Correlation functions for M^N/S_N orbifolds

TL;DR

This paper introduces a method to compute correlation functions of twist operators in orbifold conformal field theories, expressing them through partition functions on Riemann surfaces, revealing universal and genus-dependent contributions.

Contribution

It develops a universal covering space approach for correlation functions in M^N/S_N orbifolds, connecting genus expansion with 1/N expansion and analyzing fusion rules and stringy effects.

Findings

Correlation functions expressed via Riemann surface partition functions.

Genus zero contributions depend only on the central charge.

3-point functions vanish unless fusion rules are satisfied.

Abstract

We develop a method for computing correlation functions of twist operators in the bosonic 2-d CFT arising from orbifolds M^N/S_N, where M is an arbitrary manifold. The path integral with twist operators is replaced by a path integral on a covering space with no operator insertions. Thus, even though the CFT is defined on the sphere, the correlators are expressed in terms of partition functions on Riemann surfaces with a finite range of genus g. For large N, this genus expansion coincides with a 1/N expansion. The contribution from the covering space of genus zero is `universal' in the sense that it depends only on the central charge of the CFT. For 3-point functions we give an explicit form for the contribution from the sphere, and for the 4-point function we do an example which has genus zero and genus one contributions. The condition for the genus zero contribution to the 3-point functions to be non--vanishing is similar to the fusion rules for an SU(2) WZW model. We observe that the 3-point coupling becomes small compared to its large N limit when the orders of the twist operators become comparable to the square root of N - this is a manifestation of the stringy exclusion principle.