Related papers: Etale Groupoids, eta invariants and index theory

Many index theorems (both classical and in noncommutative geometry) can be interpreted in terms of a Lie groupoid acting properly on a manifold and leaving an elliptic family of pseudodifferential operators invariant. Alain Connes in his…

For a smooth family F of admissible elliptic pseudodifferential operators with differential form coefficients associated to a geometric fibration of manifolds M--> B we show that there is a natural zeta-form z(F,s) and zeta-determinant-…

We establish existence of the eta-invariant as well as of the Atiyah-Patodi-Singer and the Cheeger-Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature, the…

Using recently introduced Debord-Skandalis Blup's groupoids we study index theory for a compact foliated manifold with boundary inducing a foliation in its boundary. For this we consider first a blup groupoid whose Lie algebroid has…

We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of etale groupoid is subsumed in a natural way by that of quantale. In particular, to each etale groupoid, either localic or…

In their construction of the topological index for flat vector bundles, Atiyah, Patodi and Singer associate to each flat vector bundle a particular $\mathbb{C/Z}$-$K$-theory class. This assignment determines a map, up to weak homotopy, from…

In this paper we establish a formula, expressing the generalized Atiyah-Patodi-Singer index in terms of eta invariants of domain-wall massive Dirac operators, without assuming that the Dirac operator on the boundary is invertible. Compared…

Let $\GR \to B$ be a bundle of compact Lie groups acting on a fiber bundle $Y \to B$. In this paper we introduce and study gauge-equivariant $K$-theory groups $K_\GR^i(Y)$. These groups satisfy the usual properties of the equivariant…

We prove very general index formulae for integral Galois modules, specifically for units in rings of integers of number fields, for higher K-groups of rings of integers, and for Mordell-Weil groups of elliptic curves over number fields.…

Atiyah-Singer index theorem on a lattice without boundary is well understood owing to the seminal work by Hasenfratz et al. But its extension to the system with boundary (the so-called Atiyah- Patodi-Singer index theorem), which plays a…

This paper collects the notes of a serie of lectures given by the two authors during the summer school "Geometric and topological methods for Quantum Field Theory" at Villa de Leyva, Colombia, summer 2007. These lecture notes are mainly…

We consider the index of a Dirac operator on a compact even dimensional manifold with a domain wall. The latter is defined as a co-dimension one submanifold where the connection jumps. We formulate and prove an analog of the…

We study eta-invariants on odd dimensional manifolds with boundary. The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary. We…

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…

We prove an index theorem for families of pseudodifferential operators generalizing those studied by C. Callias, N. Anghel and others. Specifically, we consider operators on a manifold with boundary equipped with an asymptotically conic…

We show how the Atiyah-Singer family index theorem for both, usual and self-adjoint elliptic operators fits naturally into the framework of the Madsen-Tillmann-Weiss spectra. Our main theorem concerns bundles of odd-dimensional manifolds.…

Let $F$ be a $p$-adic field, $E$ be a quadratic extension of $F$, $D$ be an $F$-central division algebra of odd index and let $\theta$ be the Galois involution attached to $E/F$. Set $H=GL(m,D)$, $G=GL(m,D\otimes_F E)$, and let $P=MU$ be a…

We present a higher index theorem for a certain class of etale one-dimensional complex-analytic groupoids. The novelty is the use of the local anomaly formula established in a previous paper, which represents the bivariant Chern character…

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…

These are notes from a talk at the 2010 Talbot Workshop on Twisted K-theory and Loop Groups. This particular talk is an overview of index theory from the point of view of topological K-theory. Assuming little background in analysis, but…