Field Theory via Higher Geometry I: Smooth Sets of Fields
Abstract
The physical world is fundamentally: (1) fieldtheoretic, (2) smooth, (3) local, (4) gauged, (5) containing fermions, and last but not least: (6) nonperturbative. Tautologous as this may sound, it is remarkable that the mathematical notion of geometry which reflects all of these aspects  namely, as we will explain: "supergeometric homotopy theory"  has received little attention even by mathematicians and remains unknown to most physicists. Elaborate algebraic machinery is known for perturbative field theories, but in order to tackle the deep open questions of the subject, these will need to be lifted to a global geometry of physics. Our aim in this series is, first, to introduce inclined physicists to this theory, second to fill mathematical gaps in the existing literature, and finally to rigorously develop the full power of supergeometric homotopy theory and apply it to the analysis of fermionic (not necessarily supersymmetric) field theories. To warm up, in this first part we explain how classical bosonic Lagrangian field theory (variational EulerLagrange theory) finds a natural home in the "topos of smooth sets", thereby neatly setting the scene for the higher supergeometry discussed in later parts of the series. This introductory material will be largely known to a few experts but has never been comprehensively laid out before. A key technical point we make is to regard jet bundle geometry systematically in smooth sets instead of just its subcategories of diffeological spaces or even Fréchet manifolds  or worse simply as a formal object. Besides being more transparent and powerful, it is only on this backdrop that a reasonable supergeometric jet geometry exists, needed for satisfactory discussion of any field theory with fermions.
 Publication:

arXiv eprints
 Pub Date:
 December 2023
 DOI:
 10.48550/arXiv.2312.16301
 arXiv:
 arXiv:2312.16301
 Bibcode:
 2023arXiv231216301G
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Differential Geometry
 EPrint:
 120 pages, minor changes, clarifying material added