From Spacetime to Worldsheet: Four point correlators

TL;DR

The paper tackles the problem of obtaining a worldsheet description for strings in $AdS_5\times S^5$ by mapping large-$N$ four-point field theory amplitudes, via the Schwinger parametrization, onto the closed-string moduli space using Strebel differentials. It develops a general framework for relating Schwinger parameters to the cross ratio $\eta$ on the four-punctured sphere and demonstrates that for the $Y$-diagram contribution one can derive an explicit, crossing-symmetric, locally consistent candidate worldsheet four-point function $G^{\{J_i\}}_{\{x_i\}}(\eta,\bar{\eta})$. A perturbative expansion around the $Y$ diagram is formulated, allowing systematic access to more general correlators through controlled corrections in the Strebel lengths and the elliptic modulus $k$. The approach yields concrete expressions that resemble local worldsheet correlators and opens avenues to link the worldsheet theory to spin-field structures and Ising-model-like blocks, potentially illuminating the worldsheet CFT for the AdS/CFT dual of free ${\cal N}=4$ Yang-Mills theory.

Abstract

The Schwinger representation gives a systematic procedure for recasting large N field theory amplitudes as integrals over closed string moduli space. This procedure has recently been applied to a class of free field four point functions by Aharony, Komargodski and Razamat, to study the leading terms in the putative worldsheet OPE. Here we observe that the dictionary between Schwinger parameters and the cross ratio of the four punctured sphere actually yields an explicit expression for the full worldsheet four point correlator in many such cases. This expression has a suggestive form and obeys various properties, such as crossing symmetry and mutual locality, expected of a correlator in a two dimensional CFT. Therefore one may take this to be a candidate four point function in a worldsheet description of closed strings on highly curved AdS_5 \times S^5. The general framework, that we develop for computing the relevant Strebel differentials, also admits a systematic perturbation expansion which would be useful for studying more general four point correlators.