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In all references you can see that the Poincare coordinates are used to get the minimal area for the circular wilson loop. I want to use the metric that is used also for the D3-brane (e.g. see equation (4.37) in arXiv:hep-th/1608.02963).

The $AdS_5$ metric is (with radius $L$)

$$ds^2=L^2(du^2+\cosh^2 u(d\rho^2+\cosh^2\rho d\tau^2)+\sinh^2u ds^2_{S^2})$$

My ansatz is that $u$ depends on $\rho$ only. The other coordinates are constants and I will use $\rho$ and $\tau$ as worldsheet coordinates. The induced metric is

$$g_{ab}=L^2(\partial_au\partial_bu+\cosh^2 u\partial_a\rho\partial_b\rho+\cosh^2u\cosh^2\rho\partial_a\tau\partial_b\tau)$$

then

$$\sqrt{g}=L^2\sqrt{u'^2+\cosh^2 u}\cosh u\cosh\rho$$

So I get the equations of motion $$\frac{\cosh u(\mathcal{F}+\cosh\rho \cosh^2 u(-\sinh (2u)+u''))}{(\cosh^2 u+u'^2)^{3/2}}=0$$ Where $\mathcal{F}$ are terms proportional to derivatives. Let's say that $u$ is constant, then the equation reduces to $\sinh (2u)=0$. So the solution is

$$u=0$$ Now replacing in the action $$S=\frac{1}{2\pi\alpha'}\int d\rho d \tau\sqrt{\det g}=\frac{1}{2\pi\alpha'}\int d\rho d \tau \cosh\rho$$

Off course, my first question is: Am I right?

My second one: How should I set the integration limits?

The answer should be

$$S=-\frac{L^2}{\alpha'} +\frac{L^2a}{\alpha'\varepsilon}$$

I got this using the standar Poincare coordinates.

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