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Magnetic Brane Solutions in AdS

Eric D'Hoker, Per Kraus

TL;DR

The paper constructs magnetic brane solutions in five-dimensional Einstein-Maxwell theory with a negative cosmological constant, modeling N=4 SYM in a background magnetic field via AdS/CFT. The geometry flows from AdS5 in the UV to an IR AdS3 × T2 region (replacing by BTZ at finite temperature), yielding a low-temperature entropy linear in temperature with a central charge enhanced by a factor sqrt(4/3) relative to the free field theory. At high temperatures the gravity result reproduces the familiar S ∝ T^3 with a factor 3/4 relative to the free theory, while at low temperatures the entropy vanishes for the AdS5 magnetic brane, matching fermionic zero-mode expectations. The work generalizes to AdS_{d+1}, with odd d giving AdS3 × T^{d−1} IR behavior and even d giving AdS2-like near-horizon structure, both with central charges and entropies governed by the magnetic flux. Overall, the paper clarifies holographic thermodynamics of strongly coupled gauge theories in magnetic fields and connects gravity results to free-field limits across dimensions.

Abstract

We construct asymptotically AdS_5 solutions of Einstein-Maxwell theory dual to N=4 SYM theory on R^{3,1} in the presence of a background magnetic field. The solutions interpolate between AdS_5 and a near horizon AdS_3\times T^2. The central charge of the near horizon region, and hence low temperature entropy of the solution, is found to be \sqrt{4\over 3} times that of free N=4 SYM theory. The entropy vanishes at zero temperature. We also present the generalization of these solutions to arbitrary spacetime dimensionality.

Magnetic Brane Solutions in AdS

TL;DR

The paper constructs magnetic brane solutions in five-dimensional Einstein-Maxwell theory with a negative cosmological constant, modeling N=4 SYM in a background magnetic field via AdS/CFT. The geometry flows from AdS5 in the UV to an IR AdS3 × T2 region (replacing by BTZ at finite temperature), yielding a low-temperature entropy linear in temperature with a central charge enhanced by a factor sqrt(4/3) relative to the free field theory. At high temperatures the gravity result reproduces the familiar S ∝ T^3 with a factor 3/4 relative to the free theory, while at low temperatures the entropy vanishes for the AdS5 magnetic brane, matching fermionic zero-mode expectations. The work generalizes to AdS_{d+1}, with odd d giving AdS3 × T^{d−1} IR behavior and even d giving AdS2-like near-horizon structure, both with central charges and entropies governed by the magnetic flux. Overall, the paper clarifies holographic thermodynamics of strongly coupled gauge theories in magnetic fields and connects gravity results to free-field limits across dimensions.

Abstract

We construct asymptotically AdS_5 solutions of Einstein-Maxwell theory dual to N=4 SYM theory on R^{3,1} in the presence of a background magnetic field. The solutions interpolate between AdS_5 and a near horizon AdS_3\times T^2. The central charge of the near horizon region, and hence low temperature entropy of the solution, is found to be \sqrt{4\over 3} times that of free N=4 SYM theory. The entropy vanishes at zero temperature. We also present the generalization of these solutions to arbitrary spacetime dimensionality.

Paper Structure

This paper contains 12 sections, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Magnetic brane in AdS$_5$
  3. Zero temperature solutions
  4. Finite temperature solutions
  5. Comparison with ${\cal N}=4$ Super Yang-Mills
  6. Generalization to AdS$_{d+1}$
  7. $d$ odd
  8. $d$ even
  9. Discussion
  10. Towards Exact Solutions
  11. The large $B$ case
  12. The boost-invariant case

Figures (1)

  • Figure 1: Plot of entropy versus temperature for gravity (red) and free ${\cal N}=4$ SYM theory (blue). The inset shows the low temperature behavior. At low temperature the entropies are linear in $T$, with $S_{\rm grav} = \sqrt{4\over 3}S_{{\cal N}=4}$. At high temperature the entropies are cubic in $T$, with $S_{\rm grav} = {3\over 4}S_{{\cal N}=4}$. Gravity gives the larger entropy at low temperature by a factor of $\sqrt{4\over 3}$; as the temperature is raised the curves cross, and then asymptotically the gravitational entropy is lower by a factor of ${3\over 4}$.