Fedosov dg Foliation

• Fedosov dg foliation is a canonical differential graded structure associated with a regular foliation that encodes the leafwise de Rham complex and higher characteristic classes.
• It is constructed via a Fedosov-type dg manifold using a splitting and a torsion-free connection, ensuring invariance through unique vertical dg isomorphisms.
• This framework facilitates the transfer of L_infty and Gerstenhaber structures to the derived leaf space, underpinning formality theorems and advanced calculi.

A Fedosov dg foliation is a canonical differential graded geometric structure naturally associated to a regular foliation on a smooth manifold. Its construction formalizes and globalizes the derived geometry of the normal bundle to the foliation, encoding not only the leafwise de Rham complex but also the full range of curvature and higher characteristic classes (including all higher Atiyah classes) of the foliation. The core of the construction is a Fedosov-type dg manifold, defined via a choice of splitting and torsion-free connection on a canonical exact sequence, possessing a unique class of vertical dg isomorphisms making its structure independent of choices. Such dg foliations play a central role in derived geometry, formality theory, and the paper of characteristic classes and Tamarkin–Tsygan calculi (Chang et al., 2023, Liao et al., 16 Nov 2025, Bandiera et al., 2019, Stiénon et al., 2016).

1. Lie Pairs, Foliations, and the Normal Bundle

A Lie pair $(L, A)$ comprises a Lie algebroid $L \to M$ and a Lie subalgebroid $A \hookrightarrow L$. In the geometric setting of a foliation $F \subset TM$ on $M$, one takes $L = TM$ and $A = T_F$, the integrable distribution tangent to the leaves, so $(TM, T_F)$ forms a Lie pair. The normal bundle, $B = L/A = TM/F$, provides the transverse directions to the foliation. Sections of $A$ represent vector fields tangent to leaves, while sections of $B$ correspond to vector fields transverse to the foliation (Chang et al., 2023, Stiénon et al., 2016).

2. Fedosov dg Manifold Associated to a Lie Pair

Given a Lie pair $(L, A)$, the Fedosov dg manifold is the graded manifold $M_{\mathrm{Fed}} = L[1] \oplus B$ with algebra of functions

$$R = C^\infty(L[1] \oplus B) = \Gamma(\Lambda^\bullet L^* \otimes \widehat{S} B^*)$$

where $\widehat{S} B^*$ is the completed symmetric algebra of $B^*$. The construction utilizes:

• a splitting $j: B \rightarrow L$ of the short exact sequence $0 \to A \rightarrow L \rightarrow B \to 0$, giving $L \cong A \oplus B$;
• a torsion-free $L$-connection $\nabla$ on $B$ extending the Bott connection.

Within local coordinates, the Koszul operator $k = \sum_i \xi^i \partial_{\eta^i}$ and associated coboundary operator $\delta = [k, -]$ govern the "internal" fiberwise calculus along $B$.

A degree $+1$ homological vector field (the Fedosov differential) is then recursively constructed: $Q = -k + d_L^\nabla + X^\nabla, \quad Q^2 = 0$ with $X^\nabla$ produced by the Fedosov iteration, involving curvature and higher terms (Chang et al., 2023, Stiénon et al., 2016).

The two principal methods to realize $Q$ are:

• Iterative solution using Koszul homotopy and curvature data;
• Equivalently, via pullback of the canonical coderivation on $U(L)/U(L)\Gamma(A)$ through a Poincaré–Birkhoff–Witt (PBW) isomorphism (Stiénon et al., 2016).

3. The Fedosov dg Foliation (Vertical dg Lie Algebroid Structure)

The vertical distribution $F \subset T_{M_{\mathrm{Fed}}}$, comprising fiberwise tangent directions along $B$, inherits a natural dg Lie algebroid structure over $(M_{\mathrm{Fed}}, Q)$. The anchor is the inclusion $\rho: F \to T_{M_{\mathrm{Fed}}}$, and the bracket is the restriction of the graded Lie bracket on vector fields. The Lie derivative $L_Q$ preserves $F$, so $(F \to M_{\mathrm{Fed}}, L_Q)$ defines the dg foliation. This object canonically encodes the transverse and derived geometry of the initial foliation (Bandiera et al., 2019, Chang et al., 2023).

4. Uniqueness, Vertical Automorphisms, and Invariance

Although the construction of $(M_{\mathrm{Fed}}, Q)$ appears to depend on chosen splitting $(j)$ and connection $(\nabla)$, the main theorem asserts a unique vertical dg isomorphism between the Fedosov dg manifolds corresponding to any two choices. Explicitly, the isomorphism $\phi$ (of the form $\exp(Y)$, $Y$ a vertical vector field vanishing on linear terms) intertwines the differentials $\phi \circ Q_2 = Q_1 \circ \phi$. It is given by a universal iteration formula and can be expressed as the homotopy logarithm of $\phi$. This invariance ensures the resulting dg manifold is canonical and justifies designating it as the Fedosov dg foliation of $(M, F)$ (Chang et al., 2023).

Construction Step	Choice Dependent	Canonical Up to
Splitting and Connection	Yes	Vertical automorphism $\phi$
Fedosov Differential $Q$	Yes	dg isomorphism
Fedosov dg Foliation $F_Q$	No (canonical)	N/A

5. Homotopy Contractions and $L_\infty$ Structures

Fedosov–Dolgushev type homotopy contractions transfer $L_\infty$ algebra and Gerstenhaber structures from the horizontal (Chevalley–Eilenberg) to the vertical (Fedosov) complexes. Explicitly, the Dolgushev–Fedosov contraction and perturbative augmentations produce quasi-isomorphisms between standard complexes $(\Gamma(\Lambda^\bullet A^*), d_A)$ and the Fedosov dg foliation complex $$(\Gamma(\Lambda^\bullet L^* \otimes \widehat{S}B^*), Q)$$. This transfer realizes canonical $L_\infty$ algebras (e.g., polyvector fields, polydifferential operators) on the derived leaf space, whose Gerstenhaber structures on cohomology do not depend on auxiliary choices (Liao et al., 16 Nov 2025, Bandiera et al., 2019, Stiénon et al., 2016).

6. Compatibility with Cartan and Tamarkin–Tsygan Calculi

Natural homotopy contractions lift all algebraic structures such as wedge, Schouten, Gerstenhaber, cup, and Hochschild operations to the Fedosov dg foliation setting. The morphisms $\tau_Q, \sigma_Q, h_Q$ provide explicit chain-level isomorphisms, inducing equivalences

$$\mathcal{C}_C(M, Q) \cong \mathcal{C}_C(\mathcal{F}_Q), \qquad \mathcal{C}_H(M, Q) \cong \mathcal{C}_H(\mathcal{F}_Q)$$

between the Cartan and Tamarkin–Tsygan calculi. This enables globalizing local formality isomorphisms (e.g., Kontsevich–Tsygan) to arbitrary dg manifolds, ensuring all calculi live on the finite-dimensional Fedosov resolution of the foliation (Liao et al., 16 Nov 2025).

7. Specialization to Foliations and Applications

In the regular foliation case with $L = TM$, $A = F$, and $B = TM/F$, the Fedosov dg manifold produces a global, choice-independent resolution of the leaf space $M/F$ in the derived category. The $Q$-cohomology computes the leafwise de Rham cohomology; higher structure encodes secondary characteristic classes (Atiyah, Todd, etc.) of the foliation. As such, the Fedosov dg foliation provides a canonical derived-geometric model for formality theorems, characteristic classes, and noncommutative geometry of leaf spaces (Chang et al., 2023, Stiénon et al., 2016).

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Key References:

• "Vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair" (Chang et al., 2023)
• "Formal geometry and Tamarkin--Tsygan calculi of dg manifolds" (Liao et al., 16 Nov 2025)
• "Polyvector fields and polydifferential operators associated with Lie pairs" (Bandiera et al., 2019)
• "Fedosov dg manifolds associated with Lie pairs" (Stiénon et al., 2016)