The action functional in non-commutative geometry
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 \(\theta\in {\mathcal L}({\mathcal E}\otimes_{{\mathcal A}}{\mathcal H})\) belongs to \({\mathcal L}^{2+}\) and the value of the Dixmier trace \(Tr_{\omega}(\theta^ 2)=I(\theta),\) is independent of \(\omega\) and defines a gauge invariant positive functional I. 2. The restriction of I to each (affine space) fiber of the map \(\nabla \to \nabla_ c\) is Gaussian (i.e. a quadratic form) and one has \(\inf_{\nabla_ c=A}I(\nabla)=(16\pi^ 2)^{-1}YM(A).\)
scientific article; zbMATH DE number 4076018
scientific article