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Universality of Sparse $d>2$ Conformal Field Theory at Large $N$

Alexandre Belin, Jan de Boer, Jorrit Kruthoff, Ben Michel, Edgar Shaghoulian, Milind Shyani

TL;DR

This work addresses the universality of the free energy and the extended Cardy regime for large-$N$ CFT$_d$ on toroidal backgrounds. It establishes modular-invariance constraints on vacuum energy, derives necessary and sufficient vacuum-domination conditions across torus channels, and demonstrates how symmetric product orbifolds can realize universality only when the seed theory has a universal vacuum energy ($\tilde{f}(\mathbf{y})=0$). The analysis shows that, under these conditions, the high-energy density of states follows a higher-dimensional Cardy form with a bulk-gravity-consistent phase structure, including a Hawking-Page-like transition and a holographic match to Einstein gravity. The results extend the 2D modular bootstrap paradigm to $d>2$, and provide explicit orbifold constructions linking CFT data to semiclassical gravity via universal thermodynamics.

Abstract

We derive necessary and sufficient conditions for large $N$ conformal field theories to have a universal free energy and an extended range of validity of the higher-dimensional Cardy formula. These constraints are much tighter than in two dimensions and must be satisfied by any conformal field theory dual to Einstein gravity. We construct and analyze symmetric product orbifold theories on $\mathbb{T}^d$ and show that they only realize the necessary phase structure and extended range of validity if the seed theory is assumed to have a universal vacuum energy.

Universality of Sparse $d>2$ Conformal Field Theory at Large $N$

TL;DR

This work addresses the universality of the free energy and the extended Cardy regime for large- CFT on toroidal backgrounds. It establishes modular-invariance constraints on vacuum energy, derives necessary and sufficient vacuum-domination conditions across torus channels, and demonstrates how symmetric product orbifolds can realize universality only when the seed theory has a universal vacuum energy (). The analysis shows that, under these conditions, the high-energy density of states follows a higher-dimensional Cardy form with a bulk-gravity-consistent phase structure, including a Hawking-Page-like transition and a holographic match to Einstein gravity. The results extend the 2D modular bootstrap paradigm to , and provide explicit orbifold constructions linking CFT data to semiclassical gravity via universal thermodynamics.

Abstract

We derive necessary and sufficient conditions for large conformal field theories to have a universal free energy and an extended range of validity of the higher-dimensional Cardy formula. These constraints are much tighter than in two dimensions and must be satisfied by any conformal field theory dual to Einstein gravity. We construct and analyze symmetric product orbifold theories on and show that they only realize the necessary phase structure and extended range of validity if the seed theory is assumed to have a universal vacuum energy.

Paper Structure

This paper contains 33 sections, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generalities of CFT$_{d}$
  3. Review of higher-dimensional Cardy formulas
  4. Review of vacuum energies in CFT
  5. Normalization of vacuum energy
  6. Subextensive corrections to the vacuum energy
  7. Phase structure of toroidally compactified AdS gravity
  8. Thermal phase structure
  9. Quantum phase structure
  10. Necessary and sufficient conditions for universality
  11. Modular constraints on vacuum energy
  12. Necessary and sufficient conditions
  13. Sparseness constraints without assuming $\tilde{f}(\mathbf{y})=0$
  14. Sparseness constraints assuming $\tilde{f}(\mathbf{y})=0$
  15. Symmetric Product Orbifolds in $d>2$
  16. ...and 18 more sections

Figures (1)

  • Figure 1: The functional form of $f(L_1/L_2)$ in the vacuum energy (defined in \ref{['casimirf']}) of a free boson in $2+1$ dimensions on a two-torus $\mathbb{T}^2$ with sides $L_1$ and $L_2$. As can be seen in the plot, $f(L_1/L_2)$ is positive and monotonically increasing.