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

Grigorios Giotopoulos, Hisham Sati

TL;DR

This work builds a rigorous synthetic-differential-geometry foundation for local Lagrangian field theory by passing from smooth sets to infinitesimally thickened smooth sets, organized within the Cahiers topos. It develops thickened probes, thickened mapping spaces, and synthetic tangent bundles, and constructs thickened infinite jet bundles with a coherent horizontal-vertical decomposition. Local Lagrangian theory, the variational bicomplex, transgression, and the on-shell (Euler–Lagrange) locus all arise naturally in this setting, enabling a coordinate-invariant, perturbative interpretation as restrictions to infinitesimal neighborhoods. The appendix establishes the $\mathbb{R}$-algebraic nature of the Cahiers topos and treats manifolds with corners and Weil bundles, paving the way for applications to field theories on manifolds with boundaries and more general geometric features.

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.

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

TL;DR

This work builds a rigorous synthetic-differential-geometry foundation for local Lagrangian field theory by passing from smooth sets to infinitesimally thickened smooth sets, organized within the Cahiers topos. It develops thickened probes, thickened mapping spaces, and synthetic tangent bundles, and constructs thickened infinite jet bundles with a coherent horizontal-vertical decomposition. Local Lagrangian theory, the variational bicomplex, transgression, and the on-shell (Euler–Lagrange) locus all arise naturally in this setting, enabling a coordinate-invariant, perturbative interpretation as restrictions to infinitesimal neighborhoods. The appendix establishes the -algebraic nature of the Cahiers topos and treats manifolds with corners and Weil bundles, paving the way for applications to field theories on manifolds with boundaries and more general geometric features.

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, -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.
Paper Structure (20 sections, 49 theorems, 527 equations)

This paper contains 20 sections, 49 theorems, 527 equations.

Table of Contents

  1. Introduction
  2. The setting.
  3. Structure and outline of results.
  4. Infinitesimally thickened smooth sets
  5. Infinitesimally thickened probes
  6. Sheaves over thickened probes
  7. Thickened mapping spaces and the synthetic tangent bundle
  8. Infinitesimal neighborhoods and jets of sections
  9. Application: Formalizing perturbative field theory.
  10. Infinitesimal neighborhoods of submanifolds.
  11. Local Lagrangian field theory
  12. Geometry of the infinite jet bundle
  13. Transgression and the local bicomplex
  14. On-shell fields as a thickened smooth critical set
  15. Outlook.
  16. ...and 5 more sections

Key Result

Proposition 2.1

The functor sending a finite-dimensional smooth (second countable and Hausdorff) manifold to its function algebra is fully faithful, in that for any pair $N,M \in \mathrm{SmthMfd}$ the smooth functions $f : N \xrightarrow{\;} M$ biject onto the algebra homomorphisms $f^\ast : C^\infty(M) \xrightarrow{\;} C^\inf is fully faithful.

Theorems & Definitions (152)

  • Proposition 2.1: Smooth manifolds embed into Algebras
  • Lemma 2.2: Tangent vectors via infinitesimals
  • proof
  • Remark 2.3: Infinitesimal vs finite curves
  • Definition 2.4: Infinitesimal disks
  • Remark 2.5: On nomenclature of thickened points
  • Lemma 2.6: Hadamard's Lemma
  • Definition 2.7: Infinitesimally thickened points
  • Lemma 2.8: Infinitesimal points as subspaces of infinitesimal disks
  • proof
  • ...and 142 more