Equivariant q-action and connection in SH^*_{u,q}(M,D)

Determine the explicit [q]-action on the S^1-equivariant deformed symplectic cohomology SH^*_{u,q}(M,D) and provide a concrete description of the associated connection ∇_{u∂_q}, including how these structures relate to enumerative geometry and act on the H^*(D) z^w components.

Background

The paper defines the S^1-equivariant deformation SH^*_{u,q}(M,D), proves isomorphisms with cohomological models, and constructs a connection ∇_{u∂_q} with compatibility to the quantum connection. Despite these results, the authors state that a full determination of the equivariant [q]-action and the detailed form of the connection remains open.

Understanding these structures explicitly would clarify their enumerative meaning and strengthen applications, including those explored in related work on the quantum connection and wrapped Fukaya categories.