G_infty structure on open-closed Hochschild cochains

Establish the existence of a G_infty (homotopy Gerstenhaber) algebra structure on the open-closed Hochschild cochain space C^{\bullet,\bullet}(B; A, A) for every open-closed homotopy algebra (B, A, \mathfrak{l}, \mathfrak{q}), where \mathfrak{l} is an L_infty structure on B and \mathfrak{q} = \{\mathfrak{q}_{\ell,k}\} specifies the open-closed operations defining the OCHA.

Background

Classically, Deligne’s Hochschild cohomological conjecture and its resolutions show that the usual Hochschild cochain complex C^\bullet(A, A) admits an E_2/G_infty structure, and Tamarkin’s formality results give operadic actions realizing this homotopy Gerstenhaber structure.

This paper constructs an open-closed analogue of Hochschild cohomology for an open-closed homotopy algebra (OCHA) (B, A, \mathfrak{l}, \mathfrak{q}), proves a canonical Gerstenhaber algebra structure on the resulting cohomology, and formulates an open-closed version of Deligne’s conjecture: namely, that the cochain space C^{\bullet,\bullet}(B; A, A) itself carries a natural G_infty structure compatible with the OCHA data.