Field Theory via Higher Geometry I: Smooth Sets of Fields

Abstract

Most modern theoretical considerations of the physical world suggest that nature is: (1) field-theoretic, (2) smooth, (3) local, (4) gauged, (5) containing fermions, and (6) non-perturbative. Tautologous as this may sound to experts, it is remarkable that the mathematical notion of geometry which reflects all of these aspects - namely, ``supergeometric homotopy theory'' - has received little attention. Elaborate algebraic machinery is known for perturbative field theories both at the classical and quantum level, but 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 to introduce inclined physicists to this theory, to fill mathematical gaps in the existing literature, and to rigorously develop the full power of supergeometric homotopy theory and apply it to the analysis of fermionic (not necessarily super-symmetric) field theories. Secondarily, this will also lead to a streamlined and rigorous perspective we hope would also be desirable to mathematicians. In this first part, we explain how classical bosonic Lagrangian field theory (variational Euler-Lagrange 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 Fr{\'e}chet 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.