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Spectral theoretic characterization of the massless Dirac action

Research output: Contribution to journalArticlepeer-review

Robert James Downes, Dmitri Vassiliev

Original languageEnglish
Pages (from-to)701-718
JournalMATHEMATIKA
Volume62
Early online date8 Mar 2016
DOIs
E-pub ahead of print8 Mar 2016
Published2016

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Abstract

We consider an elliptic self-adjoint first-order differential operator LL acting on pairs (2-columns) of complex-valued half-densities over a connected compact three-dimensional manifold without boundary. The principal symbol of the operator LL 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 LL. The corresponding counting function (number of eigenvalues between zero and a positive λ휆) is known to admit, under appropriate assumptions on periodic trajectories, a two-term asymptotic expansion as λ→+∞휆→+∞ 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, non-vanishing 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.

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