Quantum fields, noncommutative spaces, and motives The Riemann zeta function and noncommutative geometry Quantum statistical mechanics and Galois symmetries Endomotives, thermodynamics, and the Weil explicit formula Appendix Bibliography Index.

Noncommutative Geometry, Quantum Fields and Motives

Illustration of key concepts in dimension d = 1.- Topological solid state systems: conjectures, experiments and models.- Observables algebras for solid state systems.- K-theory for topological solid state systems.- The topological invariants and their interrelations.- Index theorems for solid state systems.- Invariants as measurable quantities.

/pdf/bulk-and-boundary-invariants-for-complex-topological-4djtiqixhq.pdf

Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics

Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces R2N endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes–Lott functional action, are given for these noncommutative hyperplanes.

Moyal Planes are Spectral Triples

Preliminaries.- Modular Hilbert algebras.- Generalized Hilbert algebras.- The commutation theorem for modular Hilbert algebras.- Self-adjoint subalgebras of generalized Hilbert algebras.- The spectral algebra.- The modular operator ?.- The resolvent of the modular operator ?.- The one-parameter automorphisms defined by the modular operator ?.- Formulation of the modular Hilbert algebra.- Tensor product and direct sum of modular Hilbert algebras.- The standard representation of von Neumann algebras.- The modular automorphism group and the Kubo-Martin-Schwinger boundary condition.- Semi-finiteness and the modular automorphism group.

Tomita's theory of modular Hilbert algebras and its applications

We show that the first five of the axioms we had formulated on spectral triples suffice (in a slightly stronger form) to characterize the spectral triples associated to smooth compact manifolds. The algebra, which is assumed to be commutative, is shown to be isomorphic to the algebra of all smooth functions on a unique smooth oriented compact manifold, while the operator is shown to be of Dirac type and the metric to be Riemannian.

https://www.ems-ph.org/journals/show_pdf.php?issn=1661-6952&vol=7&iss=1&rank=1

On the spectral characterization of manifolds

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.

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Smoothness and Locality for Nonunital Spectral Triples

https://openresearch-repository.anu.edu.au/bitstream/1885/23275/2/01_Carey_The_local_index_formula_in_2006.pdf

The local index formula in semifinite Von Neumann algebras I: Spectral flow

The local index formula in semifinite von Neumann algebras II: the even case

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

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Adam Rennie

Papers

Smoothness and Locality for Nonunital Spectral Triples

The local index formula in semifinite Von Neumann algebras I: Spectral flow

The local index formula in semifinite von Neumann algebras II: the even case

Index Theory for Locally Compact Noncommutative Geometries

The Hochschild class of the Chern character for semifinite spectral triples