Extention Cohomological Fields Theory and Noncommutative Frobenius Manifolds

INTRODUCTION The Cohomological Field Theory was propose by Kontsevich and Manin [5] for description of Gromov-Witten Classes. They prove that Cohomological Field Theory is equivalent to Formal Frobenius manifold. Formal Frobenius manifold is defined by a formal series F , satisfying to associative equations. In points of convergence the series F defines a Frobenius algebras. The set of these points forms a Frobenius manifold [4], i.e. a space of special deformations of Frobenius algebras. Cohomological Field Theory is a system of special homomorphisms to spaces of cohomology of Deline–Mumford compactifications for moduli spaces of complex curves (Riemann spheres) with punctures. In this paper there is an extension (Stable Field Theory) of Cohomological Field Theory. It is a system of homomorphisms to some algebras generated by disks with punctures. I conjecture that they describe relative Gromov–Witten classes. Stable Field Theory is equivalent to some analog of Formal Frobenius manifold. This analog is defined by formal tensor series (Structure Series) satisfying some system of ”differential equations” (including associativity equation). In points of convergence the Structure Series define Extend Frobenius algebras. They are special noncommutative analog of Frobenius algebra. Extended Frobenius algebras describe ’Open-Closed’ Topological Field Theories [6] of genus 0 by analogy that Frobenius algebras describe Atiyah–Witten 2D Topological Field Theories. Thus Structure Series are noncommutable analogs of Formal Frobenius manifolds. In passing it is given some other proof of Kontsevich–Manin theorem about equivalence Cohomological Field Theories and Formal Frobenius manifolds. In sections 1 and 2 it is proposed a general axiomatic of Topological Field Theories over functors. This class involves, 2D Atiyah–Witten [2], [11] ’open-closed’ [6], Klein [3] Topological Field Theories and Cohomological Field Theories [5], [7]. In section 3 it is constructed and investigated a Stabilizing functor on a category of spheres with punctures, disks with punctures and its