# Minimal area for circular Wilson loops in these coordinates

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?

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