Perturbations vs deformations
Maksim Gritskov, Andrey Losev
TL;DR
The paper addresses the problem of relating perturbative geometry to deformation theory for affine varieties by introducing perturbative charts $\mathrm{Pert}_{n,k}(\lambda)$ and deformational charts $\mathrm{Def}_{n,k}(\epsilon)$. It develops a main theorem giving a 1-1 correspondence between perturbative charts and $\mathrm{S}_k$-invariant deformation maps, with an explicit embedding $\mathcal{I}_{\mathrm{S}}$ and a concrete quadric example. It then defines finite-dimensional analogs of perturbative QFT features—the beta field $\beta_g$ and the gamma-structure $\gamma_g$—and derives their relationship, linking obstructions and tangent-space operators. The results provide a unified geometric framework to derive perturbative theories from deformation data and suggest practical formulas connecting perturbative and deformational charts for broader deformation theories and QFT-inspired constructions.
Abstract
In the first part of the paper we define a perturbative (pre-formal) geometry and formulate a theorem on the relation between the construction of a perturbative neighborhood of affine varieties and the higher tangent bundles. In the second part of the paper, we discuss perturbative vector fields and related structures, which are finite-dimensional analogs of perturbation theory characteristics arising in quantum field theory.