Covering space maps for $n$-point functions with three long twists

TL;DR

This paper constructs explicit covering space maps for three long twist operators in symmetric orbifold CFTs, analyzing their limits and providing closed-form expressions for specific cases, advancing understanding of multi-twist correlation functions.

Contribution

It introduces a general class of covering maps involving three arbitrary-length twists and any number of twist-2 insertions, with explicit formulas and analysis of OPE limits.

Findings

Derived covering maps as ratios of Jacobi polynomials with symmetry properties.

Analyzed OPE limits as algebraic varieties in complex projective space.

Provided closed-form expressions for four- and five-point functions of bare twists.

Abstract

We consider correlation functions in symmetric product orbifold CFTs on the sphere, focusing on the case where all operators are single-cycle twists, and the covering surface is also a sphere. We directly construct the general class of covering space maps where there are three twists of arbitrary lengths, along with any number of twist-2 insertions. These are written as a ratio of sums of Jacobi polynomials with $\Delta N+1$ coefficients $b_N$. These coefficients have a scaling symmetry $b_N\rightarrow \lambda b_N$, making them naturally valued in $\mathbb{CP}^{\Delta N}$. We explore limits where various ramified points on the cover approach each other, which are understood as crossing channel specific OPE limits, and find that these limits are defined by algebraic varieties of $\mathbb{CP}^{\Delta N}$. We compute the expressions needed to calculate the group element representative correlation functions for bare twists. Specializing to the cases $\Delta N=1,2$, we find closed form for these expressions which define four- and five-point functions of bare twists.