Fractional Conformal Descendants and Correlators in General 2D $S_N$ Orbifold CFTs at Large $N$

TL;DR

This paper develops a method to compute correlation functions of descendant operators in large N symmetric orbifold CFTs, showing that these can be expressed in terms of ancestor correlators and are independent of covering space details.

Contribution

It introduces a universal approach to relate descendant correlators to ancestor correlators in symmetric orbifold CFTs, independent of covering space data and seed CFT details.

Findings

Descendant correlators can be expressed solely in terms of ancestor correlators.

Covering space data cancels out in the calculation of descendant correlators.

Results are universal across different seed CFTs and independent of covering space details.

Abstract

We consider correlation functions in symmetric product ($S_N$) orbifold CFTs at large $N$ with arbitrary seed CFT. Specifically, we consider correlators of descendant operators constructed using both the full Virasoro generators $L_{m}$ and fractional Virasoro generators $\ell_{m/n_i}$. Using covering space techniques, we show that correlators of descendants may be written entirely in terms of correlators of ancestors, and further that the appropriate set of ancestors are those operators that lift to conformal primaries on the cover. We argue that the covering space data should cancel out in such calculations. To back this claim, we provide some example calculations by considering a three-point function of the form (4-cycle)-(2-cycle)-(5-cycle) that lifts to a three-point function of arbitrary primaries on the cover, and descendants thereof. In these examples we show that while the covering space is used for the calculation, the final descent relations do not depend on covering space data, nor on the details of which seed CFT is used to construct the orbifold, making these results universal.