The space-time operator product expansion in string theory duals of field theories

TL;DR

This paper analyzes the operator product expansion in field theories with string duals, showing how worldsheet correlation functions encode the field theory OPE, and confirming the consistency of the duality in the space-time limit.

Contribution

It demonstrates how the worldsheet OPE reproduces the field theory OPE in the dual string description using saddle point analysis, applicable to general weakly coupled string duals.

Findings

Worldsheet OPE contains only non-physical operators, which are resummed to physical contributions.

The saddle point approximation captures the dominant contributions matching field theory expectations.

The approach applies to any local, conformal field theory with a weakly coupled string dual.

Abstract

We study the operator product expansion (OPE) limit of correlation functions in field theories which possess string theory duals, from the point of view of the string worldsheet. We show how the interesting ("single-trace") terms in the OPE of the field theory arise in this limit from the OPE of the worldsheet theory of the string dual, using a dominant saddle point which appears in computations of worldsheet correlation functions in the space-time OPE limit. The worldsheet OPE generically contains only non-physical operators, but all the non-physical contributions are resummed by the saddle point to a contribution similar to that of a physical operator, which exactly matches the field theory expectations. We verify that the OPE limit of the worldsheet theory does not have any other contributions to the OPE limit of space-time correlation functions. Our discussion is completely general and applies to any local field theory (conformal at high energies) that has a weakly coupled string theory dual (with arbitrary curvature). As a first application, we compare our results to a proposal of R. Gopakumar for the string theory dual of free gauge theories.