Comments on the $S_N$ orbifold CFT in the large $N$-limit

TL;DR

This paper reviews the symmetric orbifold conformal field theory, focusing on its operator content, large N limit, and methods for computing correlation functions, including four-point functions and their algebraic structures.

Contribution

It provides a coherent synthesis of existing results and introduces a systematic approach for calculating correlation functions involving orbifold Virasoro generators in the large N limit.

Findings

Operator content of the symmetric orbifold CFT clarified.

Selection rules for twist operator OPEs derived.

General four-point functions expressed via algebraic equations resembling scattering equations.

Abstract

We elaborate on various aspects of the conformal field theory of the symmetric orbifold. We collect various results that have appeared in the literature, and we present a coherent picture of the operator content of this CFT, relying on the orbifold extension of the Virasoro algebra. We then focus on the large $N$-limit of this theory, discuss the OPE of two twist operators, and find various selection rules. We review how to calculate four-point functions of twist operators, and we write down the most general four-point function in the covering space for large $N$. We show that it depends on some functions that obey a set of algebraic equations, that resemble the scattering equations. Finally, we provide a recipe on how to calculate correlation functions with insertions of the orbifold Virasoro generators.