Morse Homotopy and Homological Conformal Field Theory

TL;DR

This paper constructs operations on the cohomology of manifolds using Morse theory, parametrized by moduli spaces of Riemann surfaces, satisfying open homological conformal field theory axioms, and relates to Floer homology.

Contribution

It introduces a Morse theoretic framework for operations on cohomology that align with conformal field theory structures, extending previous algebraic and geometric constructions.

Findings

Operations satisfy the gluing axiom of open homological conformal field theory

Provides a Morse theoretic counterpart to pseudoholomorphic curve constructions

Connects cohomology operations with moduli spaces of Riemann surfaces

Abstract

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.