Homotopy algebra of open-closed strings

In work with Jim Stasheff, we define a strongly homotopy algebra associated with tree open-closed strings and call it an open-closed homotopy algebra (OCHA). It is known that in general tree open string theory is associated with an A_∞-structure and tree closed string theory is associated with an L_∞-structure. An OCHA is defined as an algebra which includes both an A_∞-algebra and an L_∞-algebra as subalgebras. We explain homotopy algebraic properties of OCHAs and their application to deformation theory of A_∞-algebras.