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Derived Moduli Spaces of Nonlinear PDEs II: Variational Tricomplex and BV Formalism

Jacob Kryczka, Artan Sheshmani, Shing-Tung Yau

Abstract

This paper is the second in a series of works dedicated to studying non-linear partial differential equations via derived geometric methods. We study a natural derived enhancement of the de Rham complex of a non-linear PDE via algebro-geometric techniques and examine its consequences for the functional differential calculus on the space of solutions. Applications to the BV-formalism with and without boundary conditions are discussed.

Derived Moduli Spaces of Nonlinear PDEs II: Variational Tricomplex and BV Formalism

Abstract

This paper is the second in a series of works dedicated to studying non-linear partial differential equations via derived geometric methods. We study a natural derived enhancement of the de Rham complex of a non-linear PDE via algebro-geometric techniques and examine its consequences for the functional differential calculus on the space of solutions. Applications to the BV-formalism with and without boundary conditions are discussed.

Paper Structure

This paper contains 49 sections, 48 theorems, 386 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. What do we achieve?
  3. Background and motivation
  4. Variational bicomplex.
  5. Variational tricomplexes.
  6. Homotopical PDE Geometry and (Non) Perturbative BV-Quantization
  7. Main Results.
  8. Applications to Field Theory.
  9. Notations and Conventions.
  10. Geometry of $\hbox{$\mathcal{D}$}$-Prestacks
  11. Coefficients
  12. Example: Horizontal Jets as $\mathcal{D}$-Bundles.
  13. Local Calculus.
  14. The $\hbox{$\mathcal{D}$}_X$-linear derived de Rham differential
  15. De Rham Algebras
  16. ...and 34 more sections

Key Result

Proposition \oldthetheorem

There are equivalences of categories $\mathrm{Mod}_{\mathcal{D}_X}(\mathcal{A})\simeq\mathrm{Mod}(\mathcal{A}\otimes_{\mathcal{O}_X}\mathcal{D}_X),$ and $\mathrm{Mod}_{\mathcal{D}_X^{\mathrm{op}}}(\mathcal{A}^{r})\simeq \mathrm{Mod}_{\mathcal{D}_X^{\mathrm{op}}}(\mathcal{A})$.

Figures (1)

  • Figure 1: A truncated quasi-smooth variational tri-complex. Each coloured face is a bi-complex in a fixed cohomological degree.

Theorems & Definitions (87)

  • Remark \oldthetheorem: Notation/Terminology
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • ...and 77 more