Spontaneous symmetry breaking and geometry from the spectral viewpoint
type: Item
instance of: scholarly article
title: Spontaneous symmetry breaking and geometry from the spectral viewpoint
zbMATH Open document ID: 0918.46064
DOI: 10.1016/S0393-0440(97)80001-0
author: Alain Connes
published in: Journal of Geometry and Physics
publication date: 1998-03-01T00:00:00Z
review text: This is an introduction to the main ideas of noncommutative geometry, developed by the author. It's main object, the space, is a triple \((A, H, D)\), where \(A\) is an involutive operator algebra in the Hilbert space \(H\) and \(D\) is a self-adjoint unbounded operator on \(H\). The algebra \(A\) is the set of ``continuous functions on a noncommutative space'', and the operator \(D^{-1} = ds\) corresponds to the infinitesimal length of Riemannian geometry. In this general setting, a fiber bundle is given by an \(A\)-projective module of finite type, a de Rham homology class is given by a cyclic cocycle of \(A.\) The right notion of infinitesimal calculus is developed: for example, a real variable is given by a self-adjoint operator, an infinitesimal is given by a compact operator, the integral of an infinitesimal \(T\) of order one is given by the coefficient of the logarithmic divergence of the trace of \(T\), the differential of \(g\) is given by the bracket \([F,g]\), where \(F\) is a fixed operator such that \(F^2= \text{Id}\) and \([F,a]\) is compact for any \(a\in A\). There is a pairing between \(K_*(A)\) and cyclic cohomology \(HC_*(A)\), generalizing the Chern-Weil construction of classical differential geometry. When one pairs a class \(e \in K_*(A)\) with the Chern character of \((A, H, D)\) one gets the index of a Fredholm operator and thus an integer. This result explains the quantum Hall effect in physics. A local formula in cyclic cohomology for the Chern character of \((A, H, D)\) generalizes the local index theorem for Dirac-type operators. The notion of noncommutative Poincaré duality allows to extend to finite type noncommutative spaces satisfying Poincaré duality the eigenvalue inequalities of Vafa and Witten. There are ``noncommutative manifolds'' - examples are provided by noncommutative tori and the ``Standard Model'', which proved to be useful in physics.
Mathematics Subject Classification ID: 46L85
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 81R40
zbMATH DE Number: 1112279
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: noncommutative geometry
zbMATH Keywords: involutive operator algebra
zbMATH Keywords: fiber bundle
zbMATH Keywords: \(A\)-projective module of finite type
zbMATH Keywords: de Rham homology class
zbMATH Keywords: cyclic cocycle
zbMATH Keywords: infinitesimal calculus
zbMATH Keywords: Chern-Weil construction
zbMATH Keywords: Chern character
zbMATH Keywords: index of a Fredholm operator
zbMATH Keywords: quantum Hall effect
zbMATH Keywords: local index theorem for Dirac-type operators
zbMATH Keywords: noncommutative Poincaré duality
zbMATH Keywords: eigenvalue inequalities of Vafa and Witten
zbMATH Keywords: noncommutative tori
MaRDI profile type: Publication
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full work available at URL: s0393-0440(97)80001-0
OpenAlex ID: W1979528218
description: scientific article; zbMATH DE number 1112279