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Matrix models appeared in the context of AdS/CFT while trying to calculate the Circular Wilson Loop. It was first noted by Erickson, Semenoff & Zarembo [hep-th/0003055] that the 2-loop contribution to the circular Wilson Loop canceled exactly and they conjectured that all the contributions for the ladder diagrams where equivalent to a Gaussian Matrix Model.

Later in a work by Drukker, Gross [hep-th/0010274] they showed that there was an anomaly involving the large conformal transformation mapping the line (1/2 BPS) to the circle which was the reason why the Gaussian Matrix Model arose. They solve the matrix model for all N finding:

$\langle W_\textrm{circle} \rangle = \frac{1}{N} L_{N-1}^1(-\lambda/4N) \exp[\lambda/8N]$

Where $L_n^m(x)$ is the Laguerre polynomial. All of this was later explained by Pestun in terms of localization.

At the moment I'm reading this paper by Hubeny, Semenoff where they are trying to utilize this result but for the hiperbola. In equation (11) they present the result, stating that "(the result) can be modified to obtain in the Lorentzian case":

$W[x_0, \tilde{x}_0] = N L_{N-1}^1\left(\frac{\epsilon^2}{N} (M \tau_p)^2\right) \exp\left(\frac{\epsilon^2}{2N} (M \tau_p)^2 \right)$

Where the change was: $\lambda \mapsto \frac{-\lambda \tau_p^2}{\pi^2 u_h}$

Using the definition of $\epsilon = \frac{\lambda E^2}{4\pi^2 M^4}$, $u_h = \frac{M^2}{E^2}$ and $\tau_p$ is some cutoff to avoid the divergence due to the infinite extension of the hyperbola branch.

My question is the following:

This modification is presented as obvious but how would someone come up with that?, how does that condition over lambda assures us that we are switching from the circle: $x_0 ^2 + x_1^2 = R^2$ to the hyperbola $x_0^2 - x_1^2 = R^2$?

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The non-perturbative result in [hep-th/0003055, hep-th/0010274] is based on two facts: the effective propagator is constant and all effective propagators (i.e. the ladder diagrams) resum exactly in a Laguerre polynomial in the coupling constant.

Given that I have not fully read the paper by Hubeny and Semenoff, I think that if you follow the same argument, now a different value of the constant, as written above equation (11), leads to the same Laguerre polynomial but with a rescaled coupling constant, as written in (11).

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