) and recursive relations which determine the Poincare polynomial Pt(M(P );Q). WhenM(P ) is singular, their results give generators of the equivariant cohomology groups H∗ GC(Css;Q) and formula for the equivariant Poincare series P G C t (Css;Q). Under the isomorphism A ∼= C, the top stratum Css corresponds to Ass which is the stable manifold of Nss, the set of central Yang–Mills connections, where the Yang–Mills functional achieves its absolute minimum [1,6].

Equivariant cohomology, Fock space and loop groups

Equivariant de Rham cohomology is extended to the infinite-dimensional setting of a loop subgroup acting on a loop group, using Hida supersymmetric Fock space for the Weil algebra and Malliavin test forms on the loop group. The Mathai–Quillen isomorphism (in the BRST formalism of Kalkman) is defined so that the equivalence of various models of the equivariant de Rham cohomology can be established.

On the crystalline cohomology of Deligne–Lusztig varieties

Let X → Y 0 be an abelian prime-to- p Galois covering of smooth schemes over a perfect field k of characteristic p > 0 . Let Y be a smooth compactification of Y 0 such that Y − Y 0 is a normal crossings divisor on Y. We describe a logarithmic F-crystal on Y whose rational crystalline cohomology is the rigid cohomology of X, in particular provides a natural W [ F ] -lattice inside the latter; here W is the Witt vector ring of k. If a finite group G acts compatibly on X, Y 0 and Y then our construction is G-equivariant. As an example we apply it to Deligne–Lusztig varieties. For a finite field k, if G is a connected reductive algebraic group defined over k and L a k-rational torus satisfying a certain standard condition, we obtain a meaningful equivariant W [ F ] -lattice in the cohomology ( ℓ-adic or rigid) of the corresponding Deligne–Lusztig variety and an expression of its reduction modulo p in terms of equivariant Hodge cohomology groups.

The equivariant Serre spectral sequence as an application of a spectral sequence of Spanier

E.H. Spanier (1992) has constructed, for a cohomology theory defined on a triangulated space and locally constant on each open simplex, a spectral sequence whose E 2-term consists of certain simplicial cohomology groups, converging to the cohomology of the space. In this paper we study a closed G-fibration ƒ: Y → X, where G is a finite group. We show that if the base- G-space X is equivariantly triangulated and Y is paracompact, then Spanier's spectral sequence yields an equivariant Serre spectral sequence for ƒ. The main point here is to identify the equivariant singular cohomology groups of X with appropriate simplicial cohomology groups of the orbit space X G .

The Candelas-de la Ossa-Green-Parkes formula

In this note we discuss a recent proof of the formula for the worldsheet instanton prepotential predicted by Mirror Symmetry for the quintics in P 4. One of the key ingredients in the proof is the equivariant cohomology groups on the so-called linear sigma model moduli spaces. We introduce the notion of admissible data on the equivariant cohomology groups of the linear sigma model. An admissible data may be thought of as a sequence of equivariant classes satisfying certain algebraic conditions. It arises naturally from Kontsevich's stable map compactification of moduli spaces of maps from curves into projective manifolds. The structures of admissible data help reduce many counting problems to checking certain combinatorial structure of a compactification. The mirror transformation of Candelas et al turns out to be a transformation between two admissible data associated respectively to the linear and the non-linear sigma models. As an application, we prove the formula for the worldsheet instanton prepotential in terms of hypergeometric series. At the end we also interprete an infinite dimensional transformation group, called the mirror group, acting on admissible data, as a certain duality group of the linear sigma model.

The equivariant cohomology groups of a real algebraic surface and their applications

Exact sequences connecting the equivariant cohomology groups of a real algebraic surface are constructed, and sufficient conditions for the convergence of the second spectral sequence are established. The results obtained are applied to the study of the topology of a surface and to the determination of the Brauer group.

Equivariant Bundles and Cohomology

Let $G$ be a topological group, $A$ an abelian topological group on which $G$ acts continuously and $X$ a $G$-space. We define "equivariant cohomology groups" of $X$ with coefficients in $A$, $H_G^i(X;A)$, for $i \geq 0$ which generalize Graeme Segal’s continuous cohomology of the topological group $G$ with coefficients in $A$. In particular we have $H_G^1(X;A) \simeq$ equivalence classes of principal $(G,A)$-bundles over $X$. We show that when $G$ is a compact Lie group and $A$ an abelian Lie group we have for $i > 1\;H_G^i(X;A) \simeq {H^i}(EG{ \times _G}X;\tau A)$ where $\tau A$ is the sheaf of germs of sections of the bundle $(X \times EG \times A)/G \to (X \times EG)/G$. For $i = 1$ and the trivial action of $G$ on $A$ this is a theorem of Lashof, May and Segal.