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Instability in ${\cal N}=4$ supersymmetric Yang-Mills theory on $S^3$ at finite density

Alex Buchel

Abstract

Homogeneous and isotropic equilibrium states of strongly coupled ${\cal N}=4$ supersymmetric Yang-Mills charged plasma in ${\mathbb R}^3$ with equal chemical potentials for the maximal Abelian subgroup of the $R$-symmetry group become dynamically unstable below some critical temperature. The instabilities arise in the $R$-symmetry charge transport, precisely when the equilibrium state becomes thermodynamically unstable. We study the fate these correlated instabilities when the theory is place on $S^3$. The curvature of the three-sphere affects the onset of the dynamical and the thermodynamic instabilities differently: increasing the curvature at low temperatures can stabilize its transport, but leave the plasma thermodynamically unstable. Thermodynamic stability is eventually recovered at large curvatures.

Instability in ${\cal N}=4$ supersymmetric Yang-Mills theory on $S^3$ at finite density

Abstract

Homogeneous and isotropic equilibrium states of strongly coupled supersymmetric Yang-Mills charged plasma in with equal chemical potentials for the maximal Abelian subgroup of the -symmetry group become dynamically unstable below some critical temperature. The instabilities arise in the -symmetry charge transport, precisely when the equilibrium state becomes thermodynamically unstable. We study the fate these correlated instabilities when the theory is place on . The curvature of the three-sphere affects the onset of the dynamical and the thermodynamic instabilities differently: increasing the curvature at low temperatures can stabilize its transport, but leave the plasma thermodynamically unstable. Thermodynamic stability is eventually recovered at large curvatures.

Paper Structure

This paper contains 11 sections, 87 equations, 2 figures.

Table of Contents

  1. Introduction and summary
  2. Master equations of Einstein-Maxwell-scalar black holes in $D=5$
  3. Helicity $h=0$ sector
  4. $\ell=0$
  5. $\ell=1$
  6. Helicity $h=1$ sector
  7. $\ell=1$
  8. Helicity $h=2$ sector
  9. The STU model
  10. The STU black hole background geometry and thermodynamics
  11. Helicity $h=0$ sector fluctuations of the STU black hole

Figures (2)

  • Figure 1: The left panel: we use the master field formalism of Appendix \ref{['eomshol']} to reproduce the diffusion coefficient ${\cal D}$\ref{['defcald']} of the ${\cal N}=4$ SYM plasma with equal chemical potentials in ${\mathbb R}^3$ reported in Gladden:2024ssb. The right panel: we study diffusive modes in the charged SYM plasma on $S^3$ of radius $\frac{1}{\sqrt K}$. Different $\ell$ diffusive QNMs \ref{['defell']} are stabilized at increasing values of $K=K_{crit}^{(\ell)}$, represented by the green dots. The last mode to stabilize is $\ell=1$.
  • Figure 2: The left panel: the critical values $K_{crit}^{(\ell)}$ required to stabilize the diffusive QNMs, ${\rm Im }[\mathfrak{w}^{(\ell)}(K>K_{crit}^{(\ell)})]<0$, as a function of $\kappa$\ref{['stuthermo']}. The right panel: charged ${\cal N}=4$ SYM plasma on $S^3$ of curvature $K$ is thermodynamically unstable below the red curve, and is dynamically unstable below the black curve.