arXiv research

A Chern-Weil construction for extensions of Lie-Rinehart algebras is introduced. This generalizes the classical Chern-Weil construction in differential geometry and yields characteristic classes for arbitrary extensions of Lie-Rinehart algebras. Some examples arising from spaces with singularities and from foliations a…

Constructs Chern-Weil classes for Cartan geometries.

We construct Chern-Weil classes on infinite dimensional vector bundles with structure group contained in the algebra $\cl[\leq 0](M, E)$ of non-positive order classical pseudo-differential operators acting on a finite rank vector bundle $E$ over a closed manifold $M$. Mimicking the finite dimensional Chern-Weil constru…

Constructs a universal Chern-Weil map for infinite dimensional Lie groups.

Develops Chern-Weil theory for singular foliations.

New classes for Lie algebra extensions discovered.

Develops connections and Chern-Weil theory for Lie groupoids.

Categorifies Chern-Weil theory for infinite local systems.

We study the existence of $S^1$-equivariant characteristic classes on certain natural infinite rank bundles over the loop space $LM$ of a manifold $M$. We discuss the different $S^1$-equivariant cohomology theories in the literature and clarify their relationships. We attempt to use $S^1$-equivariant Chern-Weil techniq…

A equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring. By applying the Grothedieck construction one obtains the ring K, elem…

The paper extends Chern-Weil-Lecomte map to $L_{\infty}$-algebras.

This paper extends Bott-Chern cohomology to coherent sheaves using superconnections.

We provide local expressions for Chern-Weil type forms built from superconnections associated with families of Dirac operators previously investigated in work by S. Scott and later work by S. Scott and the second author. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as …

We construct connections and characteristic forms for principal bundles over groupoids and stacks in the differentiable, holomorphic and algebraic category using Atiyah sequences associated to transversal tangential distributions.

Proves Chern-Weil theory for line bundles with a specific metric.

We describe the Cartan and Weil models of twisted equivariant cohomology together with the Cartan homomorphism among the two, and we extend the Chern-Weil homomorphism to the twisted equivariant cohomology. We clarify that in order to have a cohomology theory, the coefficients of the twisted equivariant cohomology must…

We show that Chern-Weil theory for tensor bundles over manifolds is a consequence of the existence of natural closed differential forms on total spaces of torsion free connections on frame bundles.

Paper develops techniques for singular metrics on vector bundles.

We discuss possible extensions of the classical Chern-Weil formalism to an infinite dimensional setup. This is based on joint work with Steven Rosenberg, joint work with Simon Scott and joint work with Jouko Mickelsson.

We explain how the distributional index of a transversally elliptic operator on a principal G-manifold P that is obtained by lifting a Dirac operator on P/G can serve as a link between the Duflo isomorphism and Chern-Weil forms.

Local index theorem for chiral geometric operators proved using heat kernel.

In this paper we give a Chern-Weil-type construction of characteristic classes of fiber bundles, based on homotopy theory of C-infinity algebras. Our idea is to replace a family of closed manifolds to a family of C-infinity morphisms with family of metrics.

In this paper we study the groups of contactomorphisms of a closed contact manifold from a topological viewpoint. First we construct examples of contact forms on spheres whose Reeb flow has a dense orbit. Then we show that the unitary group U(n+1) is homotopically essential in the group of contactomorphisms of the stan…

The article constructs invariants for flat connections on bundles.

The paper extends Chern-Weil theory to simplicial principal bundles.

Chern-Weil and Chern-Simons theory extend to certain infinite-rank bundles that appear in mathematical physics. We discuss what is known of the invariant theory of the corresponding infinite-dimensional Lie groups. We use these techniques to detect cohomology classes for spaces of maps between manifolds and for diffeom…

Study the spaces of flat connections for classical Lie groups using Chern-Weil theory.

We present a geometric approach, in the spirit of the Chern-Weil theory, for constructing cocycles representing the classes of the Hopf cyclic cohomology of the Hopf algebra H(n) relative to GL(n, R). This provides an explicit description of the universal Hopf cyclic Chern classes, which complements our earlier geometr…

The theory of principal $G$-bundles over a Lie groupoid is an important one, unifying the various types of principal $G$-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal $G$-bundles. In this paper, we study the differential geometry of these objects, including connections …

We show that every bad orbifold vector bundle can be realized as the restriction of a good orbifold vector bundle to a suborbifold of the base space. We give an explicit construction of this result in which the Chen-Ruan orbifold cohomology of the two base spaces are isomorphic (as additive groups). This construction i…

Bounding characteristic numbers of Riemannian manifolds via volume.

We describe two constructions giving rise to curved $A_{\infty}$-algebras. The first consists of deforming $A_{\infty}$-algebras, while the second involves transferring curved dg structures that are deformations of (ordinary) dg structures along chain contractions. As an application of the second construction, given a …

Combines techniques to remove tameness condition in Morse-Smale flows.

In this paper, we proved the Gauss-Bonnet-Chern theorem on moduli space of polarized Kahler manifolds. Using our results, we proved the rationality of the Chern-Weil forms (with respect to the Weil-Petersson metric) on CY moduli. As an application in physics, by the Ashok-Douglas theory, counting the number of flux com…

In this paper we look at two naturally occurring situations where the following question arises. When one can find a metric so that a Chern-Weil form can be represented by a given form ? The first setting is semi-stable Hartshorne-ample vector bundles on complex surfaces where we provide evidence for a conjecture of Gr…

We propose the notion of a supercategory as an alternative approach to supermathematics. We show that this setting is rich to carry out many of the basic constructions of supermathematics. We also prove generalizations of a number of results in equivariant cohomology, including the Chern-Weil theorem for an arbitrary r…

We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive struct…

Study primitive cohomology in symplectic manifolds.

Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. Kotov and Strobl have observed that this naturally generalizes from Lie algebras to dg-manifolds and dg-bundles …

A strong from of invariance under a group G is manifested in a family over the classifying space BG. We advocate a differential-geometric avatar of BG when G is a Lie group. Applied to G-equivariant connections on smooth principal or vector bundles, the equivariance-->families principle converts the G-equivariant exten…

We prove that Chern-Weil forms are the only natural differential forms associated to a connection on a principal G-bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof. Other arguments come from classical invariant theory. We identify the Weil algebr…

Develops a SageMath framework for computing characteristic classes.

We generalize some of the results of Harvey, Lawson and Latschev about transgression formulas. The focus here is on flowing forms via vertical vector fields, especially Morse-Bott-Smale vector fields. We prove a very general transgression formula including also a version covering non-compact situations. Among applicati…

After a review of the general properties of holomorphic spheres in complex surfaces we describe the local geometry in the vicinity of a CP^1 embedded with a negative normal bundle. As a by-product, we build (asymptotically locally hyperbolic) Kahler-Einstein metrics on the total spaces of the line bundles O(-m), m >= 3…

For any regular Courant algebroid, we construct a characteristic class a la Chern-Weil. This intrinsic invariant of the Courant algebroid is a degree-3 class in its naive cohomology. When the Courant algebroid is exact, it reduces to the Severa class (in H^3_{DR}(M)). On the other hand, when the Courant algebroid is a …

This paper contains the constructions of a real manifold version of relative K-theory, and of an extension of Karoubi's multiplicative K-theory suggested by U. Bunke (which I call ``free multiplicative K-theory'' in the sequel). Chern-Simons-Nadel type classes on relative K-theory are constructed, while it is proved th…

This note announces a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particular, a Chern-Weil-Simons theory for smooth bundle maps $α: E \rightarrow F$ which, for smooth connections on $E$ and $F$, establishes formulas of the type $$ φ\ = \ \text{\rm Res}_φΣ…

The study proves that certain 4-manifolds are Kähler if curvature is constant.