Rigorous perturbation theory via restriction to synthetic infinitesimal neighborhoods

Prove, in the infinite-dimensional setting of local Lagrangian field theory formulated in the Cahiers topos of thickened smooth sets, that restricting local field-theoretic functionals to synthetic infinitesimal neighborhoods D_{\phi,F} \hookrightarrow F around a field configuration \phi rigorously encodes perturbative field theory around \phi. Here F denotes the thickened smooth field space of sections \mathbold{\Gamma}_M(F), and D_{\phi,F} is the synthetic infinitesimal neighborhood of \phi in F as defined via the infinitesimal shape functor.

Background

The paper develops a synthetic differential-geometric framework for local Lagrangian field theory using the Cahiers topos of thickened smooth sets, introducing synthetic infinitesimal neighborhoods D_{p,F} that capture jets and variations intrinsically. Within this setting, the authors motivate a rigorous formulation of perturbative field theory as the restriction of local functionals to the synthetic infinitesimal neighborhood around a field configuration.

They demonstrate this correspondence in a 0-dimensional toy model and argue that, conceptually, the restriction to D_{\phi,F} should encode the standard practice of perturbative expansions. However, they explicitly leave open the proof of this claim for infinite-dimensional field spaces, noting that it requires more technical work beyond the motivating example.