arXiv research

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

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

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

Abstract: Review of Index theorem and its applications.

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

Note proves index theorem for non-elliptic Heisenberg operators.

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

This paper proves the Atiyah-Singer index theorem for Dirac operators.

Proves an index theorem for foliations using spectral triples.

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 …

Absolute index theorem for warped product manifolds.

The index theorem connects anomalies on a domain wall to global integrals.

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…

In this talk, we review the heat kernel approach to the Atiyah-Singer index theorem for Dirac operators on closed manifolds, as well as the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. We also discuss the odd dimensional counterparts of the above results. In particular, we describe…

This paper proves the Atiyah-Singer index theorem for elliptic operators.

New theorem for manifolds with boundaries calculates operator index.

Proves an index theorem for proper group actions on manifolds.

Paper proves index theorem for self-adjoint elliptic boundary problems.

We extend the Atiyah, Patodi, and Singer index theorem for first order differential operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. This theorem provides a natural complement to Taubes' Fredholm theory for general end-periodic operators. Our index theorem is expressed in t…

Study of spinors and tangent groupoid for index theorem.

Formula for spectral flow connects manifold properties to index theorem.

Surveying the signature theorem, researchers cancel anomalies on elliptic surfaces.

We give a short proof of the Morse index theorem for geodesics in semi-Riemannian manifolds by using K-theory. This makes the Morse index theorem reminiscent of the Atiyah-Singer index theorem for families of selfadjoint elliptic operators.

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

New index formula connects numerical and $K$-theoretic indices.

In his book (II.5), Connes gives a proof of the Atiyah-Singer index theorem for closed manifolds by using deformation groupoids and appropiate actions of these on R^N. Following these ideas, we prove an index theorem for manifolds with boundary.

Physicists explain a mathematical theorem about topological insulators.

The Atiyah-Singer index theorem is a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their…

We present an index theorem for certain hypoelliptic differential operators on foliated manifolds. Our proof is a development of Alain Connes tangent groupoid proof of the Atiyah-Singer index theorem. The paper is largely self-contained.

Massive fermions help understand index theorems without chiral symmetry.

Proposes a lattice formulation of APS index using η invariant.

We prove an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah-Singer index theorem for selfadjoint elliptic operators. For the special case of one-parameter families, we compare our theorem with a classical result of Salamon and Zehnder. Finally, we use the index theo…

We give a cohomological formula for the index of a fully elliptic pseudodifferential operator on a manifold with boundary. As in the classic case of Atiyah-Singer, we use an embedding into an euclidean space to express the index as the integral of a cohomology class depending in this case on a noncommutative symbol, th…

In the first part of this paper, given a smooth family of Dirac-type operators on an odd-dimensional closed manifold, we construct an abelian gerbe-with-connection whose curvature is the three-form component of the Atiyah-Singer families index theorem. In the second part of the paper, given a smooth family of Dirac-typ…

Let $Γ$ be a finitely generated discrete group satisfying the rapid decay condition. We give a new proof of the higher Atiyah-Patodi-Singer theorem on a Galois $Γ$-coverings, thus providing an explicit formula for the higher index associated to a group cocycle $c\in Z^k (Γ;\mathbb{C})$ which is of polynomial growth wit…

Constructs small bundle gerbes and proves index theorems for manifolds.

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

We prove an analogue for even dimensional manifolds of the Atiyah-Patodi-Singer twisted index theorem for trivialized flat bundles. We show that the eta invariant appearing in this result coincides with the eta invariant by Dai and Zhang up to an integer. We also obtain the odd dimensional counterpart for manifolds wit…

Introduces generalized products for pseudodifferential operators on manifolds with corners.

Study on eta and rho invariants on incomplete edge spaces.

Constructs a rigorous path integral for supersymmetric spin manifolds.

Let $D$ be a (generalized) Dirac operator on a non-compact complete Riemannian manifold $M$ acted on by a compact Lie group $G$. Let $v:M --> Lie(G)$ be an equivariant map, such that the corresponding vector field on $M$ does not vanish outside of a compact subset. These data define an element of $K$-theory of the tran…

Study shows infinitely many nonnegatively curved metric spaces on exotic 7-manifolds.

Mathematician-friendly formulation of Atiyah-Patodi-Singer index.

We construct eta- and rho-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah-Patodi-Singer index theorem in this setting, as well as its higher generalization. Applications …

We generalise the Atiyah-Segal-Singer fixed point theorem to noncompact manifolds. Using $KK$-theory, we extend the equivariant index to the noncompact setting, and obtain a fixed point formula for it. The fixed point formula is the explicit cohomological expression from Atiyah-Segal-Singer's result. In the noncompact …

This article surveys the relations among local and nonlocal invariants in Atiyah-Singer index theory. We discuss the local invariants that arise from the heat equation approach to the index theorem for geometric operators, as well as the nonlocal invariants (the eta invariant, the determinant of the Laplacian/analytic …

In this paper, we define the eta cochain form and prove its regularity when the kernel of a family of Dirac operators is a vector bundle. We decompose the eta form as a pairing of the eta cochain form with the Chern character of an idempotent matrix and we also decompose the Chern character of the index bundle for a fi…