From Free Fields to $AdS$ -- III

TL;DR

This paper develops a concrete realization of open-closed string duality by reconstructing closed string worldsheets from field theory skeleton graphs using Strebel differentials, linking Schwinger parameters to string moduli.

Contribution

It proposes a method to derive closed string worldsheets from field theory graphs via Strebel differentials, connecting Schwinger parameters to string moduli and illustrating the emergence of string correlators.

Findings

Reconstruction of closed string worldsheets from field theory graphs.

Identification of Schwinger parameters with Strebel differential edge lengths.

Extraction of worldsheet OPE directly from field theory expressions.

Abstract

In previous work we have shown that large $N$ field theory amplitudes, in Schwinger parametrised form, can be organised into integrals over the stringy moduli space ${\cal M}_{g,n}\times R_{+}^n$. Here we flesh this out into a concrete implementation of open-closed string duality. In particular, we propose that the closed string worldsheet is reconstructed from the unique Strebel quadratic differential that can be associated to (the dual of) a field theory skeleton graph. We are led, in the process, to identify the inverse Schwinger proper times ($\s_i={1\over \t_i}$) with the lengths of edges of the critical graph of the Strebel differential. Kontsevich's matrix model derivation of the intersection numbers in moduli space provides a concrete example of this identification. It also exhibits how closed string correlators very naturally emerge from the Schwinger parameter integrals. Finally, to illustrate the utility of our approach to open-closed string duality, we outline a method by which a worldsheet OPE can be directly extracted from the field theory expressions. Limits of the Strebel differential for the four punctured sphere play a key role.