NONCOMMUTATIVE HOMOTOPY ALGEBRAS ASSOCIATED WITH OPEN STRINGS

Abstract

We discuss general properties of A∞-algebras and their applications to the theory of open strings. The properties of cyclicity for A∞-algebras are examined in detail. We prove the decomposition theorem, which is a stronger version of the minimal model theorem, for A∞-algebras and cyclic A∞-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∞-isomorphic to each other. The same results hold for classical closed string field theories, whose algebraic structure is governed by cyclic L∞-algebras.