Open String Boundary Variation

Varying $X$ in [A1] while holding $h$ fixed gives $\delta S_P=-T{\int }_{\mathrm{\Sigma }}\sqrt{-h}{h}^{ab}{\mathrm{\partial }}_a{X}_{\mu }{\mathrm{\partial }}_b\delta X^{\mu }{d}^2\sigma$, because $h^{ab}$ is symmetric; [F2] fixes the positive coefficient $T=(2\pi {\alpha }^{\mathrm{\prime }}{)}^{-1}$.

Integrate the divergence ${\mathrm{\partial }}_b(\sqrt{-\det h}{h}^{ab}{\mathrm{\partial }}_a{X}_{\mu }\delta X^{\mu })$ over the coordinate rectangle. The fundamental theorem of calculus in each variable gives the interior term $T{\int }_{\mathrm{\Sigma }}\delta X_{\mu }{\mathrm{\partial }}_b(\sqrt{-\det h}{h}^{ab}{\mathrm{\partial }}_a{X}^{\mu })d^2\sigma$ and the spatial term $-T{\int }_I[C_{\mu }\delta X^{\mu }{]}_0^{\pi }d\tau$; the temporal terms vanish by the Given clause. No unit normal is used in this calculation. On a timelike face, the two-dimensional inverse formula gives $h^{\sigma \sigma }=h_{\tau \tau }\mathrm{/}\det h>0$. Thus $h^{ab}{n}_a{n}_b=1$ and $ds{n}_a=\epsilon \sqrt{-\det h}(d\sigma {)}_{a}d\tau$, proving the stated proper-time form on precisely those faces.

For each component and each endpoint, Dirichlet data make the corresponding $\delta X^{\mu }$ factor zero; Neumann data $C_{\mu }=0$ make its coordinate-density flux coefficient zero; on a timelike face the positive normalizing factors give exactly the equivalent unit-conormal condition. The sum of these endpoint integrands therefore vanishes for coordinatewise mixed choices as well.

In the conformal gauge [F3], $h_{\tau \tau }=-e^{2\varphi }<0$ and $C_{\mu }={\mathrm{\partial }}_{\sigma }{X}_{\mu }$. The unit conormal at constant $\sigma$ is $n_a=\pm e^{\varphi }(d\sigma {)}_a$, so $n_a{h}^{ab}{\mathrm{\partial }}_b{X}^{\mu }=\pm e^{-\varphi }{\mathrm{\partial }}_{\sigma }{X}^{\mu }$. Thus the natural condition is equivalent to ${\mathrm{\partial }}_{\sigma }{X}^{\mu }=0$.