Classical Open String Ending on a Flat D2-Brane

The coordinate functions are $C^{\mathrm{\infty }}$. For each of $X^1,\dots ,X^4$, the $\tau$-second derivative and $\sigma$-second derivative are both the negative of that coordinate, so ${\ddot{X}}^{\mu }-X^{\mu \mathrm{\prime }\mathrm{\prime }}=0$. $X^0$ is linear in $\tau$, and $X^5,\dots ,X^9$ are constant, so their wave equations also hold. By [F1], [A1], [F2], and [F3], this verifies the free conformal-gauge field equations in the adopted model.

At $\sigma =0,\pi$, $\sin \sigma =0$, hence $X^3=X^4=0$ and $X^5=\cdots =X^9=0$, so all seven normal coordinates are fixed on $W$. Also ${\mathrm{\partial }}_{\sigma }{X}^0=0$, ${\mathrm{\partial }}_{\sigma }{X}^1=-ϵ\cos \tau \sin \sigma =0$, and ${\mathrm{\partial }}_{\sigma }{X}^2=-ϵ\sin \tau \sin \sigma =0$ at both endpoints. These equalities hold for every real $\tau$. Restrict the smooth fields to any finite $[a,b]\times [0,\pi ]$ and take variations vanishing at its temporal faces; [F5] applies and its spatial flux vanishes by the displayed endpoint equalities. Every variation with compact time support on the full strip lies in the interior of some such interval. Thus the fixed Dirichlet and Neumann conditions in [F4]–[F5] hold locally for all time, without requiring a finite infinite-time action.

Direct differentiation gives ${\dot{X}}^2=-2{ϵ}^2+{ϵ}^2({\cos }^2\sigma +{\sin }^2\sigma )=-{ϵ}^2$, $X^{\mathrm{\prime }2}={ϵ}^2({\sin }^2\sigma +{\cos }^2\sigma )={ϵ}^2$, and $\dot{X}\cdot X^{\mathrm{\prime }}=0$, because the two rotating component pairs have cancelling cross terms. Therefore ${\dot{X}}^2+X^{\mathrm{\prime }2}=0$ and $\dot{X}\cdot X^{\mathrm{\prime }}=0$. The induced metric is ${\gamma }_{ab}={ϵ}^2\mathrm{diag}(-1,+1)$, so it is nondegenerate Lorentzian and $X$ has rank two since $ϵ>0$.

The map therefore satisfies the free Polyakov field and metric equations and the D2 endpoint data on all of $\mathbb{R}\times [0,\pi ]$. It is a classical bosonic coordinate-field example only; no fermionic, quantum, charge, or backreaction conclusion follows.