Classification of first order sesquilinear forms

Matteo Capoferri, Nikolai Saveliev, Dmitri Vassiliev

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

A natural way to obtain a system of partial differential equations on a manifold is to vary a suitably defined sesquilinear form. The sesquilinear forms we study are Hermitian forms acting on sections of the trivial n-bundle over a smooth m-dimensional manifold without boundary. More specifically, we are concerned with first order sesquilinear forms, namely, those generating first order systems. Our goal is to classify such forms up to GL(n, ) gauge equivalence. We achieve this classification in the special case of m = 4 and n = 2 by means of geometric and topological invariants (e.g., Lorentzian metric, spin/spinc structure, electromagnetic covector potential) naturally contained within the sesquilinear form - a purely analytic object. Essential to our approach is the interplay of techniques from analysis, geometry, and topology.

Original languageEnglish (US)
Article number2050027
JournalReviews in Mathematical Physics
Volume32
Issue number9
DOIs
StatePublished - Oct 1 2020

Keywords

  • Sesquilinear forms
  • first order systems
  • gauge transformations
  • spin c structure

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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