Tag Archives: Dirac operator

Dense domains, symmetric operators and spectral triples

A cautionary paper appeared on the arXiv last week

Forsyth, I., B. Mesland, and A. Rennie. Dense Domains, Symmetric Operators and Spectral Triples. ArXiv e-print, June 6, 2013. http://arxiv.org/abs/1306.1580.

The paper is about the definition of unbounded Fredholm modules and how certain, initially plausible, weakenings of the definition do not work as expected. (Some well-known authors are called out for insufficient attention to these points; I was initially worried that Analytic K-Homology appears in the bibliography, but I don’t think Nigel and I are among the criminals here.)

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“Mass endomorphism, surgery and perturbations”

That is the title of an interesting paper just posted on the arXiv. I had never heard of the “mass endomorphism” so this was new to me… Take a compact Riemannian spin manifold and suppose that the metric is flat in the neighborhood of a point p.  If there are no harmonic spinors (so that the Dirac operator is invertible) then the Dirac Green’s function, i.e. the inverse of the Dirac operator, has an asymptotic expansion near p in which the zero term is an endomorphism of the spinor bundle called the mass operator.  It is known that if the mass operator is non-zero then a solution exists to the classical Yamabe problem.  In this paper it is shown that the mass operator is “generically” non zero – using a lot of the machinery from positive-scalar-curvature land: psc surgery, results of Stolz, etc…