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Formal geometry and Tamarkin--Tsygan calculi of dg manifolds

Hsuan-Yi Liao, Mathieu Stiénon, Ping Xu

TL;DR

The paper addresses the formal geometry of dg manifolds by constructing Fedosov dg manifolds and Fedosov dg Lie algebroids, establishing contractions between spaces of polyvector fields, differential forms, polydifferential operators, and polyjets. It extends Tamarkin--Tsygan calculi to dg manifolds and dg Lie algebroids, proving that the Cartan and noncommutative calculi are preserved under Fedosov transport, yielding isomorphisms between the calculi of a dg manifold and its Fedosov model. The main advancement is a contraction-based globalization that underpins a Duflo–Kontsevich-type theory in the dg setting (to be developed in a companion paper). This formal machinery provides a robust framework for globalization of formality in derived/graded contexts and has potential applications to deformation quantization and quantum field theory on dg target spaces.

Abstract

The main goal of this paper is to study the formal geometry of dg manifolds à la Fedosov. For any dg manifold $(\mathcal{M}, Q)$, we construct a Fedosov dg foliation (or dg Lie algebroid) $\mathcal{F}_Q \to \mathcal{N}_Q$. We establish homotopy contractions between their respective spaces of polyvector fields, differential forms, polydifferential operators, and polyjets. As a consequence, we prove that their respective Cartan calculi and noncommutative calculi, in the sense of Tamarkin--Tsygan, are isomorphic.

Formal geometry and Tamarkin--Tsygan calculi of dg manifolds

TL;DR

The paper addresses the formal geometry of dg manifolds by constructing Fedosov dg manifolds and Fedosov dg Lie algebroids, establishing contractions between spaces of polyvector fields, differential forms, polydifferential operators, and polyjets. It extends Tamarkin--Tsygan calculi to dg manifolds and dg Lie algebroids, proving that the Cartan and noncommutative calculi are preserved under Fedosov transport, yielding isomorphisms between the calculi of a dg manifold and its Fedosov model. The main advancement is a contraction-based globalization that underpins a Duflo–Kontsevich-type theory in the dg setting (to be developed in a companion paper). This formal machinery provides a robust framework for globalization of formality in derived/graded contexts and has potential applications to deformation quantization and quantum field theory on dg target spaces.

Abstract

The main goal of this paper is to study the formal geometry of dg manifolds à la Fedosov. For any dg manifold , we construct a Fedosov dg foliation (or dg Lie algebroid) . We establish homotopy contractions between their respective spaces of polyvector fields, differential forms, polydifferential operators, and polyjets. As a consequence, we prove that their respective Cartan calculi and noncommutative calculi, in the sense of Tamarkin--Tsygan, are isomorphic.

Paper Structure

This paper contains 47 sections, 62 theorems, 349 equations.

Table of Contents

  1. Tamarkin--Tsygan calculi of dg manifolds
  2. Tamarkin--Tsygan calculi
  3. Cartan calculi of dg manifolds
  4. Noncommutative calculi of dg manifolds
  5. Tamarkin--Tsygan calculi of dg Lie algebroids
  6. Differential graded Lie algebroids
  7. Tamarkin--Tsygan calculus structure $\mathop{\mathrm{Cal}}\nolimits_C(\mathcal{L},\mathcal{Q})$
  8. Universal enveloping algebras of graded Lie algebroids
  9. Tamarkin--Tsygan calculus structure $\mathop{\mathrm{Cal}}\nolimits_H(\mathcal{L},\mathcal{Q})$
  10. Fedosov dg Lie algebroids of graded manifolds
  11. Fedosov dg manifolds
  12. Connections on graded manifold
  13. Fedosov formal manifold
  14. Fedosov dg formal manifolds
  15. Fedosov connection in local coordinates
  16. ...and 32 more sections

Key Result

Theorem A

Given a dg manifold $(\mathcal{M}, Q)$ and a torsion-free affine connection $\nabla$ on $\mathcal{M}$, let $\mathcal{F}_Q \to {\mathcal{N}_Q} := (\mathcal{F} \to \mathcal{N}, L_{D^\nabla + \breve{\tau}_{\natural}(Q)})$ be its corresponding Fedosov dg Lie algebroid. Then:

Theorems & Definitions (103)

  • Theorem A
  • Theorem B
  • Definition 1.1: MR2986860
  • Example 1.2
  • Example 1.3
  • Definition 1.4
  • Definition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Lemma 2.1
  • ...and 93 more