From Free Fields to AdS -- II

TL;DR

This paper demonstrates how free gauge field theory correlators can be reformulated as integrals over moduli spaces of Riemann surfaces, providing evidence for their dual string theory description, especially in the context of AdS/CFT correspondence.

Contribution

It explicitly relates free field theory correlators to integrals over moduli spaces of Riemann surfaces, supporting the emergence of string theory from gauge theories.

Findings

Planar correlators correspond to moduli space of genus zero Riemann surfaces with holes.

The integrand is compatible with a string theory on AdS.

Method generalizes to higher genus diagrams.

Abstract

We continue with the program of hep-th/0308184 to implement open-closed string duality on free gauge field theory (in the large $N$ limit). In this paper we consider correlators such as $\la \prod_{i=1}^n \Tr\Phi^{J_i}(x_i)\ra$. The Schwinger parametrisation of this $n$-point function exhibits a partial gluing up into a set of basic skeleton graphs. We argue that the moduli space of the planar skeleton graphs is exactly the same as the moduli space of genus zero Riemann surfaces with $n$ holes. In other words, we can explicitly rewrite the $n$-point (planar) free field correlator as an integral over the moduli space of a sphere with $n$ holes. A preliminary study of the integrand also indicates compatibility with a string theory on $AdS$. The details of our argument are quite insensitive to the specific form of the operators and generalise to diagrams of higher genus as well. We take this as evidence of the field theory's ability to reorganise itself into a string theory.