Bott--Cattaneo's theory defines the integral invariants of framed rational homology 3-spheres with acyclic orthogonal local systems associated to graph cocycles without self-loops. The 2-loop term of their invariants is associated with the Theta graph. Their invariants can be defined when a cohomological condition holds.

Cattaneo--Shimizu gave a refinement of the 2-loop term of Bott--Cattaneo invariants by removing this cohomological condition, their 2-loop term is associated with a linear combination of the Theta graph and the dumbbell graph that is the only 2-loop trivalent graph with self-loops.

In this talk, when an acyclic local system is given by the adjoint representation of a semi-simple Lie group composed with a representation of the fundamental group of a closed 3-manifold, we show that the associated integral of dumbbell graph can be vanished by a cohomological reason. Based on this idea, we construct a theory of graph complexes and cocycles, so that higher-loop invariants can be defined using both the graph cocycles without self-loop, as by Bott--Cattaneo, and with self-loops, as by Cattaneo--Shimizu. As a consequence, we prove that the generating series from 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-loop. This talk is based on joint work with Bingxiao Liu.