= Solution
In <conformal gauge>, variation of the flat-background <Polyakov action> gives the bulk <wave equation> and the endpoint term
$$
\delta S_{\partial}=-\frac1{2\pi\alpha'}\int d\tau\,\left[\partial_\sigma X_\mu\,\delta X^\mu\right]_{\sigma=0}^{\sigma=\pi}.
$$
For each endpoint and target-space direction, two basic choices make this term vanish. A <Neumann boundary condition> allows arbitrary $\delta X^\mu$ and requires $\partial_\sigma X^\mu=0$. A <Dirichlet boundary condition> fixes $X^\mu$ at the endpoint, hence $\delta X^\mu=0$; for a fixed endpoint location its time derivative also vanishes. Thus
$$
\boxed{\text{Neumann: }\partial_\sigma X^\mu=0;\qquad
\text{Dirichlet: }X^\mu|_{\partial\Sigma}=y^\mu.}
$$
Both are consistent with the <principle of stationary action> and zero energy flux through a stationary endpoint. Different directions can use different conditions. Background two-form or boundary gauge fields can modify tangential conditions into mixed ones; the displayed alternatives are the basic flat, unforced choices.
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