arXiv research

Generalizes Molino's theory for Riemannian foliations.

From 1980s, it is an open problem of proposing cohomologic formula for the basic index of a transversally elliptic basic differential operator on a vector bundle over a foliated manifold. In 1990s, El Kacimi-Alaoui has proprosed to use the Molino theory for study this index. Molino has proved that to every transversall…

Equivariant localization techniques give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In part…

In this paper, we prove that infinitesimal equivariant Chern-Connes characters are well-defined. We decompose an equivariant index as a pairing of infinitesimal equivariant Chern-Connes characters with the Chern character of an idempotent matrix. We compute the limit of infinitesimal equivariant Chern- Connes character…

We describe equivariant differential characters (classifying equivariant circle bundles with connections), their prequantization, and reduction.

Develops equivariant Chern characters for coherent sheaves with group actions.

We construct for an equivariant cohomology theory for proper equivariant CW-complexes an equivariant Chern character, provided that certain conditions about the coefficients are satisfied. These conditions are fulfilled if the coefficients of the equivariant cohomology theory possess a Mackey structure. Such a structur…

These notes are the first chapter of a monograph, dedicated to a detailed proof of the equivariant index theorem for transversally elliptic operators. In this preliminary chapter, we prove a certain number of natural relations in equivariant cohomology. These relations include the Thom isomorphism in equivariant cohomo…

Two constructions of Chern character for equivariant vector bundles in noncommutative geometry.

We study super parallel transport around super loops in a quotient stack, and show that this geometry constructs a global version of the equivariant Chern character.

We derive simple explicit formulas for the character of a cycle in the Connes' (b,B)-bicomplex of cyclic cohomology and give applications to the Fredholm modules and equivariant characteristic classes.

Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.

Study of non-Archimedean Hitchin map for SL2(F) characters.

The paper constructs bundles and recovers Kirillov character formula.

Compactifies character varieties for group actions.

We develop a Chern character map for twisted equivariant non-abelian cohomology.

Paper develops equivariant basic cohomology for Lie groupoids.

Study on equivariant Heegaard genus of reducible 3-manifolds with group actions.

A projection maps geodesic currents to Teichmüller space.

Study of pseudo-Anosov actions on $SU(2)$-character variety for genus 2 surfaces.

The paper introduces $μ$K-stability for polarized schemes and develops equivariant calculus.

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-Cher…

These notes form the next episode in a series of articles dedicated to a detailed proof of a cohomological index formula for transversally elliptic pseudo-differential operators and applications. The first two chapters are already available as math.DG/0702575 and arXiv:0711.3898. In this episode, we construct the relat…

The paper defines new knot genera and finds bounds for stabilization distances.

This paper is the second part of a series of papers on noncommutative geometry and conformal geometry. In this paper, we compute explicitly the Connes-Chern character of an equivariant Dirac spectral triple. The formula that we obtain for which was used in the first paper of the series. The computation has two main ste…

New invariants refine link homology, showing large genus differences.

In this paper, we give proofs of the family index formula and the equivariant family index formula by the Greiner's approach to heat kernel asymptotics. We compute equivariant family JLO characters. We also define the equivariant eta form and give a proof of its regularity.

Study on knots, genera, and algebraic concordance groups.

This paper computes the obstruction to the existence of equivariant extensions of basic gerbes over non-simply connected compact simple Lie groups. By modifying a (finite dimensional) construction of Gawȩdzki-Reis [J. Geom. Phys. 50(1):28-55, 2004], we exhibit basic equivariant bundle gerbes over non-simply connected c…

Study equivariant 4-genus of knots in symmetric 4-manifolds.

Researchers describe the dual of cohomology generators for SU(2) character varieties of surfaces.

Study knot Floer homology to create concordance invariants and slice genus bounds.

Study on cohomology of singular foliations with localization results.

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here a so-called enhanced Burnside ring $\widehat{B}(G)$ of a finite group $G$…

Study character varieties of hyperbolic 3-manifolds using bundle methods.

We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1,0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0,1) supersymmetry and, possibly, a residual flavor symmetry. …

We study some basic properties of the variety of characters in PSL(2,C) of a finitely generated group. In particular we give an interpretation of its points as characters of representations. We construct 3-manifolds whose variety of characters has arbitrarily many components that do not lift to SL(2,C). We also study t…

Elliptic Chern characters and Atiyah-Witten formula generalized to double loop spaces.

In 1993, Bismut and Zhang establish a mod Z embedding formula of Atiyah-Patodi-Singer reduced eta invariants. In this paper, we explain the hidden mod Z term as a spectral flow and extend this embedding formula to the equivariant family case. In this case, the spectral flow is generalized to the equivariant chern chara…

Proof that $SU(2)$ character variety of genus 2 surface is ${\mathbb C} P^3$.

Given a compact manifold $M$ and $g\in C^{\infty}(M,U(l;\mathbb{C}))$ we construct a Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of the equivariant (entire) cyclic Chen-normalized bar complex $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{T}))$ of $M$, and which is mapped to the odd Bismut-C…

We will give several descriptions of some basic examples of wild character varieties, including a discussion of links to work of Sibuya, Calabi and Euler, amongst others.

Develops quantum character theory for complex reductive groups.

We prove that in genus bigger than $2$, the mapping class group action on $\mathrm{Aff}(\mathbb{C})$-characters is ergodic. This implies that almost every representation $π_1 S \longrightarrow \mathrm{Aff}(\mathbb{C})$ is the holonomy of a branched affine structure on $S$, where $S$ is a closed orientable surface of ge…

Researchers create a new compactification of character varieties using geometric and algebraic methods.

The Witten class is derived from equivariant cohomology of a conformal loop space.

We construct a groupoid equivariant Kasparov class for transversely oriented foliations in all codimensions. In codimension 1 we show that the Chern character of an associated semifinite spectral triple recovers the Connes-Moscovici cyclic cocycle for the Godbillon-Vey secondary characteristic class.

We develop the theory of simplicial extensions for bundle gerbes and their characteristic classes with a view towards studying descent problems and equivariance for bundle gerbes. Equivariant bundle gerbes are important in the study of orbifold sigma models. We consider in detail two examples: the basic bundle gerbe on…