Adam Rennie

Smoothness and locality for nonunital spectral triples

To deal with technical issues in noncommutative geometry for nonunital algebras, we introduce a useful class of algebras and their modules. These algebras and modules allow us to extend all of the smoothness results for spectral triples to the nonunital case. In addition, we show that smooth spectral triples are closed under the C ∞ functional calculus of self-adjoint elements. In the final section we show that our algebras allow the formulation of Poincare Duality and that the algebras of smooth spectral triples are H-unital.

Index theory for locally compact noncommutative geometries

Introduction Pseudodifferential calculus and summability Index pairings for semifinite spectral triples The local index formula for semifinite spectral triples Applications to index theorems on open manifolds Noncommutative examples Appendix A. Estimates and technical lemmas Bibliography Index

Pseudo-Riemannian spectral triples and the harmonic oscillator

Abstract We define pseudo-Riemannian spectral triples, an analytic context broad enough to encompass a spectral description of a wide class of pseudo-Riemannian manifolds, as well as their noncommutative generalisations. Our main theorem shows that to each pseudo-Riemannian spectral triple we can associate a genuine spectral triple, and so a K -homology class. With some additional assumptions we can then apply the local index theorem. We give a range of examples and some applications. The example of the harmonic oscillator in particular shows that our main theorem applies to much more than just classical pseudo-Riemannian manifolds.

Riemannian Manifolds in Noncommutative Geometry

Abstract We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin c manifolds; and conversely, in the presence of a spin c structure. We also show how to obtain an analogue of Kasparov’s fundamental class for a Riemannian manifold, and the associated notion of Poincare duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.

A non-commutative framework for topological insulators

We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of non-commutative index theory of operator algebras. In particular, we formulate the index problems using Kasparov theory, both complex and real. We show that the periodic table of topological insulators and superconductors can be realized as a real or complex index pairing of a Kasparov module capturing internal symmetries of the Hamiltonian with a spectral triple encoding the geometry of the sample’s (possibly non-commutative) Brillouin zone.

Nonunital spectral triples and metric completeness in unbounded KK-theory

Abstract We consider the general properties of bounded approximate units in non-self-adjoint operator algebras. Such algebras arise naturally from the differential structure of spectral triples and unbounded Kasparov modules. Our results allow us to present a unified approach to characterising completeness of spectral metric spaces, existence of connections on modules, self-adjointness and regularity of induced operators on tensor product C ⁎ -modules and the lifting of Kasparov products to the unbounded category. In particular, we prove novel existence results for quasicentral approximate units in non-self-adjoint operator algebras, allowing us to strengthen Kasparovs technical theorem and extend it to this realm. Finally, we show that given any two composable KK-classes, we can find unbounded representatives whose product can be constructed to yield an unbounded representative of the Kasparov product.

Spectral flow invariants and twisted cyclic theory for the Haar state on SUq(2)

Abstract In [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605 ], we presented a K -theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely S U q ( 2 ) and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the Haar state. In contrast to the Cuntz algebras studied in [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605 ], the computations are considerably more complex and interesting, because there are non-trivial ‘eta’ contributions to this index.

The Bulk-Edge Correspondence for the Quantum Hall Effect in Kasparov Theory

We prove the bulk-edge correspondence in K-theory for the quantum Hall effect by constructing an unbounded Kasparov module from a short exact sequence that links the bulk and boundary algebras. This approach allows us to represent bulk topological invariants explicitly as a Kasparov product of boundary invariants with the extension class linking the algebras. This paper focuses on the example of the discrete integer quantum Hall effect, though our general method potentially has much wider applications.

The chern character of semifinite spectral triples

In previous work we generalised both the odd and even local index formula of Connes and Moscovici to the case of spectral triples for a -subalgebra A of a general semifinite von Neumann algebra. Our proofs are novel even in the setting of the original theorem and rely on the introduction of a function valued cocycle (called the resolvent cocycle) which is ‘almost’ a .b; B/-cocycle in the cyclic cohomology of A. In this paper we show that this resolvent cocycle ‘almost’ represents the Chern character and assuming analytic continuation properties for zeta functions we show that the associated residue cocycle, which appears in our statement of the local index theorem does represent the Chern character. Mathematics Subject Classification (2000). Primary: 19K56, 46L80; Secondary: 58B30, 46L87.

The K -Theoretic Bulk–Edge Correspondence for Topological Insulators

We study the application of Kasparov theory to topological insulator systems and the bulk–edge correspondence. We consider observable algebras as modelled by crossed products, where bulk and edge systems may be linked by a short exact sequence. We construct unbounded Kasparov modules encoding the dynamics of the crossed product. We then link bulk and edge Kasparov modules using the Kasparov product. Because of the anti-linear symmetries that occur in topological insulator models, real