 Solution

In conformal gauge, variation of the flat-background Polyakov action gives the bulk wave equation and the endpoint termFor 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. ThusBoth 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.