Narrow Results

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.

In this paper, we calculate the low order relations of $B_{\infty }$-algebra and introduce the bibrace algebra. It can be applied to the $B_{\infty }$-algebras of the (co)Hochschild cochain complex and the singular Hochschild complex of an algebra.