Maurer–Cartan description of ambient deformation via S^1-equivariant symplectic cochains

Construct, under the hypothesis that the boundary ∂D of the Liouville domain D has no contractible Reeb orbits, a Maurer–Cartan element x in the S^1-equivariant symplectic cochains SC^*_{S^1}(D) (well-defined up to gauge equivalence) such that the ambient symplectic cochain complex SC^*_M(D) of the embedding D ⊂ M is quasi-isomorphic to the twist of the intrinsic symplectic cochain complex SC^*(D) by the pushed-forward element ρ_*(x) along the Gysin map ρ: SC^*_{S^1}(D) → SC^{*+1}(D).

Background

A central theme of the paper is relating the ambient symplectic cohomology SC^*_M(D) of a Liouville domain D embedded in a symplectic manifold M to its intrinsic symplectic cohomology SC^*(D). The authors propose a neck-stretching perspective in which contributions from holomorphic planes in M\D are encoded by a Maurer–Cartan element that twists the intrinsic complex.

In particular, using the framed E_2 structure and S^1-equivariant enhancements of symplectic cochains, a Gysin map ρ: SC^*_{S^1}(D) → SC^{*+1}(D) induces a pushforward on Maurer–Cartan moduli. The conjecture asserts the existence of x ∈ SC^*_{S^1}(D) (up to gauge) that fully captures the deformation from SC^*(D) to SC^*_M(D) by twisting. The hypothesis excludes contractible Reeb orbits on ∂D to ensure the necessary mappings and formality structures.

This characterization would provide a conceptual mechanism to identify and control ambient deformations via equivariant algebraic data, aligning with ideas attributed to Cieliebak–Latschev and related SFT-type constructions.