# 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://journals.aps.org/prd/abstract/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 $$\phi$$.