New moduli spaces of pointed curves and pencils of flat connections.

It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad (H∗(M0,n+1)) of the moduli spaces of n–pointed stable curves of genus zero. In this paper we establish a similar relationship between the pencils of formal flat connections (or solutions to the Commutativity Equations) and homology of a new series Ln of pointed stable curves of genus zero. Whereas M0,n+1 parametrizes trees of P 1 ’s with pairwise distinct nonsingular marked points, Ln parametrizes strings of P 1 ’s stabilized by marked points of two types. The union of all Ln’s forms a semigroup rather than operad, and the role of operadic algebras is taken over by the representations of the appropriately twisted homology algebra of this union. 0. Introduction and plan of the paper One of the remarkable basic results in the theory of the Associativity Equations (or Frobenius manifolds) is the fact that their formal solutions are the same as cyclic algebras over the homology operad (H∗(M0,n+1)) of the moduli spaces of n– pointed stable curves of genus zero. This connection was discovered by physicists,