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Spectral theoretic characterization of the massless Dirac action

Downes, RJ; Vassiliev, D; (2016) Spectral theoretic characterization of the massless Dirac action. Mathematika , 62 pp. 701-718. 10.1112/S0025579315000509. Green open access

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Abstract

We consider an elliptic self-adjoint first order differential operator L acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of the operator L is assumed to be trace-free and the subprincipal symbol is assumed to be zero. Given a positive scalar weight function, we study the weighted eigenvalue problem for the operator L. The corresponding counting function (number of eigenvalues between zero and a positive lambda) is known to admit, under appropriate assumptions on periodic trajectories, a two-term asymptotic expansion as lambda tends to plus infinity and we have recently derived an explicit formula for the second asymptotic coefficient. The purpose of this paper is to establish the geometric meaning of the second asymptotic coefficient. To this end, we identify the geometric objects encoded within our eigenvalue problem - metric, nonvanishing spinor field and topological charge - and express our asymptotic coefficients in terms of these geometric objects. We prove that the second asymptotic coefficient of the counting function has the geometric meaning of the massless Dirac action.

Type: Article
Title: Spectral theoretic characterization of the massless Dirac action
Open access status: An open access version is available from UCL Discovery
DOI: 10.1112/S0025579315000509
Publisher version: http://dx.doi.org/10.1112/S0025579315000509
Language: English
Additional information: Copyright © University College London 2016 This article is distributed with Open Access under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided that the original work is properly cited.
Keywords: math.SP, math.SP, gr-qc, math.DG, 35P20 (primary), 35J46, 35R01, 35Q41 (secondary)
UCL classification: UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Medical Sciences
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Medical Sciences > Cancer Institute
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Medical Sciences > Cancer Institute > Research Department of Cancer Bio
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/1416527
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