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Phase transitions in symmetric orbifold CFTs and universality

Christoph A. Keller

TL;DR

The paper shows that the thermodynamics of holographic duals to AdS$_3$ gravity, when modeled by symmetric orbifolds of general 2D CFTs, exhibits universal behavior in the large $N$ limit. By leveraging modular invariance, Cardy growth, and Hecke operators, it derives universal state counts $ ilde{d}_ ty(m,ar{m}) o e^{4\,\pi\sqrt{m\bar{m}}}$, a universal Hagedorn temperature at $eta=2\,\pi$, and a first-order Hawking-Page transition at $T_H= rac{1}{2\pi}$, with a phase diagram for the rescaled free energy $f( au)$ that is theory-independent at leading order. The analysis extends to holomorphic sectors via the elliptic genus and introduces order parameters that quantify contributions from twisted sectors, demonstrating universal critical behavior with exponent 1 for certain observables. It further discusses moving away from the orbifold point and outlines criteria under which general CFT families can share the same universal phase structure, with implications for classifying CFTs that admit AdS$_3$ gravity duals and for extremal/CFT constructions. Overall, the work links microscopic state counting, modular properties, and holographic thermodynamics to impose universal constraints on CFTs compatible with gravity duals.

Abstract

Since many thermodynamic properties of black holes are universal, the thermodynamics of their holographic duals should be universal too. We show how this universality is exhibited in the example of symmetric orbifolds of general two dimensional CFTs. We discuss the free energies and phase diagrams of such theories and show that they are indeed universal in the large N limit. We also comment on the implications of our results for the classification of CFTs that can have an interpretation as holographic duals to gravity theories on AdS(3).

Phase transitions in symmetric orbifold CFTs and universality

TL;DR

The paper shows that the thermodynamics of holographic duals to AdS gravity, when modeled by symmetric orbifolds of general 2D CFTs, exhibits universal behavior in the large limit. By leveraging modular invariance, Cardy growth, and Hecke operators, it derives universal state counts , a universal Hagedorn temperature at , and a first-order Hawking-Page transition at , with a phase diagram for the rescaled free energy that is theory-independent at leading order. The analysis extends to holomorphic sectors via the elliptic genus and introduces order parameters that quantify contributions from twisted sectors, demonstrating universal critical behavior with exponent 1 for certain observables. It further discusses moving away from the orbifold point and outlines criteria under which general CFT families can share the same universal phase structure, with implications for classifying CFTs that admit AdS gravity duals and for extremal/CFT constructions. Overall, the work links microscopic state counting, modular properties, and holographic thermodynamics to impose universal constraints on CFTs compatible with gravity duals.

Abstract

Since many thermodynamic properties of black holes are universal, the thermodynamics of their holographic duals should be universal too. We show how this universality is exhibited in the example of symmetric orbifolds of general two dimensional CFTs. We discuss the free energies and phase diagrams of such theories and show that they are indeed universal in the large N limit. We also comment on the implications of our results for the classification of CFTs that can have an interpretation as holographic duals to gravity theories on AdS(3).

Paper Structure

This paper contains 19 sections, 89 equations, 3 figures.

Table of Contents

  1. Introduction and overview
  2. Introduction
  3. Some general remarks on two dimensional CFTs
  4. Hagedorn transitions and Hawking-Page transitions
  5. The phase diagram
  6. Universal properties of large $N$ symmetric orbifolds
  7. Symmetric orbifolds
  8. Universal properties
  9. $SU(N)$ gauge theories: comparison to $d>2$
  10. Elliptic genus
  11. Order parameters
  12. General families
  13. Moving away from the orbifold point
  14. Holomorphic partition functions
  15. Appendix
  16. ...and 4 more sections

Figures (3)

  • Figure 1: A generic CFT (gray line) has no phase transition and is universal only for very high and very low temperatures. The symmetric orbifold for $N=\infty$ (black line) is universal at all temperatures and has a first order phase transition at $T_H=\frac{1}{2\pi}$.
  • Figure 2: To the left, the tessellation of the upper half plane into fundamental regions of $SL(2,{\mathbb{Z}})$. To the right, the proposed phase diagram, taking into account the invariance under $T$. (Figures taken from Maloney:2007ud.)
  • Figure 3: The free boson at self-dual radius: $\langle P_{\textrm{utw}}\rangle_\infty$ as a function of $T$ (left), $\langle P_{(n)}\rangle_\infty$ for $n=2,3,4$ (right).