Integral invariants for framed 3-manifolds associated to trivalent graphs possibly with self-loops

Abstract

Bott--Cattaneo's theory defines the integral invariants for a framed rational homology 3-sphere equipped with an acyclic orthogonal local system, in terms of graph cocycles without self-loops. The 2-loop term of their invariants is associated with the theta graph. Their definition requires a cohomological condition. Cattaneo--Shimizu removed this cohomological condition and gave a 2-loop invariant associated with a linear combination of the theta graph and the dumbbell graph, the 2-loop trivalent graph with self-loops. In this paper, we are concerned with an acyclic local system given by the adjoint representation of a semi-simple Lie group composed with a representation of the fundamental group of a closed 3-manifold, and we show that through a cohomological construction eventually the integral associated with the dumbbell graph vanishes. Based on this idea, we construct a theory of graph complexes and cocycles, so that higher-loop invariants can be defined by two different but equivalent methods: the graph cocycles without self-loops as in Bott--Cattaneo's theory, and the ones with self-loops that extend Cattaneo--Shimizu's 2-loop invariants. As a consequence, we prove that the generating series of Chern--Simons perturbation theory gives rise to topological invariants for framed 3-manifolds in our setting, which admits a formula in terms of only trivalent graphs without self-loops.