Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds I. Invariants of homology 3-spheres

TL;DR

This paper extends Fukaya's Morse homotopy approach to define new invariants of integral homology 3-spheres using 3-valent graphs with multiple loops, generalizing previous 2-loop theories.

Contribution

It introduces a higher-order generalization of Fukaya's invariants, constructing a sequence of graph-valued invariants for homology 3-spheres with arbitrary loop numbers.

Findings

Constructed invariants using 3-valent graphs satisfying differential equations

Generalized 2-loop Chern--Simons perturbation theory to higher loops

Established a new framework for graph-based 3-manifold invariants

Abstract

We give a generalization of Fukaya's Morse homotopy theoretic approach for 2-loop Chern--Simons perturbation theory to 3-valent graphs with arbitrary number of loops at least 2. We construct a sequence of invariants of integral homology 3-spheres with values in a space of 3-valent graphs (Jacobi diagrams or Feynman diagrams) by counting graphs in an integral homology 3-sphere satisfying certain condition described by a set of ordinary differential equations.