arXiv research

Atiyah and Todd classes of Lie algebroids respect their Atiyah sequence.

Established a version of the Atiyah-Floer conjecture for SO(3)-bundles.

Atiyah reviewed holomorphic vector bundles and gauge theories.

Paper computes Atiyah class for DG manifolds of amplitude +1.

Study connections on Lie groupoids and stacks using Atiyah sequences.

Paper develops a unified framework for Lie algebroid connections on various bundles.

The subject of this paper is strongly homotopy (SH) Lie algebras, also known as $L_\infty$-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an (SH) Lie algebra $A$ when it is extended to $L$. In fact, given such an SH Lie pair $(L, A)$, and any $A$-module $E$, there ass…

Atiyah-Singer theorem links math fields, predicts topological insights.

Derives Atiyah sequence for noncommutative bundles.

Established equivalence of Atiyah classes for generalized holomorphic vector bundles.

Lattice formulation captures Atiyah-Patodi-Singer index.

Constructs a triple on an Atiyah algebroid with connection.

We provide new conditions for the Strong Atiyah conjecture to lift to finite group extensions. In particular, we show cocompact special groups satisfy these conditions, so the Strong Atiyah conjecture holds for virtually cocompact special groups.

We prove Atiyah's conjecture for two special types of configurations of N points in the three-dimensional Euclidean space. For one of these types, it is shown that the stronger conjecture of Atiyah and Sutcliffe is valid.

Proves a lattice version of the Atiyah-Singer index theorem.

We state and prove a condition under which the strong Atiyah Conjecture carries over to subgroups. Moreover, we show that if a group satisfies the (strong) Atiyah Conjecture then any quotient with finite kernel does.

We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah-Patodi-Singer index of our previous paper. Our viewpoint sheds some new light on the interplay among the Atiyah-Patodi-Singer boundary condition, domain-wall fermions, and edge modes.

For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\mathbb{Z}$-graded manifold $\mathcal M=L[1]\oplus L/A$ such that the inclusion $ι: A[1] \to \mathcal M$ and the projection $p:\mathcal M\to L[1]$ are morphisms of dg-manifolds. The vertical tangent bundle $T^p\mathcal M$ then inherit…

Paper establishes a new formula for Atiyah-Patodi-Singer index using eta invariants.

In [Wu], the noncommutative Atiyah-Patodi-Singer index theorem was proved. In this paper, we extend this theorem to the equivariant case.

Gravitational instantons are constructed as superpositions of Atiyah-Hitchin and Taub-NUT geometries.

Researchers construct an index map for contact manifolds using K-theory.

We present the details of our embedding proof of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary.

Constructs a model for differential KO-theory using Clifford modules.

We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field $Q$ admits a structure of L-infinity algebra with the Lie derivative $L_Q$ as unary …

The index theorem, discovered by Atiyah and Singer in 1963, is one of most important results in the twentieth century mathematics. It found numerous applications in analysis, geometry and physics. Since it was discovered numerous attempts to generalize it were made, see for example [5, 3, 4, 16, 12] to mention a few; s…

Cohomological and homological spectral sequences are shown to be isomorphic.

We give a new short proof of the index formula of Atiyah and Singer based on combining Getzler's rescaling with Greiner's approach of the heat kernel asymptotics. As application we can easily compute the Connes-Moscovici cyclic cocycle of even and odd Dirac spectral triples, and then recover the Atiyah-Singer index for…

This short note gives a geometric interpretation of the Atiyah class of a Lie pair. It proves that it vanishes if the subalgebroid is the kernel of a fibration of Lie algebroids. In other words, the Atiyah class of a Lie pair vanishes if the subalgebroid is the fiber of an ideal system in the Lie algebroid. In order to…

Develops theory of d-holomorphic connections on Klein surfaces.

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.

In this paper, we study the Atiyah class and Todd class of the DG manifold $(F[1],d_F)$ corresponding to an integrable distribution $F \subset T_{\mathbb{K}} M = TM \otimes_{\mathbb{R}} \mathbb{K}$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. We show that these two classes are canonically identical to those of the…

We prove an Atiyah-Patodi-Singer index theorem for Dirac operators twisted by C*-vector bundles. We use it to derive a general product formula for eta-forms and to define and study new rho-invariants generalizing Lott's higher rho-form. The higher Atiyah-Patodi-Singer index theorem of Leichtnam-Piazza can be recovered …

We prove a generalized version of the Strong Atiyah Conjecture for the infinite dihedral group W, replacing the group von Neumann algebra NW with the Hecke-von Neumann algebra N_qW.

The paper studies Atiyah classes for Lie algebroid representations and homotopies.

Atiyah classes of DG manifolds of positive amplitude are invariant under weak equivalences.

We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.

Paper introduces Atiyah sequence for Lie groupoids and studies gauge transformations.

Explains a 1978 construction for Yang-Mills instantons.

In this note we explain how the computation of the spectrum of the lamplighter group from \cite{Grigorchuk-Zuk(2000)} yields a counterexample to a strong version of the Atiyah conjectures about the range of $L^2$-Betti numbers of closed manifolds.

Study circle actions on 4-manifolds, deriving formulas and graphs.

We study the Cauchy data spaces of the strongly Callias-type operators using maximal domain on manifolds with non-compact boundary, with the aim of understanding the Atiyah-Patodi-Singer index and elliptic boundary value problems.

Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic manifolds. Soon after that, Atiyah conjectured that the two theories should be related to each other and Lagrangian Floer homology…

Paper unifies three invariants for flat bundles over surfaces with boundary.

We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of $GL(n,C)$ is isomorphic to the deformation of the $D_2$-singularity if $n=2$, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if $n=3$, and to the Atiyah-Hitchin manifold itself if $n=4$. For higher $n$, such…

Under two boundary conditions, the generalized Atiyah-Patodi-Singer boundary condition and the modified generalized -Atiyah-Patodi-Singer boundary condition, we get the lower bounds for the eigenvalues of the fundamental Dirac operator on compact spin manifolds with nonempty boundary.

New construction of Atiyah and Todd classes for Lie pair pullback dg Lie algebroids.

We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic, then the corresponding base manifolds are necessarily diffeomorphic. Further, w…