\epsilon)=1$ and $\tau \sim \tau+2\pi$. They applied the results of~\cite{Closset-ml-2012ru} to look for supersymmetric regularized spheres. They observed that this background can be supersymmetric by turning on a connection coupling to the $\U(1)$ R symmetry \begin{equation} H=-\frac{i}{f_{\epsilon}}; ~~~~~~ A^{(R)}=\bigg(\frac{n}{ f_{\epsilon}}-1\bigg) \frac{d\tau}{2}+\bigg(\frac{1}{ f_{\epsilon}}-1\bigg) \frac{d\phi}{2} \end{equation} where the killing spinor equation $D_{\mu} \xi=-\frac{1}{2} H \gamma_{\mu} \xi$ is satisfied for two constant spinors with opposite R charges. For $n \not = 1$ the background is different that the original one but if $n\rightarrow 1,f_{\epsilon} \rightarrow 1$ we get the three sphere without any extra field turned on. After localizing a Chern-Simons matter theory in this background they observed that the matrix model that they obtain is equivalent to the one of the squashed sphere~\cite{Hama-ml-2011ea} for $b=\sqrt{n}$. In~\cite{Martelli-ml-2011fu} it was also observed that in the localization computation nothing depends on the precise shape of $f$. Now we would like to consider our Wilson loop in the $S^3$ and compute its entanglement entropy. The branched sphere is conformal to the hyperboloid, the loop we have been considering in these coordinates is extended along $\tau$ and sits at $\theta=\frac{\pi}{2}$. Wilson loops in the $b$-deformed sphere were considered in~\cite{Kapustin-ml-2013hpk,Tanaka-ml-2012nr}. We can now compute the entanglement entropy for this Wilson loop~\cite{Nishioka-ml-2013haa} \begin{equation} S_W= \left. (1-n\partial_{n}) \log |W(S^3_{b=\sqrt{n}})| \right|_{n=1} \end{equation} Here we have taken the absolute value to remove an unexpected phase. The phase is unexpected because the computation of a Wilson loop expectation value has an interpretation as the norm of a state. This is the state produced by Euclidean evolution on a half sphere with the insertion of a half circular Wilson line. The entanglement entropy computation also has a similar interpretation. A similar phase is also present in the computation of the sphere partition function. It was shown in~\cite{Closset-ml-2012vg} that it arises from a local counterterm because the regularization that is implicitly used in performing the localization procedure does not respect the unitarity of the theory. Here we expect a similar interpretation. For this reason we are always going to ignore such phases. For the pure Chern Simons theory this goes under the name of ``framing ambiguity''. The supersymmetric Wilson loop that we are considering on the squashed sphere is \begin{equation} W(S^3_b)={\text Tr} {\cal P} e^{\oint i A+\sigma} \end{equation} Where $\sigma$ is the scalar partner in the vector multiplet. These theories in the squashed sphere localize to a matrix model~\cite{Hama-ml-2011ea}, so the expectation value of the Wilson loop is just the expectation value in a matrix model: $\langle W(S^3_b) \rangle= \langle e^{2 \pi b \sigma}\rangle_{S^3_b,MM}$. An analytic expression seems complicated to find. However, because we are only interested in the first derivative at $b=1$, we can make the problem easier. This is because the Chern-Simons and matter terms in the matrix model are symmetric under $b \rightarrow b^{-1}$ and thus $b$ even to first order. This means that the only contribution will come from the localization of the Wilson loop, ie we can set $b=1$ for the other factors and compute the expectation value of the winding Wilson loop in $S^3$. So we only have to compute the expected value of the winding Wilson loop, do the analytical continuation and take the derivative: $\partial_b W(S_b^3) | _{b=1} =\partial_m \langle e^{2 \pi m \sigma} \rangle_{S^3,MM}|_{m=1}$ . We write $m$ so it is clear that it is a loop wound $m$ times in a great circle of the $S^3$. Before exploring the case of ABJM, we will analyze its relation with $h_w$. ]]>