Classes on compactifications of the moduli space of curves through solutions to the quantum master equation

In this paper we describe a construction which produces classes in compactifications of the moduli space of curves. This construction extends a construction of Kontsevich which produces classes in the open moduli space from the initial data of a cyclic A_∞-algebra. The initial data for our construction are what we call a 'quantum A_∞-algebra', which arises as a type of deformation of a cyclic A_∞-algebra. The deformation theory for these structures is described explicitly. We construct a family of examples of quantum A_∞-algebras which extend a family of cyclic A_∞-algebras, introduced by Kontsevich, which are known to produce all the kappa classes using his construction.