I've started to read something about strings and I feel a little confused with the Polyakov action. The reason is because in this action you get two metrics, one of them is the induced metric over the world-sheet and the other is an arbitrary metric on every point of this world-sheet.
This is the definition I have of the Polyakov action: $\int d\tau d\sigma(-\gamma)^{1/2}\gamma^{ab}h_{ab}$
Here $h_{ab}=\partial_{a}X^{\mu}\partial_{b}X^{nu}g_{\mu\nu}$ and is called induced metric, because you can get it from the metric of the spacetime. On the other hand, the second one $\gamma^{ab}$ is called metric as well, but is dynamical and arbitrary, because is as a field smeared on worldsheet, this new dynamical metric don't have any relation to $h_{ab}$, this $\gamma^{ab}$ get a dependence on $h^{ab}$ only when we start to work with the EOM with regard to this metric.
So now I'm confused. Which of this metrics $h^{ab}$ or $\gamma^{ab}$ I must use to rising and lowering indices in a tensor living in the world-sheet?
My answer would be $h^{ab}$, because it has a geometric meaning , but then why you give the name of metric to the other field $\gamma^{ab}$