Weak Field Expansion of Gravity: Graphs, Matrices and Topology

TL;DR

This paper develops graphical, matrix, and topological methods to analyze weak field expansions in gravity, focusing on constructing and classifying SO(n) invariants for classical and quantum gravity perturbations.

Contribution

It introduces a comprehensive framework combining graphs, matrices, and topology to systematically analyze invariants in weak field gravity expansions, including applications to Weyl anomalies.

Findings

Constructed global SO(n) invariants systematically.

Applied methods to analyze independence of invariants.

Explored implications for Weyl anomalies in various dimensions.

Abstract

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 $(\pl)^d h_\ab$, which generally appears in the weak field expansion around the flat space: $g_\mn=\del_\mn+h_\mn$. 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 $\pl\pl h-, {and} (\pl\pl h)^2-$ 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).