Field Theory via Higher Geometry II: Thickened Smooth Sets as Synthetic Foundations

TL;DR

This paper develops a rigorous synthetic foundation for field theory using thickened smooth sets, extending previous work to include infinitesimal structures and interactions with jet bundles, thus unifying and formalizing key aspects of Lagrangian field theory.

Contribution

It introduces the category ThickenedSmoothSets and formulates local Lagrangian field theory within this synthetic differential geometric framework, unifying previous constructions and incorporating infinitesimal and boundary considerations.

Findings

Established a rigorous synthetic foundation for field theory.

Reproduced tangent bundles as infinitesimal curves in the new setting.

Formalized variational principles as intersections of thickened smooth sets.

Abstract

This is the second in a series of papers that aim to develop rigorous and most encompassing foundations for field theory, where in the first installment, we laid out the natural formulation of bosonic variational field theory via the functorial geometry of smooth sets. Here, we extend this to the category ThickenedSmoothSets of infinitesimally thickened smooth sets. We first describe the Cahiers topos in a simplified, but fully rigorous, $\mathbb{R}$-algebraic setting -- which should serve as a more accessible introduction to the theory of Synthetic Differential Geometry to both physicists and mathematicians. Then, we formulate local Lagrangian field theory in this rigorous setting in which infinitesimal spaces exist and interact correctly with the field-theoretic spaces of infinite jet bundles, off-shell and on-shell spaces of fields etc. This setting subsumes all previous constructions and further recovers all the relevant tangent bundles of traditional (off-shell and on-shell) field theory considerations via the synthetic tangent bundle construction, i.e., as ``infinitesimal curves'' in those spaces, which were previously defined only in an ad-hoc manner. Beyond finally establishing a firm foundation for such aspects of the theory, this approach recognizes the variational principle of local Lagrangian field theory, equivalently, as the intersection of thickened smooth sets. It also suggests the rigorous formalization of perturbative field theory as the restriction to a (synthetic) infinitesimal neighborhood around a field configuration. Furthermore, our context naturally accommodates more general, rigorous considerations, in which the manifolds may have boundaries and corners, a situation that has recently been attracting greater attention in the field-theoretical literature.