Weak Field Expansion of Gravity : Graphs, Matrices and Topology

We present some approaches to the perturbative analysis of the classical and quantum gravity. First we introduce a graphical representation for a global SO(n) tensor (∂)hαβ , which generally appears in the weak field expansion around the flat space: gμν = δμν +hμν . Making use of this representation, we explain 1) Generating function of graphs (Feynman diagram approach), 2) Adjacency matrix (Matrix approach), 3) Graphical classification in terms of ”topology indices” (Topology approach), 4) The Young tableau ( Symmetric group approach). We systematically construct the global SO(n) invariants. How to show the independence and completeness of those invariants is the main theme. We explain it taking simple examples of ∂∂h−, and (∂∂h)2− invariants in the text. The results are applied to the analysis of the independence of general invariants and (the leading order of ) the Weyl anomalies of scalar-gravity theories in ”diverse” dimensions (2,4,6,8,10 dimensions). •