Complex Kerr-AdS Black Holes

TL;DR

The paper investigates thermodynamics of five-dimensional AdS with rotation via the Euclidean path integral, addressing a tension: complex Kerr-AdS saddles can have lower real on-shell action than the rotating thermal gas at low temperature. By combining mini-superspace truncations with Picard-Lefschetz analysis, it shows that these complex and spurious saddles do not contribute to the partition function, preserving the rotating thermal AdS background as the correct dual description in the low-temperature, finite-rotation regime. A corollary is that the unstable small rotating black hole does not contribute at any temperature. Overall, the work reinforces the confinement-deconfinement interpretation in the dual gauge theory under rotation and provides a practical method to assess the role of complex saddles in holographic thermodynamics, with potential extensions to other boundary conditions and finite-cutoff holography.

Abstract

We revisit thermodynamics of five-dimensional AdS spacetime at finite temperature and rotation using the Euclidean path integral. It is generally believed that at low temperatures and finite rotation, the bulk saddle point that governs the thermodynamics describes a rotating gas of thermal radiation. Consequently, the dual gauge theory at low temperatures is in a confined thermal state. We demonstrate that this holographic expectation is at odds with the fact that, even at low temperatures, there exist saddles of the bulk path integral with real part of on-shell action smaller than that of the thermal rotating gas. The usual Kerr-AdS black holes but with complex parameters are examples of such saddles. Using mini-superspace ideas and steepest descent, we argue that these additional saddles do not actually feature in the low temperature partition function. This saves the original claim that rotating thermal gas is indeed the correct background for understanding the dual gauge theory at low temperatures. As a corollary, we also find that the unstable small rotating black hole does not contribute to the partition function at any temperature, even in a suppressed manner.

Figures (6)

• Figure 1: The phase diagram of AdS$_{5}$ in the $\beta$--$\Omega$ space. We only show the region $0 \leq \Omega < 1$ since the plot for negative values of $\Omega$ simply mirrors this. The black hole phase is indicated by the green shaded region while the thermal phase is the complementary region shaded in orange. The Hawking-Page transition curve (blue) separates the two phases.
• Figure 2: We plot the curve $\beta_{\rm max}(\Omega)$ (red) in $\beta$--$\Omega$ space. The Hawking-Page transition curve (blue) is shown for comparison. The orange shaded part corresponds to $\beta > \beta_{\rm max}(\Omega)$, the region for which rotating black holes are complex. So, black holes become subdominant saddles before they become complex.
• Figure 3: A region on the real plane ($\Im(r_{+}) = 0, \Im(a) = 0$) showing downward flows from the thermal saddle $p_{0}$ (thick black dot) for $\beta = \pi > \beta_{\rm max}(\Omega)$ and $\Omega = 0.1$ . The thimble $\mathcal{J}_{0}$ is a union of trajectories (blue) that start at $p_{0}$ and asymptote towards $r_{+} = \pm \infty, a = \pm 1$.
• Figure 4: A region on the real plane ($\Im(r_{+}) = 0, \Im(a) = 0$) showing flow lines emanating from the thermal saddle $p_{0}$ and the large black hole saddle $p_{+}$ (thick black dots) for $\beta = \frac{\pi}{2} < \beta_{\rm max}(\Omega)$ and $\Omega = 0.1$ . The thimble $\mathcal{J}_{0}$ is a union of trajectories (blue) that start at $p_{0}$ and asymptote towards $r_{+} = -\infty, a = \pm 1$, while the thimble $\mathcal{J}_{+}$ is a union of trajectories (red) that start at $p_{+}$ and asymptote towards $r_{+} = \infty, a = \pm 1$. The small black hole saddle $p_{-}$ is marked by a green dot. Flow lines from $p_{-}$ go in the $\Im(r_{+})$ direction.
• Figure 5: Projection on $r_{+}$--plane of downward flow lines from the three saddles with $\Im(G) < 0$ (left) and $\Im(G) > 0$ (right) in the case $\beta = \frac{\pi}{2} < \beta_{\rm max}(\Omega)$ and $\Omega = 0.1$ . Each saddle $p_{\sigma}$ is marked by a thick black dot. The flow lines are colour coded according to their asymptotic behaviour.