Four-point functions with fractional R-symmetry excitations in the D1-D5 CFT

TL;DR

This paper analyzes four-point correlation functions with fractional R-symmetry excitations in the D1-D5 CFT, deriving explicit formulas and exploring their relation to covering maps, Hurwitz blocks, and fusion rules.

Contribution

It provides explicit formulas for four-point functions involving fractional excitations and connects these to covering maps and permutation classes in the D1-D5 CFT.

Findings

Derived explicit formulas for specific four-point functions.

Showed how fractional excitations lift to the covering surface.

Linked correlation functions to Hurwitz blocks and fusion rules.

Abstract

We study correlation functions with fractional-mode excitations of the R-symmetry currents in D1-D5 CFT. We show how fractional-mode excitations lift to the covering surface associated with correlation functions as a specific sum of integer-mode excitations, with coefficients that can be determined exactly from the covering map in terms of Bell polynomials. We consider the four-point functions of fractional excitations of two chiral/anti-chiral NS fields, Ramond ground states and the twist-two scalar modulus deformation operator that drives the CFT away from the free point. We derive explicit formulas for classes of these functions with twist structures $(n)$-$(2)$-$(2)$-$(n)$ and $(n_1)(n_2)$-$(2)$-$(2)$-$(n_1)(n_2)$, the latter involving double-cycle fields. The final answer for the four-point functions always depends only on the lift of the base-space cross-ratio. We discuss how this relates to Hurwitz blocks associated with different conjugacy classes of permutations, the corresponding OPE channels and fusion rules.