= Solution
Varying the <Polyakov action> with respect to the worldsheet metric gives
$$
T_{\alpha\beta}=T\left(\partial_\alpha X\cdot\partial_\beta X
-\frac12g_{\alpha\beta}g^{\gamma\delta}\partial_\gamma X\cdot\partial_\delta X\right)=0.
$$
In <conformal gauge>, variation of $X$ gives the wave equation
$$
(\partial_\tau^2-\partial_\sigma^2)X^\mu=0,
$$
while the <Virasoro constraint>[Virasoro constraints] become $\dot X\cdot X'=0$ and $\dot X^2+X'^2=0$. The boundary variation is $-T\int d\tau,X'\cdot\delta X$ at each endpoint. It vanishes through a <Neumann boundary condition> $X'^\mu=0$, a <Dirichlet boundary condition> $\delta X^\mu=0$, or a direction-by-direction mixture of the two.
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