Related papers: The Equivariant Noncommutative Atiyah-Patodi-Singe…

We use the G-signature theorem to define an invariant of strongly invertible knots analogous to the knot signature.

We show that the Wilson Dirac operator in lattice gauge theory can be identified as a mathematical object in $K$-theory and that its associated spectral flow is equal to the index. In comparison to the standard lattice Dirac operator index,…

Recently, the Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. Although it is widely applied to physics, the mathematical set-up in the original APS…

We give a proof of the cobordism invariance of the index of elliptic pseudodifferential operators on sigma-compact manifolds, where, in the non-compact case, the operators are assumed to be multiplication outside a compact set. We show…

It is known that the Atiyah-Patodi-Singer index can be reformulated as the eta invariant of the Dirac operators with a domain wall mass which plays a key role in the anomaly inflow of the topological insulator with boundary. In this paper,…

We describe how Lie groupoids are used in singular analysis, index theory and non-commutative geometry and give a brief overview of the theory. We also expose groupoid proofs of the Atiyah-Singer index theorem and discuss the Baum-Connes…

We study the index problem for the d-bar operators subject to Atiyah- Patodi-Singer boundary conditions on noncommutative disk and annulus.

In this paper, an equality between the Hochs-Mathai type index and the Atiyah-Patodi-Singer type index is established when the manifold and the group action are both non-compact, which generalizes a result of Ma and Zhang for compact group…

This expository paper is an introductory text on topological K-theory and the Atiyah-Singer index theorem, suitable for graduate students or advanced undegraduates already possessing a background in algebraic topology. The bulk of the…

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…

The present paper gives calculations in detail to prove several special cases of Atiyah-Singer theorem through supersymmetric $\sigma$-models. Some technical tricks are employed to calculate the determinants of fluctuation operators of the…

Let $X_0$ be a compact Riemannian manifold with boundary endowed with a oriented, measured even dimensional foliation with purely transverse boundary. Let $X$ be the manifold with cylinder attached and extended foliation. We prove that the…

The Atiyah-Patodi-Singer (APS) index theorem relates the index of a Dirac operator to an integral of the Pontryagin density in the bulk (which is equal to global chiral anomaly) and an $\eta$ invariant on the boundary (which defines the…

Let $\Gamma$ 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 $\Gamma$-coverings, thus providing an explicit formula for the higher index…

In [20] Esnault asked whether on a general quotient surface singularity the rank and the first Chern class distinguish isomorphism classes of indecomposable reflexive modules. Wunram gave a contraexample in [46] showing two different full…

It is shown that a novel anomaly associated with transverse Ward-Takahashi identity exists for pseudo-tensor current in QED, and the anomaly gives rise to a topological index of Dirac operator in terms of Atiyah-Singer index theorem.

Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in…

Motivated by the work of Vishik on the analytic torsion we introduce a new class of generalized Atiyah-Patodi-Singer boundary value problems. We are able to derive a full heat expansion for this class of operators generalizing earlier work…

In this paper, we formulate and prove the so-called $p$-adic non-commutative analytic subgroup theorem. This result is seen as the $p$-adic analogue of a recent theorem given by Yafaev.

Given two metrics of positive scalar curvature metrics on a closed spin manifold, there is a secondary index invariant in real $K$-theory. There exist two definitions of this invariant, one of homotopical flavour, the other one defined by a…