arXiv subjects
Publications and source records attributed to Joseph C. Varilly.
The Dutch scientist Christiaan Huygens refined Archimedes' celebrated geometrical computation of $\pi$ to its highest point. Yet the rich content of his beautiful treatise \emph{De circuli magnitudine inventa} (1654) has apparently never been presented in modern form. Here we offer a detailed and contemporary development of several of his most striking results. We also make a historical conjecture concerning Archimedes' trisection figure.
The fuzzy sphere, as a quantum metric space, carries a sequence of metrics which we describe in detail. We show that the Bloch coherent states, with these spectral distances, form a sequence of metric spaces that converge to the round sphere in the high-spin limit.
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 Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.
We examine the status of massive gauge theories, such as those usually obtained by spontaneous symmetry breakdown, from the viewpoint of causal (Epstein-Glaser) renormalization. The BRS formulation of gauge invariance in this framework, starting from canonical quantization of massive (as well as massless) vector bosons as fundamental entities, and proceeding perturbatively, allows one to rederive the reductive group symmetry of interactions, the need for scalar fields in gauge theory, and the covariant derivative. Thus the presence of higgs particles is explained without recourse to a Higgs(-Englert-Brout-Guralnik-Hagen-Kibble) mechanism. Along the way, we dispel doubts about the compatibility of causal gauge invariance with grand unified theories.
We show that the algebra A of a commutative unital spectral triple (A,H,D) satisfying several additional conditions, slightly stronger than those proposed by Connes, is the algebra of smooth functions on a compact spin manifold.
We find the Stratonovich-Weyl quantizer for the nonunimodular affine group of the line. A noncommutative product of functions on the half-plane, underlying a noncompact spectral triple in the sense of Connes, is obtained from it. The corresponding Wigner functions reproduce the time-frequency distributions of signal processing. The same construction leads to scalar Fourier transformations on the affine group, simplifying and extending the Fourier transformation proposed by Kirillov.
In this note we show that the crucial orientation condition for commutative geometries fails for the natural spectral triple of an orbifold M/G.
This is the introduction and bibliography for lecture notes of a course given at the Summer School on Noncommutative Geometry and Applications, sponsored by the European Mathematical Society, at Monsaraz and Lisboa, Portugal, September 1-10, 1997. In the published version, an epilogue of recent developments and many new references from 1998-2006 have been added. 1. Commutative geometry from the noncommutative point of view. 2. Spectral triples on the Riemann sphere. 3. Real spectral triples, the axiomatic foundation. 4. Geometries on the noncommutative torus. 5. The noncommutative integral. 6. Quantization and the tangent groupoid. 7. Equivalence of geometries. 8. Action functionals. 9. Epilogue: new directions.
We extend the isospectral deformations of Connes, Landi and Dubois-Violette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group $R^l$. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of functions on the manifold. We show that this relation persists for actions of $R^l$, under mild restrictions on the geometry of the manifold which guarantee the Dixmier traceability of those operators.
We discuss the local index formula of Connes-Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.
We construct a 3^+ summable spectral triple (A(SU_q(2)),H,D) over the quantum group SU_q(2) which is equivariant with respect to a left and a right action of U_q(su(2)). The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.
The progress of noncommutative geometry has been crucially influenced, from the beginning, by quantum physics: we review this development in recent years. The Standard Model, with its central role for the Dirac operator, led to several formulations culminating in the concept of a real spectral triple. String theory then came into contact with NCG, leading to an emphasis on Moyal-like algebras and formulations of quantum field theory on noncommutative spaces. Hopf algebras have yielded an unexpected link between the noncommutative geometry of foliations and perturbative quantum field theory. The quest for a suitable foundation of quantum gravity continues to promote fruitful ideas, among them the spectral action principle and the search for a better understanding of "noncommutative spaces".
We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We show its relation to the algebra of transverse differential operators introduced by Connes and Moscovici in order to compute a local index formula in cyclic cohomology, and to the several Hopf algebras defined by Connes and Kreimer to simplify the combinatorics of perturbative renormalization. We explain how characteristic classes for a Hopf module algebra can be obtained from the cyclic cohomology of the Hopf algebra which acts on it. Finally, we discuss the theory of noncommutative spherical manifolds and show how they arise as homogeneous spaces of certain compact quantum groups.
We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for certain compact quantum groups. We give a general construction of homogeneous spaces which support noncommutative spin geometries.
We examine some recent developments in noncommutative geometry, including spin geometries on noncommutative tori and their quantization by the Shale-Stinespring procedure, as well as the emergence of Hopf algebras as a tool linking index theory and renormalization calculations
We address one of the open problems in quantization theory recently listed by Rieffel. By developping in detail Connes' tangent groupoid principle and using previous work by Landsman, we show how to construct a strict, flabby quantization, which is moreover an asymptotic morphism and satisfies the reality and traciality constraints, on any oriented Riemannian manifold. That construction generalizes the standard Moyal rule. The paper can be considered as an introduction to quantization theory from Connes' point of view.
By exploiting the relation between Fredholm modules and the Segal-Shale-Stinespring version of canonical quantization, and taking as starting point the first-quantized fields described by Connes' axioms for noncommutative spin geometries, a Hamiltonian framework for fermion quantum fields over noncommutative manifolds is introduced. We analyze the ultraviolet behaviour of second-quantized fields over noncommutative 3-tori, and discuss what behaviour should be expected on other noncommutative spin manifolds.
We render a thorough, physicist's account of the formulation of the Standard Model (SM) of particle physics within the framework of noncommutative differential geometry (NCG). We work in Minkowski spacetime rather than in Euclidean space. We lay the stress on the physical ideas both underlying and coming out of the noncommutative derivation of the SM, while we provide the necessary mathematical tools. Postdiction of most of the main characteristics of the SM is shown within the NCG framework. This framework, plus standard renormalization technique at the one-loop level, suggest that the Higgs and top masses should verify 1.3 m_top \lesssim m_H \lesssim 1.73 m_top.