Target space Poincare symmetry of Nambu-Goto action

How do I show invariance under the target space Poincare transformations of the action for a relativistic string,

$$S=-\frac{1}{2 \pi \alpha'} \int{\text{d}^2 \zeta}\sqrt{-\det(\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu \nu})}~~?$$

This is specifically taken from West's "Introduction to strings and branes" and the world sheet reparameterisation invariance is explained; however this one isn't.

• It looks to me like Lorentz symmetry is manifest while the presence of derivatives only implies translation invariance. That's all you need. – Danu May 27 '14 at 10:11
• – Qmechanic Jan 24 '16 at 5:23

Lorentz invariance covers boosts and rotations, and translation invariance is immediately apparent if you realize the the expression consists of spacetime derivatives of $X^\mu$ only. Therefore, the expression is Poincare invariant.
Poincare symmetry group of d-dimensional flat spacetime (isometries of flat space) consists of ${\Lambda^{\mu}}_{\nu}$ (Lorentz transformations, i.e. satisfying $SO(d-1,1)$ algebra) and $K^{\mu}$ (translations i.e. commuting algebra). In Nambu-Goto action one interprets $X^{\mu}$ as "flat" spacetime vectors, so the action should be invariant under $X'^{\mu} = {\Lambda^{\mu}}_{\nu} X^{\nu} + K^{\mu}$. To check this note that $\partial_{\alpha} X'^{\mu} = {\Lambda^{\mu}}_{\nu} \partial_{\beta} X^{\nu}$, therefore: $$\eta_{\mu \nu} \partial_{\alpha} X'^{\mu} \partial_{\beta} X'^{\nu} = {\Lambda^{\mu}}_{\gamma} \eta_{\mu \nu} {\Lambda^{\nu}}_{\sigma}\partial_{\alpha} X^{\gamma} \partial_{\beta} X^{\sigma} = \eta_{\gamma \sigma} \partial_{\alpha} X^{\gamma} \partial_{\beta} X^{\sigma}$$ where the last equality comes from ${\Lambda^{\mu}}_{\gamma} \eta_{\mu \nu} {\Lambda^{\nu}}_{\sigma} = \eta_{\gamma \sigma}$ which is the definition of Lorentz transformations.
It is interesting to note that in Nambu-Goto action the above transformations should be considered as internal symmetries of fields $X^{\mu}(\sigma, \tau)$ on the world-sheet.