# A question about Poincare invariance of Polyakov action

I have a question the variation of the Polyakov action, related to this Phys.SE post.

For Polyakov action $$S_p[X,\gamma]=-\frac{1}{4 \pi \alpha'} \int_{-\infty}^{\infty} d \tau \int_0^l d \sigma (-\gamma)^{1/2} \gamma^{ab} \partial_a X^{\mu} \partial_b X_{\mu}.$$

Consider the variation (p. 14 in Polchinski's string theory book)

$$\delta S_p~=~ \frac{1}{2\pi \alpha'} \int_{-\infty}^{\infty} d \tau \int_0^l d \sigma (-\gamma)^{1/2} \delta X_{\mu} \nabla^2X^{\mu}$$ $$- \frac{1}{2\pi \alpha'} \int_{-\infty}^{\infty} d \tau (-\gamma)^{1/2} \delta X_{\mu} \partial^{\sigma} X^{\mu} |^{\sigma=l}_{\sigma=0} \tag{1.2.27}$$

The open-string and closed-string boundary conditions

$$\partial^{\sigma} X^{\mu}(\tau,0)=\partial^{\sigma} X^{\mu}(\tau,l)=0 \tag{1.2.28}$$ $$X^{\mu}(\tau,l) = X^{\mu}(\tau,0), \partial^{\sigma} X^{\mu}(\tau,0)=\partial^{\sigma} X^{\mu}(\tau,l), \gamma_{ab}(\tau,l)= \gamma_{ab}(\tau,0) \tag{1.2.30}$$

will make the "surface term" (the second term in the RHS of Eq. (1.2.27)) vanishes. It is said (p. 14-15 of Polchinski)

The open-string boundary condition (1.2.28) and closed string boundary condition (1.2.30) are the only possibilities consistent with $D-$dimensional Poincare invariance and the equation of motion.

The boundary condition (1.2.28)/(1.2.30) will make the surface term vanish. The resulting equation of motion is tensor equation.
$$(-\gamma)^{1/2} \nabla^2 X^{\mu}=0$$ My question is, why non-vanishing surface term will break the Poincare invariance? E.g. a Lorentz transformation $X_{\mu} \rightarrow \Lambda_{\mu}^{\nu} X_{\nu}$ will not affect $\sigma,\tau$ space

For instance, If we have one or more Dirichlet conditions for $\sigma = 0$ and/or $\sigma = l$, this means that one or two of the extremities of the open string is ending on D-branes, so the Poincaré invariance is broken (for instance, the translation invariance is broken, because the D-brane has specific position in target space-time).