Larger Twists and Higher $n$-Point Functions with Fractional Conformal Descendants in $S_N$ Orbifold CFTs at Large $N$

TL;DR

This paper develops universal descent relations for correlation functions involving fractional conformal descendants in large $N$ symmetric orbifold CFTs, independent of seed CFT details and covering space data, using covering space techniques.

Contribution

It introduces universal descent relations for $S_N$ orbifold CFTs involving fractional Virasoro generators, applicable to arbitrary primaries and descendants, independent of seed CFT and covering space specifics.

Findings

Descent relations for three-point functions are universal and independent of seed CFT.

Four-point function descent relations depend only on base space data and a single parameter.

The methods extend to arbitrary primaries and descendants in large $N$ orbifold CFTs.

Abstract

We consider correlation functions in symmetric product ($S_N$) orbifold CFTs at large $N$ with arbitrary seed CFT, expanding on our earlier work arXiv:2211.04633 . Using covering space techniques, we calculate descent relations using fractional Virasoro generators in correlators, writing correlators of descendants in terms of correlators of ancestors. We first consider the case three-point functions of the form ($m$-cycle)-($n$-cycle)-($q$-cycle) which lift to arbitrary primaries on the cover, and descendants thereof. In these examples we show that the final descent relations do not depend on the covering space data, nor on the specific details of the seed CFT. This makes these descent relations universal in all $S_N$ orbifold CFTs. Next, we explore four-point functions of the form (2-cycle)-($n$-cycle)-($n$-cycle)-(2-cycle) which lift to arbitrary primaries on the cover, and descendants thereof. In such cases a single parameter in the map $s$ parameterizes both the base space cross ratio $\zeta_z$ and the covering space cross ratio $\zeta_t$. We find that the descent relations for the four point functions depend only on base space data and the parameter $s$, which we argue is tantamount to writing the descent relations in terms of the base space data and the base space cross ratio. These descent relations again do not depend on the covering space data, nor the specifics of the seed CFT, making these universal as well.