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Conformal field theories in anti-de Sitter space

Ofer Aharony, Donald Marolf, Mukund Rangamani

TL;DR

<3-5 sentence high-level summary>The paper studies conformal field theories on $AdS_{4}$, with a focus on ${ m cb N}=4$ SYM, and shows that boundary conditions for gauge fields crucially control the dynamics and phase structure, including a weak-coupling large-$N$ Hagedorn transition under Neumann data and a richer strong-coupling picture accessible via S-duality and holography for special boundary conditions. It identifies classes of supersymmetric boundary conditions, especially quotient constructions, that admit controlled holographic duals—often realized as $AdS_{5}$ geometries with a $bZ_2$ quotient—leading to $O(1)$ light degrees of freedom and clear finite-temperature phase transitions analogous to deconfinement. The results illuminate how boundary data and dualities constrain the spectrum, thermodynamics, and possible gravity duals of ${ m cb N}=4$ SYM on $AdS_{4}$, and point to rich open directions such as less-supersymmetric boundaries and indices for these theories. These findings advance understanding of AdS/CFT in curved boundaries and the role of boundary conditions in holographic duals of gauge theories.

Abstract

In this paper we discuss the dynamics of conformal field theories on anti-de Sitter space, focussing on the special case of the N=4 supersymmetric Yang-Mills theory on AdS_4. We argue that the choice of boundary conditions, in particular for the gauge field, has a large effect on the dynamics. For example, for weak coupling, one of two natural choices of boundary conditions for the gauge field leads to a large N deconfinement phase transition as a function of the temperature, while the other does not. For boundary conditions that preserve supersymmetry, the strong coupling dynamics can be analyzed using S-duality (relevant for g_{YM} >> 1), utilizing results of Gaiotto and Witten, as well as by using the AdS/CFT correspondence (relevant for large N and large 't Hooft coupling). We argue that some very specific choices of boundary conditions lead to a simple dual gravitational description for this theory, while for most choices the gravitational dual is not known. In the cases where the gravitational dual is known, we discuss the phase structure at large 't Hooft coupling.

Conformal field theories in anti-de Sitter space

TL;DR

<3-5 sentence high-level summary>The paper studies conformal field theories on , with a focus on SYM, and shows that boundary conditions for gauge fields crucially control the dynamics and phase structure, including a weak-coupling large- Hagedorn transition under Neumann data and a richer strong-coupling picture accessible via S-duality and holography for special boundary conditions. It identifies classes of supersymmetric boundary conditions, especially quotient constructions, that admit controlled holographic duals—often realized as geometries with a quotient—leading to light degrees of freedom and clear finite-temperature phase transitions analogous to deconfinement. The results illuminate how boundary data and dualities constrain the spectrum, thermodynamics, and possible gravity duals of SYM on , and point to rich open directions such as less-supersymmetric boundaries and indices for these theories. These findings advance understanding of AdS/CFT in curved boundaries and the role of boundary conditions in holographic duals of gauge theories.

Abstract

In this paper we discuss the dynamics of conformal field theories on anti-de Sitter space, focussing on the special case of the N=4 supersymmetric Yang-Mills theory on AdS_4. We argue that the choice of boundary conditions, in particular for the gauge field, has a large effect on the dynamics. For example, for weak coupling, one of two natural choices of boundary conditions for the gauge field leads to a large N deconfinement phase transition as a function of the temperature, while the other does not. For boundary conditions that preserve supersymmetry, the strong coupling dynamics can be analyzed using S-duality (relevant for g_{YM} >> 1), utilizing results of Gaiotto and Witten, as well as by using the AdS/CFT correspondence (relevant for large N and large 't Hooft coupling). We argue that some very specific choices of boundary conditions lead to a simple dual gravitational description for this theory, while for most choices the gravitational dual is not known. In the cases where the gravitational dual is known, we discuss the phase structure at large 't Hooft coupling.

Paper Structure

This paper contains 34 sections, 92 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction and summary of results
  2. Free quantum fields on AdS$_{d}$
  3. Scalars ($s =0$):
  4. Vectors ($s=1$):
  5. Fermions ($s=\frac{1}{2}$):
  6. Four dimensional gauge theories on AdS$_{4}$
  7. ${\cal N} =4$ SYM on AdS$_{4}$ at weak coupling
  8. Spectrum of free ${\cal N} =4$ SYM on AdS$_{4}$
  9. Free ${\cal N} = 4$ SYM partition functions
  10. Single particle partition sum
  11. Multi-particle partition sum
  12. Multi-particle partition function for Dirichlet bc:
  13. Multi-particle partition function for Neumann bc:
  14. Multi-particle partition function for Neumann boundary condition for $H \subset G$:
  15. ${\cal N}=4$ SYM on AdS$_4$ at strong coupling
  16. ...and 19 more sections

Figures (1)

  • Figure 1: Poincaré disk representation of AdS$_{d}$ slices of AdS$_{d+1}$ in the $(R,r)$ coordinates used in (\ref{['adsinads']}). The vertical curves (coloured) denote surfaces of constant $R$ which are AdS$_{d}$ geometries, with the color coding showing the UV (boundary) and IR (interior). The boundary of the spacetime is the edge of the disk, which is attained as $R\to \pm \infty$, and comprises of two copies of AdS$_{d}$ joined together at their respective boundaries ($r\to \infty$). The horizontal curves (gray) are constant $r$ surfaces.