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Detachable circles and temperature-inversion dualities for CFT$_d$

Gary T. Horowitz, Edgar Shaghoulian

TL;DR

The authors establish a Weyl-based temperature-inversion duality for CFT_d by detaching a circle in S^1 × S^{d-1} to obtain S^1 × H^{d-1}/Z, enabling mapping of thermal physics between frames and revealing Hawking-Page-type transitions on hyperbolic spaces. They compute thermodynamics in dual frames, construct smooth bulk solutions with conical boundaries, and discuss implications for the Eguchi-Kawai mechanism on curved manifolds. The work extends to higher dimensions, angular momentum, and detaching circles from other manifolds, outlining a rich set of generalizations and open questions about modular-like structures and phase structure in curved backgrounds. Together, these results provide a novel framework for finite-temperature CFTs on curved spaces and suggest new avenues for holographic dualities and large-N volume independence.

Abstract

We use a Weyl transformation between $S^1 \times S^{d-1}$ and $S^1 \times \mathcal{H}^{d-1}/\mathbb{Z}$ to relate a conformal field theory at arbitrary temperature on $S^{d-1}$ to itself at the inverse temperature on $\mathcal{H}^{d-1}/\mathbb{Z}$. We use this equivalence to deduce a confining phase transition at finite temperature for large-$N$ gauge theories on hyperbolic space. In the context of gauge/gravity duality, this equivalence provides new examples of smooth bulk solutions which asymptote to conically singular geometries at the AdS boundary. We also discuss implications for the Eguchi-Kawai mechanism and a high-temperature/low-temperature duality on $S^{d-1}$.

Detachable circles and temperature-inversion dualities for CFT$_d$

TL;DR

The authors establish a Weyl-based temperature-inversion duality for CFT_d by detaching a circle in S^1 × S^{d-1} to obtain S^1 × H^{d-1}/Z, enabling mapping of thermal physics between frames and revealing Hawking-Page-type transitions on hyperbolic spaces. They compute thermodynamics in dual frames, construct smooth bulk solutions with conical boundaries, and discuss implications for the Eguchi-Kawai mechanism on curved manifolds. The work extends to higher dimensions, angular momentum, and detaching circles from other manifolds, outlining a rich set of generalizations and open questions about modular-like structures and phase structure in curved backgrounds. Together, these results provide a novel framework for finite-temperature CFTs on curved spaces and suggest new avenues for holographic dualities and large-N volume independence.

Abstract

We use a Weyl transformation between and to relate a conformal field theory at arbitrary temperature on to itself at the inverse temperature on . We use this equivalence to deduce a confining phase transition at finite temperature for large- gauge theories on hyperbolic space. In the context of gauge/gravity duality, this equivalence provides new examples of smooth bulk solutions which asymptote to conically singular geometries at the AdS boundary. We also discuss implications for the Eguchi-Kawai mechanism and a high-temperature/low-temperature duality on .

Paper Structure

This paper contains 13 sections, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The gravitational story
  3. Free energy, total energy, and entropy
  4. A thermal phase transition on $S^1\times\mathcal{H}^{d-1}/\mathbb{Z}$
  5. Constructing smooth bulk solutions with conical boundaries
  6. Free energy of hyperbolic black hole in spherical slices
  7. Implications for the Eguchi-Kawai mechanism on curved manifolds
  8. Generalizations
  9. Higher dimensions
  10. Detaching circles from other manifolds
  11. Adding angular momentum
  12. Returning to $S^{d-1}$
  13. Open questions

Figures (2)

  • Figure 1: Left: hyperbolic slicing of spherical black hole with $r_h =3$. Right: spherical slicing of hyperbolic black hole with $\rho_h = 2$.
  • Figure 2: A conjectured phase diagram on $S^1 \times \mathcal{H}^{d-1}/\mathbb{Z}$. The solid curve meets the dashed curve at the calculable transition point $L_\phi = 2\pi$, $L_\chi = {2\pi}/(d-1)$.