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Correlators in W_N Minimal Model Revisited

Chi-Ming Chang, Xi Yin

TL;DR

This work analyzes sphere and torus correlators in the $W_N$ minimal model at large $N$, revealing large-$N$ factorization of a broad class of sphere three-point functions and enabling identification of elementary bulk particles and their bound states. It develops a Coulomb gas/affine Toda framework to compute sphere three- and four-point functions, and presents an exact torus two-point function via conformal blocks with analytic continuation to Lorentzian thermal correlators. The results clarify the bulk spectrum and interactions in the higher-spin holographic dual, including both massive and light states and their bound-state structure, and provide tools to probe bulk thermodynamics through boundary correlators. Together, these findings connect exact CFT data to bulk particle content and dynamics in $AdS_3$ higher-spin gravity.

Abstract

In this paper, we study a class of sphere and torus correlation functions in the W_N minimal model. In particular, we show that a large class of exact sphere three-point functions of W_N primaries, derived using affine Toda theory, exhibit large N factorization. This allows us to identify some fundamental particles and their bound states in the holographic dual, including light states. We also derive the torus two-point function of basic primaries, by directly constructing the relevant conformal blocks. The result can then be analytically continued to give the Lorentzian thermal two-point functions.

Correlators in W_N Minimal Model Revisited

TL;DR

This work analyzes sphere and torus correlators in the minimal model at large , revealing large- factorization of a broad class of sphere three-point functions and enabling identification of elementary bulk particles and their bound states. It develops a Coulomb gas/affine Toda framework to compute sphere three- and four-point functions, and presents an exact torus two-point function via conformal blocks with analytic continuation to Lorentzian thermal correlators. The results clarify the bulk spectrum and interactions in the higher-spin holographic dual, including both massive and light states and their bound-state structure, and provide tools to probe bulk thermodynamics through boundary correlators. Together, these findings connect exact CFT data to bulk particle content and dynamics in higher-spin gravity.

Abstract

In this paper, we study a class of sphere and torus correlation functions in the W_N minimal model. In particular, we show that a large class of exact sphere three-point functions of W_N primaries, derived using affine Toda theory, exhibit large N factorization. This allows us to identify some fundamental particles and their bound states in the holographic dual, including light states. We also derive the torus two-point function of basic primaries, by directly constructing the relevant conformal blocks. The result can then be analytically continued to give the Lorentzian thermal two-point functions.

Paper Structure

This paper contains 30 sections, 229 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and conventions for the $W_N$ minimal model
  3. Coulomb gas formalism
  4. Rewriting free boson characters
  5. $W_N$ characters and partition function
  6. Coulomb gas representation of vertex operators and screening charge
  7. Sphere three-point function
  8. Two point function and normalization
  9. Extracting correlation functions from affine Toda theory
  10. Large $N$ factorization
  11. Massive scalars and their bound states
  12. Light states
  13. Light states bound to massive scalars
  14. Sphere four-point function
  15. Screening charges
  16. ...and 15 more sections

Figures (3)

  • Figure 1: The modulus of the two-point function $\langle {\cal O}(0,t) {\cal O}(0,0)\rangle_\beta$ (normalized to 1 at $t=0$) at inverse temperature $\beta=0.3$ is plotted at integer values of time $t=0,1,2,\cdots,10$. The results for Virasoro minimal models with $k=1,2,\cdots,14$ are shown in colors ranging from red to green and then to blue. For each $k$, the values of the modulus of the two-point function at integer times before PoincarĂ© recurrence are connected with straight lines, for the purpose of illustration only.
  • Figure 2: The modulus of the two-point function $\langle {\cal O}(0,t) {\cal O}(0,0)\rangle_\beta$ (normalized to 1 at $t=0$) at inverse temperature $\beta=0.3$ is plotted at integer values of time $t=0,1,2,\cdots,40$, in Virasoro minimal models of $k=10,20,30$ (shown in red, green, and blue).
  • Figure 3: Plots of the modulus of the two-point function $\langle {\cal O}(0,t) {\cal O}(0,0)\rangle_\beta$ (normalized to 1 at $t=0$) in the $k=4$ Virasoro minimal model, at integer values of time $t=0,1,\cdots,4$ (connected with fictitious straight lines for illustration only), at different values of the temperature $T=1/\beta$. $T$ ranges from $\sim 0.05$ to $20$ (depicted in colors ranging from blue to red), evenly spaced in logarithmic scale.