Twisted equivariant quasi-elliptic cohomology and M-brane charge

Chern–Weil homomorphism in twisted equivariant cohomology

This paper touches upon two big themes, equivariant cohomology and current algebras. Our first main result is as follows: we define a pair of current algebra functor which assigns Lie algebras (current algebras) CA(M,A) and SA(M,A) to a manifold M and a differential graded Lie algebra (DGLA) A. The functors CA and SA are contravariant with respect to M and covariant with respect to A. If A = C𝔤, the cone of a Lie algebra 𝔤 spanned by Lie derivatives L(x) and contractions I(x)(x ∈ 𝔤) and satisfying the Cartan's magic formula [d, I(x)] = L(x), the corresponding current algebras coincide, and they are equal to CA(M,Cg)=SA(M,Cg)≅C∞(M,g), the space of smooth 𝔤-valued functions on M with the pointwise Lie bracket. Other examples include affine Lie algebras on the circle and Faddeev–Mickelsson–Shatashvili (FMS) extensions of higher-dimensional current algebras. The second set of results is related to the construction of a new DGLA D𝔤 assigned to a Lie algebra 𝔤. It is generated by L(x) and I(x) (similar to C𝔤) and by higher contractions I(x2), I(x3) etc. Similar to C𝔤, D𝔤 can be used in building differential models of equivariant cohomology. In particular, twisted equivariant cohomology (including twists by 3-cocycles and higher odd cocycles) finds a natural place in this new framework. The DGLA D𝔤 admits a family of central extensions Dp𝔤 parametrized by homogeneous invariant polynomials p∈(Sg∗)g. There is a Lie homomorphism from CA(M,Dpg) to the FMS current algebra defined by p. Let G be a Lie group integrating the Lie algebra 𝔤. The current algebras SA(M,Dg) and SA(M,Dpg) integrate to groups DG(M) and DpG(M). As a topological application, we consider principal G-bundles, and for every homogeneous polynomial p∈(Sg∗)g we pose a lifting problem (defined in terms of DG(M) and DpG(M)) with the only obstruction the Chern–Weil class cw(p). When M is a sphere, we study integration of the current algebra CA(M,Dpg). It turns out that the corresponding group is a central extension of DG(M). Under certain conditions on the polynomial p, this is a central extension by a circle.

We define exotic twisted $S^1$-equivariant cohomology for the loop space $LZ$ of a smooth manifold $Z$ via the invariant differential forms on $LZ$ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Chern character form, a loop space refinement of the twisted Chern character form, which represent classes in the completed periodic exotic twisted $S^1$-equivariant cohomology of $LZ$. We establish a localisation theorem for the completed periodic exotic twisted $S^1$-equivariant cohomology for loop spaces and apply it to establish T-duality in a background flux in type II String Theory from a loop space perspective.

Topology of generalized complex quotients

Twisted noncommutative equivariant cohomology: Weil and Cartan models

Equivariant simplicial cohomology with local coefficients and its classification

We give a geometric interpretation for superconformal quantum mechanics defined on a hyper-Kähler cone which has an equivariant symplectic resolution. BPS states are identified with certain twisted Dolbeault cohomology classes on the resolved space and their index degeneracies can also be related to the Euler characteristic computed in equivariant sheaf cohomology. In the special case of the Hilbert scheme of K points on ℂ2, we obtain a rigorous estimate for the exponential growth of the index degeneracies of BPS states as K → ∞. This growth serves as a toy model for our recently proposed duality between a seven dimensional black hole and superconformal quantum mechanics.

Using methods originating in the theory of intersection spaces, specifically a de Rham type description of the real cohomology of these spaces by a complex of global differential forms, we show that the Leray–Serre spectral sequence with real coefficients of a flat fiber bundle of smooth manifolds collapses if the fiber is Riemannian and the structure group acts isometrically. The proof is largely topological and does not need a metric on the base or total space. We use this result to show further that if the fundamental group of a smooth aspherical manifold acts isometrically on a Riemannian manifold, then the equivariant real cohomology of the Riemannian manifold can be computed as a direct sum over the cohomology of the group with coefficients in the (generally twisted) cohomology modules of the manifold. Our results have consequences for the Euler class of flat sphere bundles. Several examples are discussed in detail.

The character map in twisted equivariant nonabelian cohomology

We establish an equivariant generalization of the Novikov inequalities which allow to estimate the topology of the set of critical points of a closed basic invariant form by means of twisted equivariant cohomology of the manifold.We apply these inequalities to study cohomology of the fixed points set of a symplectic torus action.We show that in this case our inequalities are perfect, i.e. they are in fact equalities.

It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kahler manifolds is in perfect agreement with the physical notion of general (2, 2) gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on H-twisted generalized complex manifolds. If the manifold satisfies the \({\overline{\partial} \partial}\)-lemma, we establish the equivariant formality theorem. If in addition, the manifold satisfies the generalized Kahler condition, we prove the Kirwan injectivity in this setting. We then consider the Hamiltonian action of a torus on an H-twisted generalized Calabi-Yau manifold and extend to this case the Duistermaat-Heckman theorem for the push-forward measure.

We establish an equivariant generalization of the Novikov inequalities which allow to estimate the topology of the set of critical points of a closed basic invariant form by means of twisted equivariant cohomology of the manifold. We apply these inequalities to study cohomology of the fixed points set of a symplectic torus action. We show that in this case our inequalities are perfect, i.e. they are in fact equalities.