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Correlation functions for M^N/S_N orbifolds

Oleg Lunin, Samir D. Mathur

TL;DR

The paper develops a covering-space Liouville-action framework to compute twist-operator correlators in bosonic M^N/S_N CFTs, recasting correlation functions as partition functions on Riemann surfaces with genus-controlled contributions. It establishes universal genus-zero results that depend only on the central charge c, obtaining the twist-operator dimensions and, in certain cases, the fusion coefficients via explicit g=0 maps and Liouville actions; higher-genus contributions are shown to be suppressed at large N. The analysis reveals a close parallel between the genus-zero fusion rules and SU(2) WZW fusion rules, and it uncovers a stringy exclusion principle-type suppression in 3-point couplings as twist orders grow toward √N. The work provides a robust framework for comparing orbifold CFT data to dual descriptions (e.g., D1-D5 systems and AdS3/CFT) and sets the stage for extending the method to supersymmetric cases and more intricate dualities.

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.

Correlation functions for M^N/S_N orbifolds

TL;DR

The paper develops a covering-space Liouville-action framework to compute twist-operator correlators in bosonic M^N/S_N CFTs, recasting correlation functions as partition functions on Riemann surfaces with genus-controlled contributions. It establishes universal genus-zero results that depend only on the central charge c, obtaining the twist-operator dimensions and, in certain cases, the fusion coefficients via explicit g=0 maps and Liouville actions; higher-genus contributions are shown to be suppressed at large N. The analysis reveals a close parallel between the genus-zero fusion rules and SU(2) WZW fusion rules, and it uncovers a stringy exclusion principle-type suppression in 3-point couplings as twist orders grow toward √N. The work provides a robust framework for comparing orbifold CFT data to dual descriptions (e.g., D1-D5 systems and AdS3/CFT) and sets the stage for extending the method to supersymmetric cases and more intricate dualities.

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.

Paper Structure

This paper contains 36 sections, 213 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Computing correlation functions through the Liouville action
  3. The twist operators
  4. Path integral on the covering space
  5. Closing the punctures
  6. The method of calculation
  7. Contributions to $S_L$
  8. The correlator in terms of the Liouville field
  9. The 2-point function
  10. The calculation
  11. 'Universality' of the 2-point function
  12. The map for the 3-point function
  13. Genus of the covering surface
  14. The map for the case $g=0$
  15. The Liouville action for the 3-point function
  16. ...and 21 more sections

Figures (1)

  • Figure 1: The factor (\ref{['FactorToPlot']}) as function of $n$ for $\bar{N}=1000$.