The action functional in non-commutative geometry
type: Item
instance of: scholarly article
title: The action functional in non-commutative geometry
zbMATH Open document ID: 0658.53068
DOI: 10.1007/BF01218391
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathcal H}\) be a Hilbert space and \({\mathcal L}^{1+}({\mathcal H})\) be the Macaev ideal of all compact operators T whose characteristic values satisfy \[ \sup_{N>1}\frac{1}{\log N}\sum^{N}_{n=1}\mu_ n(T)<\infty. \] Gifted with the obvious norm it is a nonseparable Banach space. Dixmier showed that for any mean \(\omega\) on the amenable group of upper triangular (2,2)-matrices, one gets a trace on \({\mathcal L}^{1+}\), given by the formula \[ Tr_{\omega}(T)=\lim_{\omega}\frac{1}{\log N}\sum^{N}_{n=1}\lambda_ n(T) \] when T is a positive operator, \(T\in {\mathcal L}^{1+}\), with eigenvalues \(\lambda_ n(T)\) in decreasing order and \(\lim_{\omega}\) is a well defined linear form on bounded sequences using \(\omega\). Given a pseudodifferential operator P on a closed \(M^ n\), Res P is the coefficient of log t in the asymptotic expansion of \(Tr(Pe^{-t\Delta}).\) Theorem. Let \(E\to M\) be a complex vector bundle, P a pseudodifferential operator of order -n. Then the corresponding operator in \({\mathcal H}=L_ 2(M,E)\) belongs to the Macaev ideal \({\mathcal L}^{1+}({\mathcal H})\) and one has \(Tr_{\omega}(P)=(1/n)Res(P)\) for any invariant mean \(\omega\). After that the author applies this result to show how one can deduce ordinary differential forms and the natural conformal invariant norm on them from the quantized forms. Then the author discusses the analogue of the Yang-Mills action in the context of non-commutative differential geometry and shows, as expected, that 4 is the critical dimension, for \(n=4\) the leading divergency of the action is the usual local Yang-Mills action. Theorem. Let \(M^ 4\) be a closed Riemannian \(Spin^ c\) manifold, \({\mathcal H}=L_ 2(M,S)\), \(F=D| D|^{-1}\), E a Hermitian vector bundle over M, \({\mathcal E}=C^{\infty}(M,E)\), \({\mathcal A}=C^{\infty}(M)\). 1. For every compatible q-connection \(\nabla\) on \({\mathcal E}\), the curvature \(\theta\in {\mathcal L}({\mathcal E}\otimes_{{\mathcal A}}{\mathcal H})\) b…
Mathematics Subject Classification ID: 53C99
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 58J40
zbMATH DE Number: 4076018
zbMATH Keywords: trace on ideals
zbMATH Keywords: quantized differential forms
zbMATH Keywords: Macaev ideal
zbMATH Keywords: pseudodifferential operator
zbMATH Keywords: complex vector bundle
zbMATH Keywords: Yang-Mills action
zbMATH Keywords: non-commutative differential geometry
author: Alain Connes
reviewed by: Jürgen Eichhorn
MaRDI profile type: Publication
Wikidata QID: Q56689636
cites work: On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-deVries type equations
cites work: Non-commutative differential geometry
cites work: Q3771069
cites work: Compact metric spaces, Fredholm modules, and hyperfiniteness
cites work: Q5515745
cites work: Algebraic aspects of nonlinear differential equations
cites work: Q3754832
cites work: Loop groups and equations of KdV type
cites work: String theory and non-commutative gauge theory
cites work: Local invariants of spectral asymmetry
cites work: A new proof of Weyl's formula on the asymptotic distribution of eigenvalues
full work available at URL: https://doi.org/10.1007/bf01218391
OpenAlex ID: W2102879602
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description: scientific article; zbMATH DE number 4076018