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

Bulk and Boundary Invariants for Complex Topological Insulators

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 search for strong topological phases in generic aperiodic materials and meta-materials is now vigorously pursued by the condensed matter physics community. In this work, we first introduce the concept of patterned resonators as a unifying theoretical framework for topological electronic, photonic, phononic etc. (aperiodic) systems. We then discuss, in physical terms, the philosophy behind an operator theoretic analysis used to systematize such systems. A model calculation of the Hall conductance of a 2-dimensional amorphous lattice is given, where we present numerical evidence of its quantization in the mobility gap regime. Motivated by such facts, we then present the main result of our work, which is the extension of the Chern number formulas to Hamiltonians associated to lattices without a canonical labeling of the sites, together with index theorems that assure the quantization and stability of these Chern numbers in the mobility gap regime. Our results cover a broad range of applications, in particular, those involving quasi-crystalline, amorphous as well as synthetic (i.e. algorithmically generated) lattices.

https://iopscience.iop.org/article/10.1088/1751-8121/aac093/pdf

Non-commutative Chern numbers for generic aperiodic discrete systems

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 $C^*$-algebras and KKO-theory must be used.

/pdf/the-k-theoretic-bulk-edge-correspondence-for-topological-lc4nqunhtk.pdf

The $K$-theoretic bulk-edge correspondence for topological insulators

We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer-Fredholm operators in a type $I_{\infty}$ or $II_\infty$ von Neumann algebra ${\mathcal N}$ The framework is that of {\it odd unbounded} $\theta$-{\it summable} {\it Breuer-Fredholm modules} for a unital Banach *-algebra, $\mathcal A$ In the type $II_{\infty}$ case spectral flow is real-valued, has no topological definition as an intersection number and our formulae encompass all that is known We borrow Ezra Getzler's idea (suggested by I M Singer) of considering spectral flow (and eta invariants) as the integral of a closed one-form on an affine space Applications in both the type I and type II cases include a general formula for the relative index of two projections, representing truncated eta functions as integrals of one forms and expressing spectral flow in terms of the JLO cocycle to give the pairing of the $K$-homology and $K$-theory of $\mathcal A$

Spectral flow in Fredholm modules, eta invariants and the JLO cocycle

We provide a proof of Connes' formula for a representative of the Hochschild class of the Chern character for (p,\infty)-summable spectral triples. Our proof is valid for all semifinite von Neumann algebras, and all integral p\geq 1. We employ the minimum possible hypotheses on the spectral triples.

The Hochschild Class of the Chern Character For Semifinite Spectral Triples

We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer-Fredholm operators in a type $I_{\infty}$ or $II_\infty$ von Neumann algebra ${\mathcal N}$. The framework is that of {\it odd unbounded} $\theta$-{\it summable} {\it Breuer-Fredholm modules} for a unital Banach *-algebra, $\mathcal A$. In the type $II_{\infty}$ case spectral flow is real-valued, has no topological definition as an intersection number and our formulae encompass all that is known. We borrow Ezra Getzler's idea (suggested by I. M. Singer) of considering spectral flow (and eta invariants) as the integral of a closed one-form on an affine space. Applications in both the type I and type II cases include a general formula for the relative index of two projections, representing truncated eta functions as integrals of one forms and expressing spectral flow in terms of the JLO cocycle to give the pairing of the $K$-homology and $K$-theory of $\mathcal A$.

We generalise the 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. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula.

pdf/the-local-index-formula-in-semifinite-von-neumann-algebras-2dyg77nan7.pdf

The Local Index Formula in Semifinite von Neumann Algebras II: The Even Case

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Papers

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

Spectral flow in Fredholm modules, eta invariants and the JLO cocycle

The Hochschild Class of the Chern Character For Semifinite Spectral Triples

Spectral flow in Fredholm modules, eta invariants and the JLO cocycle

The Local Index Formula in Semifinite von Neumann Algebras II: The Even Case