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I am slightly confused concerning the euclidean section of a Kerr black hole. In page 5 of the following paper https://arxiv.org/abs/hep-th/9908022 it is said that in order to get the euclidean section, we need to set $t \to i \tau$ and $a \to i a$. (They consider general Kerr-Newman-AdS black holes but I am simply interested in Kerr asymptotically flat.) This makes sense because we want to keep the $dt \otimes d\phi$ components of the euclidean metric real. What confuses me is that if we do the analysis of the conical singularities as they mention, we will get the following periodicity for $\tau$ and $\phi$

\begin{equation} \tau \sim \tau +\beta \end{equation} \begin{equation} \phi\ \sim \phi+i\beta\Omega_H \end{equation} with $\beta$ the inverse temperature and $\Omega_H$ the angular velocity of the event horizon, namely \begin{equation} \Omega_H=\frac{a}{r_{+}^2+a^2} \end{equation} where $r_{+}$ is the event horizon and $a$ is the rotation parameter of the black hole. What is strange to me is that if we take $a \to 0$ in Boyer-Lindquist coordinates, we get that \begin{equation} \phi \sim \phi \end{equation} because $\Omega_H$ vanishes. This becomes a trivial identification and it does not tell us anything about the periodicity of the $\phi$ coordinate. However, we also know that if we take the $a \to 0$ limit, we get the Schwarzschild black hole in Schwarzschild coordinates. In Schwarzschild Euclidean, we should take the $\phi$ coordinate to have period \begin{equation} \phi \sim \phi+2\pi \end{equation} and even though the Boyer-Lindquist $\phi$ is different than the $\phi$ in Schwarzschild, they match in the limit I am considering $a \to 0$. What does this imply? Does this mean that even though Kerr goes to Schwarzschild in the limit $a \to 0$ as a lorentzian geometry, their euclidean sections are not connected continuously somehow?

Edit1: I also have the notion that in lorentzian Kerr, the $\phi$ coordinate has periodicity $2\pi$. When we go to Euclidean, we seem to get this other periodicity: but shouldn't the periodicity of $2\pi$ be preserved as well? At least that is what happens in Schwarzschild. So we would have both \begin{equation} \phi\ \sim \phi+i\beta\Omega_H \end{equation} \begin{equation} \phi\ \sim \phi + 2\pi \end{equation} It also confuses me that this manipulations are usually done based on the coordinate systems and therefore it is harder to get a notion of what it means to 'euclideanize' in a coordinate invariant way. If someone has a coordinate invariant way to talk about this analytic continuation, I would like to hear it.

Edit2: If we see what really is the expression in the identification of $\phi$, we get \begin{equation} i\beta \Omega_H=i4\pi \frac{r_{+}a}{r_{+}^2\left(1-\frac{a^2}{r_{+}^2}\right)} \end{equation} By doing the analytic continuation $a \to ia$, we have \begin{equation} i\beta \Omega_H=-4\pi \frac{r_{+}a}{r_{+}^2\left(1+\frac{a^2}{r_{+}^2}\right)} \end{equation} we see that it is alway less then $2\pi$ because \begin{equation} r_{+}=a+\sqrt{2}a \end{equation} defines extremality assuming the fact that we set $a \to ia$. So it seems to make the $\phi$ direction smaller in general. But if I try to compute the action on-shell \begin{equation} I=\int_{\partial \mathcal{M}}K-K_0 \end{equation} I have to integrate from $0$ to $2\pi$ along $\phi$ to get the right result mentioned in https://doi.org/10.1103/PhysRevD.15.2752 because since we are sending the boundary to infinity only the leading order of $1/r$ matters which is the same as in Schwarzschild. So I am confused what kind of geometry we have along $\phi$.

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There is a subtlety in expressing the periodicities of variables, because sometimes, there may be a "shift" or "twist" in one variable as we go along a cycle in the other. This is indeed what is happening here.

enter image description here

The picture above is a torus in the complex plane where $z = x + i y$. The torus is defined by two equivelence relations, \begin{align} z \sim z + 1 \\ z \sim z + \tau. \end{align} If we write $$ \tau = \tau_x + i \tau_y $$ then we can write the equivalence relations as \begin{align} (x, y) &\sim (x + 1, y) \\ (x, y) &\sim (x + \tau_x, y + \tau_y). \end{align} There is a "shift" in $x$ as $y$ goes from $0$ to $\tau_y$. Note that the range of x is $1$ and the range of $y$ is $\tau_y$. $\tau_x$ gives the shift.

This is exactly what is happening in the Euclidean Kerr metric. If one defines the variable $$ \Phi \equiv \phi - \Omega t $$ where $t$ is euclidean time, then the equivalence relations are \begin{align} (t, \Phi) &\sim (t + \beta, \Phi) \\ (t, \Phi) &\sim (t, \Phi + 2 \pi ). \end{align} If one uses $\phi = \Phi + \Omega t$, then this becomes \begin{align} (t, \phi) &\sim (t + \beta, \phi + \beta \Omega) \\ (t, \phi) &\sim (t, \phi + 2 \pi ). \end{align} $\beta \Omega$ is therefore not giving the periodicity of $\phi$ but rather the "shift" in $\phi$ as one makes a full cycle in $t$.

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