Periodicity trick for Kerr Black Holes 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 $\phi$.
 A: 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.

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 $t$.
