Noncommutative homotopy algebras associated with open strings

We discuss general properties of $A_{\mathrm{\infty }}$-algebras and their applications to the theory of open strings. The properties of cyclicity for $A_{\mathrm{\infty }}$-algebras are examined in detail. We prove the decomposition theorem, which is a stronger version of the minimal model theorem, for $A_{\mathrm{\infty }}$-algebras and cyclic $A_{\mathrm{\infty }}$-algebras and discuss various consequences of it. In particular it is applied to classical open string field theories and it is shown that all classical open string field theories on a fixed conformal background are cyclic $A_{\mathrm{\infty }}$-isomorphic to each other. The same results hold for classical closed string field theories, whose algebraic structure is governed by cyclic $L_{\mathrm{\infty }}$-algebras.