Noncommutative homotopy algebras associated with open strings

Abstract

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

92 pages, 16 figuers; based on Ph.D thesis submitted to Graduate School of Mathematical Sciences, Univ. of Tokyo on January, 2003; v2: explanation improved, references added, published version

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