Multi-trace Correlators from Permutations as Moduli Space

TL;DR

This paper develops a group-theoretic approach to compute multi-trace correlators in gauge theories, revealing connections to string interactions and defining a new moduli space related to skeleton-reduced graphs.

Contribution

It introduces formulae for n-point functions valid at all orders in 1/N_c and defines a novel moduli space from skeleton-reduced graphs in gauge theory.

Findings

Sum over Feynman graphs as a topological partition function

Skeleton reduction generates connected ribbon graphs

Defined a new moduli space stratified by genus

Abstract

We study the $n$-point functions of scalar multi-trace operators in the $U(N_c)$ gauge theory with adjacent scalars, such as ${\cal N}=4$ super Yang-Mills, at tree-level by using finite group methods. We derive a set of formulae of the general $n$-point functions, valid for general $n$ and to all orders of $1/N_c$. In one formula, the sum over Feynman graphs becomes a topological partition function on $\Sigma_{0,n}$ with a discrete gauge group, which resembles closed string interactions. In another formula, a new skeleton reduction of Feynman graphs generates connected ribbon graphs, which resembles open string interaction. We define the moduli space ${\cal M}_{g,n}^{\rm gauge}$ from the space of skeleton-reduced graphs in the connected $n$-point function of gauge theory. This moduli space is a proper subset of ${\cal M}_{g,n}$ stratified by the genus, and its top component gives a simple triangulation of $\Sigma_{g,n}$.