Equivariant Cohomology Classes Research Articles

Canonical equivariant cohomology classes generating zeta values of totally real fields

It is known that the special values at nonpositive integers of a Dirichlet L L -function may be expressed using the generalized Bernoulli numbers, which are defined by a generating function. The purpose of this article is to consider the generalization of this classical result to the case of Hecke L L -functions of totally real fields. Hecke L L -functions may be expressed canonically as a finite sum of zeta functions of Lerch type. By combining the non-canonical multivariable generating functions constructed by Shintani [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), pp. 393–417], we newly construct a canonical class, which we call the Shintani generating class, in the equivariant cohomology of an algebraic torus associated to the totally real field. Our main result states that the specializations at torsion points of the derivatives of the Shintani generating class give values at nonpositive integers of the zeta functions of Lerch type. This result gives the insight that the correct framework in the higher dimensional case is to consider higher equivariant cohomology classes instead of functions.

A Tableau Formula for Vexillary Schubert Polynomials in Type C

Ikeda-Mihalcea-Naruse's double Schubert polynomials [Adv. Math. 226 (2011)] represent the equivariant cohomology classes of Schubert varieties in the type C flag varieties. The goal of this paper is to obtain a new tableau formula of these polynomials associated to vexillary signed permutations introduced by Anderson-Fulton. To achieve that goal, we introduce flagged factorial (Schur) $Q$-functions, combinatorially defined functions in terms of marked shifted tableaux for flagged strict partitions, and prove their Schur-Pfaffian formula. As an application, we also obtain a new combinatorial formula of factorial $Q$-functions of Ivanov in which monomials bijectively corresponds to flagged marked shifted tableaux.

Bouquets revisited and equivariant elliptic cohomology

Let [Formula: see text] be an even-dimensional spin manifold with action of a compact Lie group [Formula: see text]. We review the construction of bouquets of analytic equivariant cohomology classes and of [Formula: see text]-integration. Given an elliptic curve [Formula: see text], we introduce, as in Grojnowski, elliptic bouquets of germs of holomorphic equivariant cohomology classes on [Formula: see text]. Following Bott–Taubes and Rosu, we show that integration of an elliptic bouquet is well defined. In particular, this imply Witten’s rigidity theorem. Our systematic use of the Chern–Weil construction of equivariant classes allows us to reduce proofs to algebraic identities.

Involution pipe dreams

AbstractInvolution Schubert polynomials represent cohomology classes of K-orbit closures in the complete flag variety, where K is the orthogonal or symplectic group. We show they also represent $\mathsf {T}$ -equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables $x_i + x_j$ , and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. Our formulas are analogues of the Billey–Jockusch–Stanley formula for Schubert polynomials. In Knutson and Miller’s approach to matrix Schubert varieties, pipe dream formulas reflect Gröbner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting.

Wonderful symmetric varieties and Schubert polynomials

Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of the general linear group on the flag variety. Combining this with a slight extension of results of Can, Joyce and Wyser, we arrive at a family of polynomial identities which show that certain explicit sums of Schubert polynomials factor as products of linear forms.

Equivariant classes of matrix matroid varieties

To each subset I of \{1,\dots,k\} associate an integer r(I) . Denote by X the collection of those n\times k matrices for which the rank of a union of columns corresponding to a subset I is r(I) , for all I . We study the equivariant cohomology class represented by the Zariski closure Y=\overline{X} . This class is an invariant of the underlying matroid structure. Its calculation incorporates challenges similar to the calculation of the ideal of Y , namely, the determination of the geometric theorems for the matroid. This class also gives information on the degenerations and hierarchy of matroids. New developments in the theory of Thom polynomials of contact singularities (namely, a recently found stability property) help us to calculate these classes and present their basic properties. We also show that the coefficients of this class are solutions to problems in enumerative geometry, which are natural generalization of the linear Gromov–Witten invariants of projective spaces.

LOOP GROUPS, STRING CLASSES AND EQUIVARIANT COHOMOLOGY

AbstractWe give a classifying theory for LG-bundles, where LG is the loop group of a compact Lie group G, and present a calculation for the string class of the universal LG-bundle. We show that this class is in fact an equivariant cohomology class and give an equivariant differential form representing it. We then use the caloron correspondence to define (higher) characteristic classes for LG-bundles and to prove a result for characteristic classes for based loop groups for the free loop group. These classes have a natural interpretation in equivariant cohomology and we give equivariant differential form representatives for the universal case in all odd dimensions.

The biinvariant diagonal class for Hamiltonian torus actions

Suppose that an algebraic torus G acts algebraically on a projective manifold X with generically trivial stabilizers. Then the Zariski closure of the set of pairs { ( x , y ) ∈ X × X | y = g x for some g ∈ G } defines a nonzero equivariant cohomology class [ Δ G ] ∈ H G × G ∗ ( X × X ) . We give an analogue of this construction in the case where X is a compact symplectic manifold endowed with a Hamiltonian action of a torus, whose complexification plays the role of G. We also prove that the Kirwan map sends the class [ Δ G ] to the class of the diagonal in each symplectic quotient. This allows to define a canonical right inverse of the Kirwan map.

Witten's nonabelian localization for noncompact Hamiltonian spaces

For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, show that certain integrals of equivariant cohomology classes localize as a sum of contributions from these compact critical sets, and bound the contribution from each critical set. In the case (1) that the contribution from higher critical sets grows slowly enough that the overall integral converges rapidly and (2) that 0 is a regular value of the moment map, we recover Witten's result [E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303–368; http://xxx.lanl.gov/abs/hep-th/9204083] identifying the polynomial part of these integrals as the ordinary integral of the image of the class under the Kirwan map to the symplectic quotient.

The residue formula and the Tolman-Weitsman theorem

We give a simple direct proof (for the case of Hamiltonian circle actions with isolated fixed points) that Tolman and Weitsman's description of the kernel of the Kirwan map (in other words the sum of those equivariant cohomology classes vanishing on one side of a collection of hyperplanes) is equivalent to the characterization of this kernel given by the residue theorem of Jeffrey and Kirwan.

Some invariant and equivariant cohomology classes of the space of Kähler metrics

Invariant and equivariant cohomology classes on the space of Kähler forms are defined. Relations to the obstructions to the existence of Kähler-Einstein metrics and Kähler metrics of harmonic Chern forms are discussed.

LIFTS OF SMOOTH GROUP ACTIONS TO LINE BUNDLES

Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let L → X be a complex line bundle. Using the Cartan complex for equivariant cohomology, we give a new proof of a theorem of Hattori and Yoshida which says that the action of G lifts to L if and only if the first Chern class c1(L) of L can be lifted to an integral equivariant cohomology class in H2G(X; ℤ), and that the different lifts of the action are classified by the lifts of c1(L) to H2G(X; ℤ). As a corollary of our method of proof, we prove that, if the action is Hamiltonian and ∇ is a connection on L which is unitary for some metric on L, and which has a G-invariant curvature, then there is a lift of the action to a certain power Ld (where d is independent of L) which leaves fixed the induced metric on Ld and the connection ∇[otimes ]d. This generalises to symplectic geometry a well-known result in geometric invariant theory.

Arrangement of hyperplanes. I: Rational functions and Jeffrey-Kirwan residue

Consider the space R Δ of rational functions of several variables with poles on a fixed arrangement Δ of hyperplanes. We obtain a decomposition of R Δ as a module over the ring of differential operators with constant coefficients. We generalize the notions of principal part and of residue to the space R Δ, and we describe their relations to Laplace transforms of locally polynomial functions. This explains algebraic aspects of the work by L. Jeffrey and F. Kirwan about integrals of equivariant cohomology classes on Hamiltonian manifolds. As another application, we will construct multidimensional versions of Eisenstein series in a subsequent article, and we will obtain another proof of a residue formula of A. Szenes for Witten zeta functions.

Some remarks on topological 4D-gravity

We show that the method of Wu [J. Geom. Phys. 12 (1993) 205] to study topological 4D-gravity can be understood within a standard method now designed to produce equivariant cohomology classes. Next, this general framework is applied to produce some observables of the topological 4D-gravity.

Representatives of the Thom class of a vector bundle

After a review of several methods designed to produce equivariant cohomology classes, we apply one introduced by Berline et al. (1992) to get a family of representatives of the universal Thom class of a vector bundle. Surprisingly, this family does not contain the representative given by Mathaï and Quillen (1986). However, it contains the particularly simple and symmetric representative of Harvey and Lawson (1993).

A note on the Jeffrey-Kirwan-Witten localisation formula

LET A4 be a compact symplectic manifold provided with a Hamiltonian action of a compact Lie group G with Lie algebra g. We note by (M, cr, p) such a data where 0 is the symplectic form of M and ~1: M + g* is the moment map. Let us assume that the action of G on p-‘(O) is free. We can then consider the symplectic manifold Mred = G\p‘(0). It is a symplectic manifold, called the Marsden-Weinstein reduction of M, with symplectic form &d. It is important to be able to compute the integral jm,,d v,,d of a de Rham cohomology class v,,d on Mred . By a theorem of Kirwan [8], any cohomology class v,,d of Mrcd is obtained from an equivariant cohomology class v on M by restriction and reduction. In [12], Witten proposed a formula relating the integral over Mred of v,,d and an integral over M x g of an equivariant cohomology class given in terms of v and the equivariant symplectic form. Witten’s formula has been proven by Kalkman [7], Wu [13] in the case of circle actions and by Jeffrey and Kirwan [6] in the general case. As the localisation formula [l] is an efficient tool to compute integrals over M of equivariant cohomology classes, the formula of Witten can be used to compute H*(Mred) in some cases [7,6]. Let us explain Witten’s statement. Let a be a G-equivariant differential form on M, that is, tl is an equivariant map from g to the space d(M) of differential forms on M. Assume that for X E g, a(X) = ei”~‘X)/?(X) wh ere /I is a closed G-equivariant form on M depending polynomially on the variable X E g and a&X) = p(X) + cr is the value at X E g of the equivariant symplectic form of M. Let a,,d = eiOrcd Bred be the de Rham cohomology class of Mred determined by ~1. We denote by jM a the P-function on g such that its value at X E g is the integral of a(X) over M:

Observables in the topological quantum field theory of monopoles

The topological quantum field theory of monopoles in three spacetime dimensions is formulated in a Yang-Mills invariant manner. The BRST algebra is found and its cohomology is shown to be trivial. Nontrivial observables representing nonvanishing equivariant cohomology classes are constructed.