Finding the maps inside of the Nambo-Goto action

Question

Let's sps that $\phi:M \to N$ is a diffeomorphism where $M$ and $N$ are two separate manifolds. Let $f$ be a function s.t. $f:M \to R$ where $R$ is the set of all real numbers. The composition of our diffeomorphism mapping and the function $f$ defined on $N$ follows as $f [ \phi]:N \to R$. This can be defined as our pullback on $f$ by $\phi$ denoted $\phi^*f$. Now consider the Nambu-Goto action describing the propagation of a 1-brane/string through the ambient spacetime $$S=-T\int_Ed^2E\sqrt{-\det h_{\mu\nu}}$$ $$=-T \int_Ed^2E\sqrt{(\dot X \cdot X')^2-\dot X^2 X'^2}$$ where $\dot X$ is the derivative w.r.t $\tau$ and $X'$ is the derivative w.r.t $\sigma$. How do are the functions $X$ found such that we can compute the action for a certain case?

Answer 1

In the Nambu-Goto action the worldsheet metric tensor $$h=X^{\ast}G~\in~ \Gamma\left( {\rm Sym}^2(T^{\ast}E)\right)$$ is the pullback of the target space metric tensor $$G~\in~ \Gamma\left( {\rm Sym}^2(T^{\ast}N)\right)$$ via the string map $$ E\stackrel{X}{\longrightarrow} N.$$