TY - JOUR

T1 - Classification of first order sesquilinear forms

AU - Capoferri, Matteo

AU - Saveliev, Nikolai

AU - Vassiliev, Dmitri

N1 - Funding Information:
NS was supported by the LMS grant 21420, the EPSRC grant EP/M000079/1, and the Simons Collaboration Grant 426269. DV was supported by the EPSRC grant EP/M000079/1.
Publisher Copyright:
© 2020 World Scientific Publishing Company.

PY - 2020/10

Y1 - 2020/10

N2 - 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.

AB - 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.

KW - first order systems

KW - gauge transformations

KW - Sesquilinear forms

KW - spin c structure

UR - http://www.scopus.com/inward/record.url?scp=85082046624&partnerID=8YFLogxK

U2 - 10.1142/S0129055X20500270

DO - 10.1142/S0129055X20500270

M3 - Article

AN - SCOPUS:85082046624

SN - 0129-055X

VL - 32

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

IS - 9

M1 - 2050027

ER -