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The limit of N=(2,2) superconformal minimal models

Stefan Fredenhagen, Cosimo Restuccia, Rui Sun

TL;DR

This work analyzes the $c\to3$ limit of $N=(2,2)$ superconformal minimal models, showing the limit theory is non-rational with a continuum of charged primary fields $\Phi_{q,n}$ and a discrete-Ramond sector, and providing explicit spectrum and correlation data. By taking $k\to\infty$ and averaging over appropriate quantum numbers, the authors derive finite two- and three-point functions on the sphere, along with disc one-point functions for two families of A-type boundaries and continuous boundary conditions. A second, distinct chargeless sector generated by fields with $q=0$, denoted $\tilde{\Phi}_p$, is shown to decouple from the charged sector, suggesting the existence of two separate limiting theories. The results support the existence of a well-behaved non-rational limit and point toward potential connections with free or Liouville-type theories and higher-spin holography, while leaving open the issue of crossing symmetry and a full geometric interpretation. This advances understanding of limit theories in AdS/CFT contexts and provides concrete CFT data for the charged sector at $c=3$.

Abstract

The limit of families of two-dimensional conformal field theories has recently attracted attention in the context of AdS/CFT dualities. In our work we analyse the limit of N=(2,2) superconformal minimal models when the central charge approaches c=3. The limiting theory is a non-rational N=(2,2) superconformal theory, in which there is a continuum of chiral primary fields. We determine the spectrum of the theory, the three-point functions on the sphere, and the disc one-point functions.

The limit of N=(2,2) superconformal minimal models

TL;DR

This work analyzes the limit of superconformal minimal models, showing the limit theory is non-rational with a continuum of charged primary fields and a discrete-Ramond sector, and providing explicit spectrum and correlation data. By taking and averaging over appropriate quantum numbers, the authors derive finite two- and three-point functions on the sphere, along with disc one-point functions for two families of A-type boundaries and continuous boundary conditions. A second, distinct chargeless sector generated by fields with , denoted , is shown to decouple from the charged sector, suggesting the existence of two separate limiting theories. The results support the existence of a well-behaved non-rational limit and point toward potential connections with free or Liouville-type theories and higher-spin holography, while leaving open the issue of crossing symmetry and a full geometric interpretation. This advances understanding of limit theories in AdS/CFT contexts and provides concrete CFT data for the charged sector at .

Abstract

The limit of families of two-dimensional conformal field theories has recently attracted attention in the context of AdS/CFT dualities. In our work we analyse the limit of N=(2,2) superconformal minimal models when the central charge approaches c=3. The limiting theory is a non-rational N=(2,2) superconformal theory, in which there is a continuum of chiral primary fields. We determine the spectrum of the theory, the three-point functions on the sphere, and the disc one-point functions.

Paper Structure

This paper contains 25 sections, 162 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The spectrum
  3. Minimal models
  4. Taking the limit
  5. Fields and correlators
  6. Three-point functions
  7. Correlators of primary fields
  8. Correlators involving superdescendants
  9. Boundary conditions and one-point functions
  10. Discrete boundary conditions
  11. Continuous boundary conditions
  12. Fields of charge zero
  13. First ansatz: average of approximately chargeless fields
  14. Second ansatz starting from exactly chargeless fields
  15. Boundary conditions
  16. ...and 10 more sections

Figures (3)

  • Figure 1: Behaviour of the spectrum of primary fields in the Neveu-Schwarz sector for large levels $k$: when one plots the values of the conformal weight $h$ and of the $U (1)$ charge $q$ as dots in the $h$-$q$-plane, one observes that the points assemble along straight lines starting from the origin. Notice that we only plotted the points corresponding to positive charge $q$ (the negative charged part is just the mirror picture) and we truncated the conformal weights by $h\leq 3$.
  • Figure 2: An illustration of the function $f$ defined in \ref{['def_f']}.
  • Figure 3: Wigner approximation.