1 Introduction↩︎

Black holes can be thought of as statistical systems with a temperature and thermodynamics associated to them [1]. One way to arrive at this conclusion is through the Euclidean path integral introduced by Gibbons and Hawking [2] (see section 6 of [3] for a recent review). The idea is that certain partition functions are computable by performing a path integral over Euclidean geometries while keeping the induced metric on a codimension one asymptotic boundary fixed. If a black hole solution (saddle) contributes and dominates the path integral then the thermodynamics of the solution can be obtained from the partition function using usual statistical methods.
A general path integral can have multiple saddles contributing in which case more can be deduced about the system. An important example is due to Hawking and Page [4], who analyzed the path integral with boundary conditions induced from a Euclidean AdS-Schwarzschild black hole. This path integral computes the partition function at a temperature fixed by the black hole parameters. A peculiar feature of AdS spacetimes was realized in [4] — there is a critical temperature corresponding to a phase transition between thermal AdS and the spherical black hole geometry. This point is commonly referred to as the Hawking-Page phase transition. From the perspective of the path integral, the phase transition is simply an exchange of dominance between the thermal and black hole saddles as the temperature is varied. A similar result can be derived for the case with rotation. There is a Hawking-Page phase transition between a rotating thermal gas in AdS and the rotating black hole geometry. This is captured by the partition function computed from a Euclidean path integral with boundary conditions set by a Kerr-AdS black hole.
Soon after the original proposal of the AdS/CFT correspondence [5]–[7], it was argued in [8] that the phase transition in the bulk spacetime should be understood as a confinement-deconfinement phase transition in the boundary theory. Through semi-classical computations in the thermal AdS and black hole backgrounds, the correspondence has revealed interesting features of the dual gauge theory in the confined and deconfined phases, respectively.
Recently, it was pointed out in [9] for the Schwarzschild case that there is a gap in the entire reasoning above stemming from the fact that at lower temperatures the black holes exist as complex saddles of the path integral. In fact, at very low temperatures, the real part of the on-shell action of these complex black holes is smaller than that of the thermal AdS solution.¹ Naïvely including these saddles lead to an inconsistency with the expectation that the dual gauge theory is confined at very low temperatures [12]. The fact that these saddles are complex does not a priori mean that we can discard them. Indeed, complex geometries have proven useful in quantum cosmology [10], [13]–[20], gravitational description of quantum chaos [21]–[24], real-time holography [25], [26], as well as computing supersymmetric indices [27]–[30].
The first objective of this paper is to show the existence of complex black hole saddles in five-dimensional AdS spacetime that can contribute to the ensemble at finite temperature and rotation. Analogous to the Schwarzschild case, the continuation of the small and large Kerr-AdS black holes to complex parameters are examples of such complex saddles at low temperatures. We also find additional saddles at finite rotation which do not have an analogue in the case without rotation. The remainder of this paper is devoted towards restoring the claim that at lower temperatures and finite rotation, the correct saddle that captures the dual gauge theory is indeed the rotating thermal AdS solution.
The plan of the paper is as follows. In section 2, we collect classic results for the five-dimensional Kerr-AdS black hole with a single rotation. The thermodynamic properties of this solution is reviewed in section 2.2. We identify the complex black holes that can contribute to the partition function at finite temperature and rotation in section 3. In section 4, we use mini-superspace methods and Picard-Lefshetz theory to determine which saddles actually contribute to the partition function at different values of the thermodynamic variables. Finally, we conclude in section 5 with a brief discussion.

2 Rotating AdS Black Holes in Five Dimensions↩︎

We start by reviewing the standard thermodynamical relations for a rotating black hole. The Lorentzian metric of a five-dimensional Kerr-AdS black hole with a single rotation is conveniently written as [31], [32] \[\begin{align} \begin{aligned} \label{Kerr32metric321} ds^{2} & = -\frac{\Delta_{r}}{\rho^{2}}\l(dt - \frac{a}{\Xi}\sin^{2}\theta d\phi\r)^{2} + \frac{\rho^{2}}{\Delta_{r}}dr^{2} + \frac{\rho^{2}}{\Delta_{\theta}}d\theta^{2} \\ & \qquad + \frac{\Delta_{\theta}\sin^{2}\theta}{\rho^{2}}\l(adt - \frac{r^{2} + a^{2}}{\Xi}d\phi\r)^{2} + r^{2}\cos^{2}\theta d\psi^{2}. \end{aligned} \end{align}\tag{1}\] Here, the \(\theta\) coordinate ranges from \(0\) to \(\frac{\pi}{2}\) while the \(\phi,\psi\) coordinates are \(2\pi\) periodic. The functions appearing in 1 are defined as \[\begin{gather} \tag{2} \Delta_{r}(r) = (r^{2} + a^{2})(r^{2} + 1) - (r_{+}^{2} + a^{2})(r_{+}^{2} + 1), \\ \tag{3} \rho^{2}(r,\theta) = r^{2} + a^{2}\cos^{2}\theta, \quad \Delta_{\theta}(\theta) = 1 - a^{2}\cos^{2}\theta, \quad \Xi = 1 - a^{2}. \end{gather}\] We have set the AdS length to unity. Going to Euclidean time (\(\tau = it\)) gives the following metric \[\begin{gather} \label{Kerr32metric324} ds^{2} = \frac{\Delta_{r}}{\rho^{2}}\l(d\tau - i\frac{a}{\Xi}\sin^{2}\theta d\phi\r)^{2} + \frac{\rho^{2}}{\Delta_{r}}dr^{2} + \frac{\Delta_{\theta}\sin^{2}\theta}{\rho^{2}}\l(iad\tau + \frac{r^{2} + a^{2}}{\Xi}d\phi\r)^{2}, \end{gather}\tag{4}\] where the \(\theta\) and \(\psi\) directions have been suppressed. For real \(r_{+}\) and \(\abs{a} < 1\), the metric 4 is called quasi-Euclidean.
To identify the inverse temperature (\(\beta\)) and the angular velocity (\(\O\)) that enter the thermodynamics, we need to analyze the near horizon metric. Taking \(r \to r_{+}\) in 4 yields \[\begin{align} \begin{aligned} \label{Kerr32metric325} ds^{2} & = \frac{(r - r_{+})\Delta_{+}'}{\rho_{+}^{2}}\l(d\tau - i\frac{a}{\Xi}\sin^{2}\theta d\phi\r)^{2} \\ & \qquad + \frac{\rho_{+}^{2}}{(r - r_{+})\Delta_{+}'}dr^{2} + \frac{\Delta_{\theta}\sin^{2}\theta}{\rho_{+}^{2}}\l(iad\tau + \frac{r_{+}^{2} + a^{2}}{\Xi}d\phi\r)^{2}, \end{aligned} \end{align}\tag{5}\] where we have defined \(\require{physics} \Delta_{+}' \equiv \eval{\frac{d\Delta_{r}}{dr}}_{r = r_{+}}\) and \(\rho_{+} \equiv \rho(r_{+})\). We define the following angular coordinate \[\begin{gather} \label{Kerr32metric326} \wh{\phi} = \phi + i\frac{a\Xi}{r_{+}^{2} + a^{2}}\tau \end{gather}\tag{6}\] and write 5 as \[\begin{align} \begin{aligned} \label{Kerr32metric327} ds^{2} & = \frac{(r - r_{+})\Delta_{+}'\rho_{+}^{2}}{(r_{+}^{2} + a^{2})^{2}}\l(d\tau - i\frac{a(r_{+}^{2} + a^{2})}{\Xi\rho_{+}^{2}}\sin^{2}\theta d\wh{\phi}\r)^{2} \\ & \qquad + \frac{\rho_{+}^{2}}{(r - r_{+})\Delta_{+}'}dr^{2} + \frac{\Delta_{\theta}(r_{+}^{2} + a^{2})^{2}\sin^{2}\theta}{\rho_{+}^{2}\Xi^{2}}d\wh{\phi}^{2}. \end{aligned} \end{align}\tag{7}\] The inverse temperature is easily read off in the \((\tau,\wh{\phi})\) coordinates \[\begin{gather} \label{Kerr32temperature321} \beta = \frac{4\pi(r_{+}^{2} + a^{2})}{\Delta_{+}'} = \frac{2\pi(r_{+}^{2} + a^{2})}{r_{+}(2r_{+}^{2} + a^{2} + 1)}. \end{gather}\tag{8}\] Now, along with the \(2\pi\) periodicity in \(\phi\), we have the following periodic identifications in the \((\tau,\wh{\phi})\) coordinates \[\begin{gather} \label{Kerr32periodicity321} (\tau,\wh{\phi}) \sim (\tau + \beta,\wh{\phi}) \sim (\tau,\wh{\phi} + 2\pi). \end{gather}\tag{9}\] In the original \((\tau,\phi)\) coordinates, the periodicity is stated as \[\begin{gather} \label{Kerr32periodicity322} (\tau,\phi) \sim \l(\tau + \beta,\phi + i\beta\frac{a\Xi}{r_{+}^{2} + a^{2}}\r) \sim (\tau,\phi + 2\pi). \end{gather}\tag{10}\]

It is tempting to identify the shift of \(\phi\) in the first identification in 10 with \(i\beta\O\). But this is not correct. As explained in [32], in the \(\phi\) coordinate the asymptotic boundary of AdS itself rotates. To see this, we analyze the metric in 4 at large \(r\) \[\begin{align} \notag ds^{2} & = r^{2}\l(d\tau - i\frac{a}{\Xi}\sin^{2}\theta d\phi\r)^{2} + \frac{dr^{2}}{r^{2}} + \frac{r^{2}\Delta_{\theta}\sin^{2}\theta}{\Xi^{2}}d\phi^{2} \\ \label{Kerr32metric328} \Rightarrow ds^{2} & = \frac{r^{2}\Delta_{\theta}}{\Xi}d\tau^{2} + \frac{dr^{2}}{r^{2}} + \frac{r^{2}\sin^{2}\theta}{\Xi}(d\phi - iad\tau)^{2}. \end{align}\tag{11}\] We note that the asymptotic metric contains a \(d\phi d\tau\) term, indicating the fact that the boundary is rotating in these coordinates. The correct variable with respect to which we define the rotation of the black hole is given by \[\begin{gather} \label{Coordinate32change321} \wt{\phi} = \phi - ia\tau. \end{gather}\tag{12}\] The periodicity 10 can be restated in the \((\tau,\wt{\phi})\) coordinates as \[\begin{gather} \label{Kerr32periodicity323} (\tau,\wt{\phi}) \sim (\tau + \beta,\wt{\phi} + i\beta\O) \sim (\tau,\wt{\phi} + 2\pi), \end{gather}\tag{13}\] where we have identified the angular velocity \[\begin{gather} \label{Kerr32angular32velocity321} \O = \frac{a\Xi}{r_{+}^{2} + a^{2}} + a = \frac{a(r_{+}^{2} + 1)}{r_{+}^{2} + a^{2}}. \end{gather}\tag{14}\]

2.1 Euclidean path integral and on-shell action↩︎

Having obtained the expressions for the inverse temperature and angular velocity, we now compute the partition function defined by the following Euclidean path integral \[\begin{gather} \label{Kerr32partition32function321} \mcz = \int{[Dg]}\exp(-I). \end{gather}\tag{15}\] The Gibbons-Hawking prescription [2] tells us to integrate over Euclidean geometries that have the same induced metric at the asymptotic boundary as the one obtained from the rotating black hole metric in 4 . We cannot simply define \(I\) as the usual gravity action because of the infinite volume divergence of AdS space. We regulate this divergence by taking the asymptotic boundary at a cutoff radius \(r = R_{c}\) and defining \(I\) as the background subtracted action² \[\begin{gather} \label{Background32subtracted32action321} I = S_{g} - S_{\rm AdS}. \end{gather}\tag{16}\] Here, \(S_{g}\) is the usual Euclidean gravity action \[\begin{gather} \label{Background32subtracted32action322} S_{g} = -\frac{1}{16\pi G}\int{d^{5}x}\sqrt{g}(R + 12) \end{gather}\tag{17}\] and \(S_{\rm AdS}\) is its value for the thermal AdS solution satisfying the correct boundary conditions. Note that in defining 17 , we have only kept the Einstein-Hilbert term along with the negative cosmological constant. In general, we also need the GHY boundary term but it is easy to check that it does not contribute to the background subtracted action. This is because the leading contribution of the boundary term is same for asymptotically AdS solutions as \(R_{c}\) is taken to be large.
Let us now evaluate the on-shell action for the rotating black hole solution in 4 . Since there is no matter source, the Einstein equations fix the curvature scalar to \(R = -20\). The action is then evaluated as \[\begin{align} \notag S_{\rm BH} & = -\frac{1}{16\pi G}\int{d\O_{3}}\int_{0}^{\beta}{d\tau}\int_{r_{+}}^{R_{c}}{dr}\frac{(r^{2} + a^{2}\cos^{2}\theta)r}{\Xi}(-20 + 12), \\ \label{Kerr32action321} \Rightarrow S_{\rm BH} & = \frac{\pi\beta}{4\Xi G}(R_{c}^{4} - r_{+}^{4} + a^{2}R_{c}^{2} - a^{2}r_{+}^{2}). \end{align}\tag{18}\] Note that we have used Hopf coordinates for the integration on \(S^{3}\) so that \(\int{d\O_{3}} = \int_{0}^{2\pi}{d\phi}\int_{0}^{2\pi}{d\psi}\int_{0}^{\frac{\pi}{2}}{d\theta}\sin\theta\cos\theta\).
To compute the background subtracted action for the black hole we need the on-shell action for the appropriate AdS solution. The solution with the correct induced metric at the asymptotic boundary can be obtained as follows. First consider the Euclidean AdS metric in global coordinates \((\tau,y,\vartheta,\Phi,\psi)\) with radial coordinate \(y\) and the boundary at \(y \to \infty\). A transformation to coordinates \((\tau,r,\theta,\phi,\psi)\) given by \[\begin{align} \tag{19} y^{2}\sin^{2}\vartheta & = \frac{(r^{2} + a^{2})\sin^{2}\theta}{\Xi}, \\ \tag{20} y^{2}\cos^{2}\vartheta & = r^{2}\cos^{2}\theta, \\ \tag{21} \Phi & = \phi - ia\tau, \end{align}\] brings the AdS metric to the form 4 with the function \(\Delta_{r}(r)\) replaced by \[\begin{gather} \label{Kerr32metric329} \Delta_{r}^{\rm AdS}(r) = (r^{2} + a^{2})(r^{2} + 1). \end{gather}\tag{22}\] Note that the \(\Phi\) coordinate is simply what we defined as \(\wt{\phi}\) in 12 .
We compute the AdS action directly in the \((\tau,r,\theta,\phi,\psi)\) coordinates. Setting \(y = 0\) in 19 –20 gives the lower limit of the \(r\)–integration as \(r = 0\). The upper limit is simply the cutoff radius \(r = R_{c}\). The period of \(\tau\) integration in the AdS metric is taken as \(\beta'\). Matching the induced metric at the cutoff surface for the AdS and black hole solutions relates \(\beta\) and \(\beta'\) as follows \[\begin{gather} \label{Kerr32temperature322} \sqrt{\Delta_{r}^{\rm AdS}(R_{c})}\beta' = \sqrt{\Delta_{r}(R_{c})}\beta. \end{gather}\tag{23}\] For large \(R_{c}\), this simplifies to \[\begin{gather} \label{Kerr32temperature323} \beta' = \beta - \frac{\beta}{2R_{c}^{4}}(r_{+}^{2} + a^{2})(r_{+}^{2} + 1) + O\l(\frac{1}{R_{c}^{5}}\r). \end{gather}\tag{24}\] Putting everything together, the action for the AdS solution reads \[\begin{align} \notag S_{\rm AdS} & = -\frac{1}{16\pi G}\int{d\O_{3}}\int_{0}^{\beta'}{d\tau}\int_{0}^{R_{c}}{dr}\frac{(r^{2} + a^{2}\cos^{2}\theta)r}{\Xi}(-20 + 12), \\ \label{Kerr32action322} \Rightarrow S_{\rm AdS} & = \frac{\pi\beta'}{4\Xi G}(R_{c}^{4} + a^{2}R_{c}^{2}). \end{align}\tag{25}\] Finally, to get the background subtracted action for the rotating black hole, we subtract 25 from 18 , use the relation 24 and then take the large \(R_{c}\) limit \[\begin{gather} \label{Kerr32action323} I_{\rm BH} = \lim_{R_{c} \to \infty}(S_{\rm BH} - S_{\rm AdS}) = \frac{\pi\beta}{8\Xi G}(r_{+}^{2} + a^{2})(1 - r_{+}^{2}). \end{gather}\tag{26}\]

In the following sections, we will use the background subtracted action 26 . But first, let us comment on a subtlety. The spatial slices at constant \(y\) are spheres while the slices at constant \(r\) are spheroids. In particular, a fixed \(y\) surface is described by a complicated relation between \(r\) and the angular variables. To study black hole thermodynamics, it is natural to define the CFT on a non-rotating sphere at large \(y\) instead of a surface at large \(r\). Despite this, taking the cutoff surface in the \(r\) variable works because at large radius the spheroids are almost spheres and the difference between the two is not seen in background subtraction. But care is required when using counter-term regularization. The finite contribution of holographic counter-terms to the on-shell action gives the Casimir energy of the vacuum, which differs between the two choices of asymptotic boundaries [35]. At large \(y\), the renormalized action differs from the background subtracted one by the constant \(\frac{3\pi\beta}{32G}\). In contrast, the large \(r\) surface is rotating and the induced metric depends on the parameter \(a\). In turn, the Casimir energy also depends on \(a\).

2.2 Thermodynamics and Hawking-Page phase transition↩︎

In the ensemble we are considering, we fix the inverse temperature, \(\beta\), and rotation, \(\O\), by specifying the boundary conditions for the path integral. From the dual CFT, we know that the partition function makes sense for \(\abs{\O} < 1\) for all values of \(\beta > 0\).³ In the AdS bulk, the rotating thermal AdS solution also exists for these values of \(\beta\) and \(\O\). Additionally, for small enough \(\beta\), we also get two quasi-Euclidean black hole saddles. These are usually referred to as small and large black holes. For now, we restrict our attention to these three solutions. Other (possibly complex) saddles will be discussed in detail in section 3.
To recover the thermodynamics of the rotating black holes, we can use the on-shell action computed in 26 . We write its contribution to the partition function as \[\begin{gather} \label{BH32partition32function321} \mcz_{\rm BH} = \exp(-I_{\rm BH}) \times (\cdots), \end{gather}\tag{27}\] where \(\cdots\) denotes perturbative loop corrections around the saddle. Taking appropriate derivatives of \(\log\mcz_{\rm BH}\) with respect to \(\beta\) and \(\O\) reproduces the correct leading order expressions for \(E, J\) and \(S\) \[\begin{align} \tag{28} E & = \l(-\frac{\p}{\p\beta}+ \frac{\O}{\beta}\frac{\p}{\p\O}\r)\log\mcz_{\rm BH} = \frac{\pi(\Xi + 2)}{8\Xi^{2}G}(r_{+}^{2} + 1)(r_{+}^{2} + a^{2}), \\ \tag{29} J & = \frac{1}{\beta}\frac{\p}{\p\O}\log\mcz_{\rm BH} = \frac{\pi a}{4\Xi^{2}G}(r_{+}^{2} + 1)(r_{+}^{2} + a^{2}), \\ \tag{30} S & = \l(-\beta\frac{\p}{\p\beta} + 1\r)\log\mcz_{\rm BH} = \frac{\pi^{2}r_{+}}{2\Xi G}(r_{+}^{2} + a^{2}). \end{align}\] The above expressions hold for both the small and large Kerr-AdS black holes, as well as the complex black holes we will discuss later. Notably, all extensive quantities for the black hole go as \(\frac{1}{G}\). In particular, the entropy matches with Bekenstein-Hawking formula, \(S = \frac{{\rm Area}}{4G}\).
For the region in \(\beta\)–\(\O\) space where a Kerr-AdS saddle dominates the full path integral, we say that the bulk is in the black hole phase. In contrast, if the rotating thermal AdS saddle dominates, we call it the thermal phase. From the perspective of the dual CFT, these are the deconfined and confined phases, respectively. One way to see this is that the entropy in the deconfined phase goes as \(N^{2} \sim \frac{1}{G}\).
In the situation where both a black hole saddle and the thermal saddle contribute, the one that dominates is decided by which has a smaller on-shell action. We have already normalized the action in a way that it vanishes for the thermal solution. So, the black holes dominates for those values of \(\beta\) and \(\O\) for which \(I_{\rm BH} < 0\). From equation 26 , we see that this condition is met when \(r_{+} > 1\). Conversely, the thermal solution dominates when \(r_{+} < 1\). Thus, at \(r_{+} = 1\) we have a first order phase transition between the black hole and thermal phases. We interpret this as a confinement-deconfinement phase transition in the dual gauge theory [8]. Substituting \(r_{+} = 1\) in the thermodynamics relations 8 and 14 gives the following Hawking-Page transition curve \[\begin{gather} \label{HP32transition321} \beta_{\rm HP}(\O) = \frac{2\pi(2 - \sqrt{1 - \O^{2}})}{3 + \O^{2}}. \end{gather}\tag{31}\] We see that the Hawking-Page (inverse) temperature nicely interpolates between the Schwarzschild value, \(\beta_{\rm HP} = \frac{2\pi}{3}\) at \(\O = 0\), and \(\beta_{\rm HP} = \pi\) at \(\O = 1\). The phase structure of AdS spacetime at finite temperature and rotation is shown in figure 1.

3 Complex Kerr-AdS Solutions↩︎

We now come to the main point of this paper. In the previous section, we derived the thermodynamic features of AdS spacetime by only considering quasi-Euclidean black holes and the thermal AdS solution. This would be fine if these were the only saddles of the path integral. But this is not the case. We now show that the small and large Kerr-AdS black holes exist as quasi-Euclidean solutions only in part of the \(\beta\)–\(\O\) space. In the remaining region, they naturally continue over to complex saddles of the path integral. In effect, this is a generalization to the case with rotation of the observation made in [9] for complex Schwarzschild black holes. But, apart from the small and large black holes, we find an additional black hole saddle at nonzero values of \(\O\) (for all \(\beta\)). We call this a spurious black hole owing to the fact that it does not have an analogue in the spherically symmetric case.
Let us collect the thermodynamic relations 8 and 14 here \[\begin{align} \tag{32} \beta & = \frac{2\pi(r_{+}^{2} + a^{2})}{r_{+}(2r_{+}^{2} + a^{2} + 1)}, \\ \tag{33} \O & = \frac{a(r_{+}^{2} + 1)}{r_{+}^{2} + a^{2}}. \end{align}\] It would be useful if we can invert these relations and write \(r_{+}\) and \(a\) directly in terms of \(\beta\) and \(\O\). In the AdS-Schwarzschild case, i.e. setting \(a = 0\), inverting the relation 32 gives two solutions for the horizon radius \[\begin{gather} \label{Schwarzschild32horizon32radius321} r_{+}^{\pm} = \frac{\pi \pm \sqrt{\pi^{2} - 2\beta^{2}}}{2\beta}. \end{gather}\tag{34}\] These are the small (\(r_{+}^{-}\)) and large (\(r_{+}^{+}\)) spherical black holes. When we turn on a finite \(\O\), we obtain the rotating counterparts of these solutions. The expressions for general \(\O\) look complicated, but for \(\O \ll 1\) we can expand around the spherical case to write \[\begin{align} \tag{35} r_{+} & = r_{0} + \frac{r_{0}^{3}}{2r_{0}^{4} + r_{0}^{2} - 1}\O^{2} + O(\O^{4}), \\ \tag{36} a & = \frac{r_{0}^{2}}{r_{0}^{2} + 1}\O + \frac{r_{0}^{4}(2r_{0}^{2} + 1)}{(r_{0}^{2} + 1)^{3}(2r_{0}^{2} - 1)}\O^{3} + O(\O^{5}). \end{align}\] Here, \(r_{0}\) takes either of the the values in 34 . The above expressions describe the small (\(r_{0} = r_{+}^{-}\)) and large (\(r_{0} = r_{+}^{+}\)) Kerr-AdS black holes.
It is clear that for small enough values of \(\beta\), the relation 34 returns real values of \(r_{+}^{\pm}\) so that the AdS-Schwarzschild black holes are real. But, as pointed out in [9], as \(\beta\) is increased all the way to \(\beta_{\rm max} = \frac{\pi}{\sqrt{2}}\), the two solutions coalesce with \(r_{+}^{\pm} = \frac{1}{\sqrt{2}}\). A further increase in \(\beta\) leads to complex solutions which were dubbed as complex AdS-Schwarzschild black holes. We now make an analogous observation in the present case of finite rotation. At fixed \(\O\), there exists a corresponding value \(\beta_{\rm max}(\O)\) at which the two rotating black holes appear as a repeated solution of equations 32 and 33 . The curve \(\beta_{\rm max}(\O)\) is determined by demanding the proportionality of tangent vectors at the solution \[\begin{gather} \label{Coalescing32saddles321} \l(\frac{\p\beta}{\p r_{+}},\frac{\p\beta}{\p a}\r) \propto \l(\frac{\p\O}{\p r_{+}},\frac{\p\O}{\p a}\r). \end{gather}\tag{37}\] This condition leads to the following parametric form for \(\beta_{\rm max}(\O)\) \[\begin{align} \tag{38} \beta_{\rm max} & = \frac{\pi}{\sqrt{2}}\frac{3a^{2} + 1}{(a^{2} + 1)^{\frac{3}{2}}}, \\ \tag{39} \O & = \frac{a(a^{2} + 3)}{3a^{2} + 1}, \end{align}\] where we take \(\abs{a} < 1\) so that \(\abs{\O} < 1\).⁴ So, for \(0 < \beta < \beta_{\rm max}({\O})\) we get quasi-Euclidean black holes while for \(\beta > \beta_{\rm max}(\O)\) the black holes are genuinely complex. The curve \(\beta_{\rm max}(\O)\) is shown in figure 2 alongside the transition curve \(\beta_{\rm HP}(\O)\).

 
Based on the analysis of section 2.2 and the discussion above, we can form the following conclusions. At a fixed value of the rotation and small enough inverse temperature, the large rotating black hole is a quasi-Euclidean solution that dominates over rotating thermal AdS. As inverse temperature is increased, the black holes first become subdominant and then complex. It is easy to see that this is actually the situation with AdS black holes in all spacetime dimensions four and higher.
However, the complex black holes are not subdominant at all the temperatures. In five spacetime dimensions (and higher), as \(\beta\) is increased beyond \(\beta_{\rm max}\), the real part of the on-shell action again becomes negative, thus implying that the complex black holes can come back to dominate over the thermal solution. In the extreme case, \(\beta \to \infty\) with \(\O\) fixed, equations 32 –33 return \[\begin{gather} \label{Extreme32case321} r_{+} = \pm i\sqrt{\frac{1 + \O^{2}}{2}}, \quad a = -\O. \end{gather}\tag{40}\] The on-shell action 26 at these values of the parameters evaluates to \[\begin{gather} \label{Extreme32case322} I_{\rm BH} = -\frac{\pi\beta(3 + \O^{2})}{32G}. \end{gather}\tag{41}\] This means that if the complex black holes do contribute to the AdS thermodynamics, then apart from the Hawking-Page phase transition there is another phase transition (at a lower temperature) back to the black hole phase. Again, by AdS/CFT, this implies an additional confinement-deconfinement phase transition in the dual gauge theory. But, this goes against the general expectation that at low enough temperatures a large–\(N\) gauge theory is confined [12].
We point out another related issue. It turns out that at a given value of the inverse temperature and rotation, there is another black hole solution apart from the two (quasi-Euclidean or complex) Kerr-AdS ones. This additional solution does not admit a nice \(\O \to 0\) limit and hence does not have an analogue in the spherically symmetric case. For this reason, we call it a spurious black hole. To construct this black hole, we look for a solution of the relations 32 –33 that has a singular \(\O \to 0\) limit. We can write the solution as an expansion in \(\O\) as follows \[\begin{align} \tag{42} r_{+} & = r_{0} - \frac{r_{0}}{r_{0}^{2} + 1}\O^{2} + O(\O^{4}), \\ \tag{43} a & = (r_{0}^{2} + 1)\frac{1}{\O} - \frac{3r_{0}^{2}}{r_{0}^{2} + 1}\O + O(\O^{3}), \end{align}\] where \(r_{0} \equiv \frac{2\pi}{\beta}\). With these values of the black hole parameters, the Kerr-AdS metric 4 is not quasi-Euclidean. Indeed, the “Lorentzian" metric 1 for this solution does not have Lorentzian signature. Rather, it describes a signature change metric.
Despite the fact the spurious black hole does not have a well-defined limit to the spherically symmetric case, it is a legitimate saddle for the path integral that computes the partition function at nonzero rotation. Plugging in the values 42 43 into 26 gives the following expansion for the on-shell action \[\begin{gather} \label{SpBH32action321} I_{\rm BH} = \frac{\pi\beta(r_{0}^{2} - 1)}{8G} - \frac{\pi\beta}{8G}\O^{2} + O(\O^{4}). \end{gather}\tag{44}\] Curiously, the action is finite in the limit \(\O \to 0\). Furthermore, from the leading term in 44 , we note that the on-shell action is negative for \(r_{0} < 1\), or equivalently, for \(\beta > 2\pi\). So, at small enough temperatures, even the spurious black hole is dominant over the thermal AdS saddle. In fact, comparing 44 against 41 in the extreme case \(\beta \to \infty\), we see that it is dominant over the complex Kerr-AdS solutions as well. This means that if the spurious black hole also contributes to the partition function, then there is a third phase transition, this time between two black hole phases. Again, this is an unexpected feature from the point of view of the dual gauge theory.
At small \(\abs{\O}\), it is easy to verify that the only solutions to the relations 32 –33 are the small and large Kerr-AdS black holes, as well as the spurious black hole. Since the number of solutions for generic values of \(\beta\) and \(\O\) remains the same, we have not missed any solution.
To remedy the puzzle with the additional phase transitions, we proceed along the lines of [9]. In section 4, we use an appropriate mini-superspace approximation for the original path integral and analyze the residual finite-dimensional integral via Picard-Lefschetz theory.

4 Mini-superspace Analysis↩︎

We want to carefully understand which saddles contribute to the partition function as the boundary conditions (values of \(\beta\) and \(\O\)) of the path integral 15 are varied. A direct approach towards understanding the full infinite-dimensional integral is not available at present. Instead, we assume that there exists a consistent truncation of the path integral to a finite-dimensional integral that can still capture some of the essential features, such as the information of which saddles contribute. The residual integral is called a mini-superspace approximation, and it can be analyzed using steepest descent, or more generally, Picard-Lefschetz theory.
For the case in hand, since we are asymptotically fixing \(\beta\) and \(\O\), the energy (\(E\)) and angular momentum (\(J\)) are integration variables in the path integral. We can then write down a mini-superspace approximation assuming that the remaining (infinitely many) variables of the path integral have already been integrated over, leaving behind the \(E\) and \(J\) integrations [36]. Alternatively, we can take \(r_{+}\) and \(a\) as a convenient choice of integration variables and write the partition function as \[\begin{gather} \label{Minisuperspace321} \mcz(\beta,\O) = \int_{0}^{\infty}{dr_{+}}\int_{-1}^{1}{da}\,\exp[S - \beta(E - \O J)], \end{gather}\tag{45}\] where the quantities \(E, J\) and \(S\) were obtained as functions of \(r_{+}\) and \(a\) in equations 28 –30 . Note that the domain of integration in the \(r_{+},a\) variables corresponds to \(0 \leq \abs{J} < E < \infty\). In writing 45 , we have only kept functions which go as \(\frac{1}{G}\) in the exponent. This is because we are keeping track of which saddles contribute but not of the one-loop (and higher) corrections.
By our very construction of the integrand in 45 , rotating thermal AdS and all the rotating black hole solutions (quasi-Euclidean, complex, and spurious) satisfying the thermodynamics relations 32 and 33 appear as saddle points of the exponent. To answer the question of which saddles contribute at a given \(\beta\) and \(\O\), we need to decompose the integration region of 45 into a collection of Lefschetz thimbles. Explicitly, we have \[\begin{gather} \label{Decomposition321} \mcc = \sum_{\sigma \in \,\text{saddles}}n_{\sigma}\mcj_{\sigma}, \end{gather}\tag{46}\] where \(\mcc\) denotes the integration region while \(\mcj_{\sigma}\) corresponds to the Lefschetz thimble of saddle \(\sigma\) with a chosen orientation. At each \(\beta\) and \(\O\), we want to know the values of \(n_{\sigma}\) to decide which saddles contribute to the integral. We know that as the parameters are varied continuously, the saddles and the thimbles also change continuously so that the numbers \(n_{\sigma}\) remain constant. These numbers only change when we hit a Stokes surface, where the thimbles also jump discontinuously. From the earlier discussion in section 3, it is clear that the curve \(\beta = \beta_{\rm max}(\O)\) for \(\abs{\O} < 1\), is (part of) the Stokes surface. So, we expect that the decomposition 46 takes one form for \(\beta > \beta_{\rm max}(\O)\) and another for \(\beta < \beta_{\rm max}(\O)\). We will study the two cases separately.
The case \(\beta > \beta_{\rm max}(\O)\) is the problematic one where the small and large Kerr-AdS solutions, as well as the spurious black hole, appear as complex saddle points that can dominate over the rotating thermal AdS saddle at large values of \(\beta\). The thermal saddle already lies on the integration region \(\mcc\). We now need to see how each thimble contributes in the decomposition 46 .
Let us briefly discuss how Lefschetz thimbles for each saddle are constructed (see [37], [38] for detailed reviews). We start by writing the exponent in \(\eqref{Minisuperspace321}\) as \[\begin{gather} \label{Exponent321} \mci(r_{+},a) = S(r_{+},a) - \beta(E(r_{+},a) - \O J(r_{+},a)). \end{gather}\tag{47}\] A saddle point \(p_{\sigma} = (r_{+,\sigma},a_{\sigma})\) is a solution to the equations \[\begin{gather} \label{Saddle32equation321} \frac{\p\mci}{\p r_{+}} = 0, \quad \frac{\p\mci}{\p a} = 0. \end{gather}\tag{48}\] The corresponding thimble is denoted as \(\mcj_{\sigma}\) and is defined as the collection of downward flow trajectories emanating from the saddle. Explicitly, the trajectories \((r_{+}(t),a(t))\) solve the flow equations \[\begin{gather} \label{Flow32equations321} \frac{du^{i}}{dt} = -\frac{\p\Re(\mci)}{\p u^{i}}, \end{gather}\tag{49}\] where \(u^{i} = (\Re(r_{+}),\Im(r_{+}),\Re(a),\Im(a))\), with the condition \((r_{+}(t),a(t)) = p_{\sigma}\) at \(t = -\infty\). Near the saddle point, the direction of the flow is determined by elements of the Hessian matrix.
Substituting in 47 the expressions 28 –30 for \(E, J\) and \(S\) gives \[\begin{gather} \label{Exponent322} \mci(r_{+},a) =\frac{\pi}{8G}\frac{r_{+}^{2} + a^{2}}{(1 - a^{2})^{2}} (4\pi r_{+}(1 - a^{2}) - \beta(r_{+}^{2} + 1)(3 - a^{2} - 2a\O)). \end{gather}\tag{50}\] The function is analytic except for poles at \(a = \pm 1\). A gradient flow (upward or downward) from a generic point will asymptote towards one of the poles.⁵ The thermal solution (\(p_{0} = (0,0)\)) automatically satisfies the saddle equations. The corresponding elements of the Hessian matrix are easily computed as \[\require{physics} \begin{gather} \label{Hessian32thermal321} \eval{\frac{\p^{2}\mci}{\p r_{+}^{2}}}_{p_{0}} = -\frac{3\pi}{4G}\beta, \quad \eval{\frac{\p^{2}\mci}{\p a^{2}}}_{p_{0}} = -\frac{3\pi}{4G}\beta, \quad \eval{\frac{\p^{2}\mci}{\p r_{+}\p a}}_{p_{0}} = 0. \end{gather}\tag{51}\] This tells us that the downward flow trajectories beginning from the thermal saddle move in the \(\Re(r_{+})\) and \(\Re(a)\) directions. Since the function \(\mci\) is “real-valued" for real \(\beta, \O\) (and \(G\)), these flow lines lie entirely on the real plane \(\Im(r_{+}) = 0, \Im(a) = 0\).
In the case \(\beta > \beta_{\rm max}(\O)\), the flow lines do not encounter another saddle and end up spanning the region described by \(-\infty < r_{+} < \infty, -1 < a < 1\). The Lefschetz thimble \(\mcj_{0}\) is precisely this subspace of the full space. We show it as a collection of flow lines emanating from \(p_{0}\) in figure 3. Since the original integration region is a part this subspace, the decomposition 46 into thimbles is \[\begin{gather} \label{Decomposition322} \mcc = \frac{1}{2}\mcj_{0}. \end{gather}\tag{52}\] This means that for any value of the rotation and for any value of the inverse temperature larger than \(\beta_{\rm max}(\O)\), the complex and the spurious black holes of section 3 do not contribute to the partition function, regardless of the value of their on-shell action. This resolves the puzzle we raised earlier.

Now, let us consider the case \(\beta < \beta_{\rm max}(\O)\), for which the small and large black holes are real saddle points. We indicated earlier that the qualitative behaviour of Lefschetz thimbles changes discontinuously as a parameter moves across a Stokes surface. This raises the following question: what happens to the decomposition 46 as the inverse temperature is lowered below \(\beta_{\rm max}(\O)\) at a given \(\O\). In other words, how do contributions of the quasi-Euclidean small and large black holes, along with that of thermal AdS, enter the partition function. Recall that the analogous question in the case without rotation was addressed in [9]. It was concluded that only thermal AdS and the large black hole contribute to the partition function.
Even in the present scenario, the Hessian elements obtained in 51 for the thermal saddle are negative (for \(G > 0\)), so the discussion surrounding the equation holds and the downward flows lie on the real plane. The small (\(p_{-}\)) and large (\(p_{+}\)) black hole saddles are also real and lie in the region \(-\infty < r_{+} < \infty, -1 < a < 1\), so there can be flow lines between two saddles. The fact that the imaginary part of the action remains constant along flow lines certainly allows for such flows, since the action is real-valued when the parameters are real. We show flow lines corresponding to thermal and large black hole saddles in figure 4. It is clear that there is a downward flow from \(p_{0}\) to \(p_{-}\) as well as one from \(p_{+}\) to \(p_{-}\). This means that the entire parameter space \(0< \beta < \beta_{\rm max}(\O)\) for \(\abs{\O} < 1\) and real \(G\) is part of the Stokes surface.

If we forget for the time being about the fact that we are on a Stokes surface, a direct attempt at decomposing the integration region \(\mcc\) in terms of the thimbles shown in figure 4 would read as \[\begin{gather} \label{Decomposition323} \mcc = \frac{1}{2}\mcj_{0} + \mcj_{+}. \end{gather}\tag{53}\] If this is true, we can conclude that the small Kerr-AdS black hole does not contribute to the partition function at fixed \(\beta\) and \(\O\), even in a suppressed manner. But since some thimbles change discontinuously, a thimble decomposition on the Stokes surface is not directly meaningful.
The usual resolution of this problem is by deforming the parameters in a way that we are no longer on the Stokes surface. We proceed by giving a small imaginary part to \(G\) (keeping \(\Re(G) > 0\)). Since it appears as an overall factor, the saddles themselves do not change, but the imaginary part of the on-shell action modifies. In particular, the imaginary part of the on-shell action is unequal for the three saddles meaning that there are no flow lines between the saddles.
The thimble decomposition of \(\mcc\) depends on the sign of \(\Im(G)\), so we treat the two cases, \(\Im(G) > 0\) and \(\Im(G) < 0\), separately. Since the thimbles are two-dimensional submanifolds of the full four-dimensional space, it is hard to visualize them directly. Instead, we show the downward flow lines from each saddle by projecting them onto the complex \(r_{+}\)–plane in figure 5 and complex \(a\)–plane in figure 6.

For saddle \(p_{\sigma}\), we denote the thimble as \(\mcj_{\sigma}^{-}\) in the case \(\Im(G) < 0\) and as \(\mcj_{\sigma}^{+}\) in the case \(\Im(G) > 0\). The thimble decomposition then reads as

Even though the decompositions [Decomposition324] and [Decomposition325] differ, the resulting value of the integral is the same in the limit of vanishing \(\Im(G)\). To see this, we take a difference of the two integrals. Since the mini-superspace integrand is continuous in the limit \(\Im(G) \to 0\), we are left with its integral over a cycle which is homologous to zero. This was expected since the original integral with \(G\) real was absolutely convergent and well-defined. In this way of thinking, the small black hole does contribute to the partition function for either sign of \(\Im(G)\).
But, in the limit \(\Im(G) \to 0\) (from either side) the small black hole contribution is purely imaginary and completely cancels off against the imaginary contribution coming from the thermal and large black hole saddles. This is because the mini-superspace integral is itself real for a real value of \(G\). To make this cancellation manifest, we use the fact that with either sign of \(\Im(G)\), the value of the integral is the same. Further, the flow lines and hence the thimbles in the two cases are complex conjugates of each other. So, if we add the two integrals, the two small black hole contributions as well as the imaginary contributions of the thermal and large black holes saddles cancel against each other, leaving behind the correct answer. In this way of describing the integral for \(\beta < \beta_{\rm max}(\O)\), we conclude that the the small black hole does not contribute.
Finally, we remark on another observation about the mini-superspace integral. The derivatives of the exponent 50 are polynomials in \(r_{+}\) and \(a\) divided by a power of \((1 - a^{2})\). Each root of the polynomials with \(a \neq 1\) constitutes a saddle for the integral. It is clear that there are many saddles in the full complex plane of which we have only considered four, viz., thermal AdS and the Kerr-AdS black holes. The “extra” saddles are also rotating black hole solutions but with a non-standard relation between temperature, rotation, and black hole parameters. Fortunately, the decomposition of the integration region does not pick these additional saddles for \(\beta > 0\) and \(\abs{\O} < 1\). Nevertheless, it would be interesting if any of these saddles, as well as the complex Kerr-AdS and spurious black hole saddles, are found directly from the gauge theory.

5 Discussion↩︎

We posed an issue concerning the phase of holographic gauge theories at low temperatures and finite rotation. The rotating AdS black holes exist as complex saddles at low temperatures with real part of on-shell action smaller than that of the thermal rotating gas. Despite this, their contribution does not enter the partition function at any temperature and rotation. We also addressed the question of whether the unstable small Kerr-AdS black hole contributes to the partition function at high temperatures. It is worthwhile to ask whether saddles which do not contribute to the partition function can be seen at all from the gauge theory side. An example of this is the map between small AdS black holes and small plasma balls in the confined phase [39].
The essential tool for our analysis was the mini-superspace approximation, which reduced the Euclidean path integral into a finite dimensional integral. The residual integral was subjected to steepest descent analysis. This approach has a limitation that it only gives qualitative information about the full path integral, such as which saddle points contribute in different ranges of the parameters. For more detailed information, like the perturbative expansion around each saddle, we must go back to the full path integral. Regardless, the simplicity of this approach makes it useful.
It would be nice if the simple methods presented in this paper can be used to discuss other novel situations. A directly related example is understanding the CFT partition function at a fixed temperature and angular momentum from a bulk perspective. The Hilbert space trace is certainly sensible in this case and the theory can exhibit different phases. Another interesting example is the bulk path integral at a finite cutoff radius and modified boundary conditions [40], [41].

I thank Raghu Mahajan and R. Loganayagam for various discussions and their comments on the draft. I am grateful for numerous interactions with Gautam Mandal, Suvrat Raju, Ashoke Sen, and the rest of the ICTS string group. I would also like to thank Ankur Barsode, Mayank Kumar Bijay, Ritwick Kumar Ghosh, Vinay Kumar, Aiswarya NS, Mithat Ünsal for helping me with some resources. I acknowledge support of the Department of Atomic Energy, Government of India, under project no. RTI4001.

References↩︎