Morse Homotopy and Homological Conformal Field Theory

By studying spaces of flow graphs in a closed oriented manifold, we construct operations on its cohomology, parametrized by the homology of the moduli spaces of compact Riemann surfaces with boundary marked points. We show that the operations satisfy the gluing axiom of an open homological conformal field theory. This complements previous constructions due to R. Cohen et al., K. Costello and M. Kontsevich and is also the Morse theoretic counterpart to a conjectural construction of operations on the Lagrangian Floer homology of the zero section of a cotangent bundle, obtained by studying uncompactified moduli spaces of higher genus pseudoholomorphic curves.