Classification of first order sesquilinear forms

Matteo Capoferri, Nikolai Saveliev, Dmitri Vassiliev

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


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
Article number2050027
JournalReviews in Mathematical Physics
Issue number9
Publication statusPublished - Oct 2020


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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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