Equivariant evaluation on free loop spaces

The main result of the paper is that the equivariant intersection cohomology of the semi-stable points on a complex projective variety , for the action of a complex reductive group, may be determined from the equivariant intersection cohomology of the semi-stable points for the action of a maximal torus. It extends the work of Brion who considered the smooth case using equivariant cohomology. Equivariant intersection cohomology is a theory due to Brylinski and the second author. As an application, a surprising relation between the intersection cohomology of Chow hypersurfaces is established in the last section of the paper.

We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite-dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for any given complex elliptic curve $\mathcal {C}$, the fiber of our construction over $\mathcal {C}$ is isomorphic to Grojnowski's equivariant elliptic cohomology theory associated to $\mathcal {C}$.

(1.1) This paper concerns three aspects of the action of a compact group K on a space X . The ®rst is concrete and the others are rather abstract. (1) Equivariantly formal spaces. These have the property that their cohomology may be computed from the structure of the zero and one dimensional orbits of the action of a maximal torus in K. (2) Koszul duality. This enables one to translate facts about equivariant cohomology into facts about its ordinary cohomology, and back. (3) Equivariant derived category. Many of the results in this paper apply not only to equivariant cohomology, but also to equivariant intersection cohomology. The equivariant derived category provides a framework in both of these may be considered simultaneously, as examples of ``equivariant sheaves''. We treat singular spaces on an equal footing with nonsingular ones. Along the way, we give a description of equivariant homology and equivariant intersection homology in terms of equivariant geometric cycles. Most of the themes in this paper have been considered by other authors in some context. In Sect. 1.7 we sketch the precursors that we know about. For most of the constructions in this paper, we consider an action of a compact connected Lie group K on a space X , however for the purposes of the introduction we will take K ˆ S1† to be a torus. Invent. math. 131, 25±83 (1998)

A Butson Hadamard matrix of order n over the kth root of unity is a square matrix H which entries are some complex kth root of unity such that HH∗ = nIn, where H∗ is the complex conjugate of H. A set of Butson Hadamard matrices of order n over the kth root of unity is denoted by BH(n,k). It is well-known that a Butson Hadamard matrices is a generalization of a Hadamard matrix. In this paper, we give some properties of Butson Hadamard matrices of order 4 which implies to the upper and the lower bounds of the number of its equivalence classes. We also showed that the entries of Butson Hadamard matrices of order 4 is 2kth root of unity for some integer k. Furthermore, we describe the equivalence classes of Butson Hadamard matrices of order 4 by constructing the representative of the class.

On the cohomology of free and twisted loop spaces

Homomorphisms of matrix algebras and constructions of Butson–Hadamard​ matrices

The S-matrix on the world-sheet theory of the string in AdS has previously been shown to admit a deformation where the symmetry algebra is replaced by the associated quantum group. The case where q is real has been identified as a particular deformation of the Green–Schwarz sigma model. An interpretation of the case with q a root of unity has, until now, been lacking. We show that the Green–Schwarz sigma model admits a discrete deformation which can be viewed as a rather simple deformation of the gauged WZW model, where . The deformation parameter q is then a kth root of unity where k is the level. The deformed theory has the same equations-of-motion as the Green–Schwarz sigma model but has a different symplectic structure. We show that the resulting theory is integrable and has just the right amount of kappa-symmetries that appear as a remnant of the fermionic part of the original gauge symmetry. This points to the existence of a fully consistent deformed string background.

In this PhD thesis, we construct an explicit algebraic model over Z of the cochains of the free loop space of a 1-connected space X. We start from an enriched Adams-Hilton model of X, which can be obtained relatively easily when X is the realisation of a simplicial set. Note it is not supposed that the Steenrod algebra acts trivially on X. The second part is dedicated to the construction of a model of the cochains of mapping spaces XY. where X is r-connected and Y is a CW-complex that has dimension less or equal to r. The space X must possess commutative models for the cochains of each Ωk X for k ≤ r. We first construct an algebraic model for the cochains of XSn ∀n ≤ r, then we then glue all of them to obtain a model of the cochains of XY. We give examples for each of these situations. The techniques used here rely heavily on the concept of a twisted bimodule. A description of this can be found in [DH99b].

Let us consider the free loop space of a manifold. It is very well known that the equivariant cohomology of the free loop spce is related to the index theorem for a finite dimensional Dirac operator. The reader can see [Bis85], [Bis86], [GJP90], [JP90] for instance.

Let X be a 1-connected compact space such that the algebra H ∗ (X; F2) is generated by one single element. We compute the cohomology of the free loop space H ∗ (�X; F2) including the Steenrod algebra action. When X is a projective space CP n , HP n , the Cayley projective plane CaP 2 or a sphere S m we obtain a splitting result for integral and mod two cohomology of the suspension spectrum � ∞ (�X)+. The splitting is in terms of � ∞ X+ and the Thom spaces Th(q�), q ≥ 0 of the q-fold Whitney sums of the tangent bundle � over X.

International audience

We show that the integer cohomology rings of the moduli spaces of stable rational marked curves are Koszul. This answers an open question of Manin. Using the machinery of Koszul spaces developed by Berglund, we compute the rational homotopy Lie algebras of those spaces, and obtain some estimates for Betti numbers of their free loop spaces in case of torsion coefficients. We also prove and conjecture some generalisations of our main result.

In this paper, we consider a Finsler sphere $(M,F)=(S^n,F)$ with the dimension $n>1$ and the flag curvature $K\equiv 1$. The action of the connected isometry group $G=I_o(M,F)$ on $M$, together with the action of $T=S^1$ shifting the parameter $t\in \mathbb{R}/\mathbb{Z}$ of the closed curve $c(t)$, define an action of $\hat{G}=G\times T$ on the free loop space $\lambda M$ of $M$. In particular, for each closed geodesic, we have a $\hat{G}$-orbit of closed geodesics. We assume the Finsler sphere $(M,F)$ described above has only finite orbits of prime closed geodesics. Our main theorem claims, if the subgroup $H$ of all isometries preserving each close geodesic has a dimension $m$, then there exists $m$ geometrically distinct orbits $\mathcal{B}_i$ of prime closed geodesics, such that for each $i$, the union ${B}_i$ of geodesics in $\mathcal{B}_i$ is a totally geodesic sub-manifold in $(M,F)$ with a non-trivial $H_o$-action. This theorem generalizes and slightly refines the one in a previous work, which only discussed the case of finite prime closed geodesics. At the end, we show that, assuming certain generic conditions, the Katok metrics, i.e. the Randers metrics on spheres with $K\equiv 1$, provide examples with the sharp estimate for our main theorem.

This paper investigates critical points of the energy functional in sub-Finsler manifolds. Critical points of the energy functional correspond to sub-Finsler geodesics, i.e., horizontal curves that are extremal for the energy, emphasizing their importance in sub-Finsler geometry. Assuming that the sub-Finsler metric is bumpy (in the Morse–Bott sense), the energy functional on the free loop space has the Morse--Bott property, and we establish a direct link between critical points and closed geodesics. We shed light on the existence of at least one closed geodesic on compact sub-Finsler manifolds.

In this chapter we review the derivation of the DuistermaatHeckman integration formula and its path integral generalizations, and explain the underlying formalism of equivariant cohomology. We evaluate the quantum mechanical partition function for a general integrable model by localizing onto an ordinary integral of an equivariant characteristic class. We also describe the Mathai-Quillen formalism and its equivariant loop space extensions. We show how certain standard relations in classical Morse theory can be derived from this formalism, and generalize these relations to the infinite-dimensional and equivariant context. We also explain how Poincare supersymmetric quantum field theories can be formulated using equivariant cohomology in the loop space.