= Solution
The <Polyakov action> has independent <worldsheet metric> $\gamma_{ab}$ and embedding fields $X^\mu$. Varying the metric before imposing a gauge gives
$$
T_{ab}=\frac1{\alpha'}\left(\partial_aX\cdot\partial_bX-\frac12\gamma_{ab}\gamma^{cd}\partial_cX\cdot\partial_dX\right)=0.
$$
<Worldsheet diffeomorphisms> and <Weyl transformations> allow <conformal gauge> $\gamma_{ab}=e^{2\omega}\eta_{ab}$ locally. The Weyl factor then cancels out of the classical action, leaving
$$
S_X=\frac1{4\pi\alpha'}\int d\tau\,d\sigma\,(\dot X^2-X'^2),\qquad (\partial_\tau^2-\partial_\sigma^2)X^\mu=0.
$$
These are free two-dimensional <scalar fields>, but the surviving metric equations require
$$
\boxed{(\dot X+X')^2=(\dot X-X')^2=0.}
$$
Equivalently, $\dot X^2+X'^2=0$ and $\dot X\cdot X'=0$. <Gauge fixing> does not license dropping these <Virasoro constraints>.
For an ordinary <open string>, the boundary conditions relate the two chiral sectors. With $\alpha_0^\mu=\sqrt{2\alpha'}k^\mu$, their independent Fourier generators are
$$
L_m=\frac12\sum_{r\in\mathbb Z}\alpha_{m-r}\cdot\alpha_r.
$$
The classical oscillator <Poisson brackets> $\{\alpha_m^\mu,\alpha_n^\nu\}=-im\eta^{\mu\nu}\delta_{m+n,0}$ give the <classical Virasoro constraint algebra>
$$
\{L_m,L_n\}=-i(m-n)L_{m+n}.
$$
Thus they are <first-class constraints>. In the <quantum theory> one applies <normal ordering> to the product and uses $[\alpha_m^\mu,\alpha_n^\nu]=m\eta^{\mu\nu}\delta_{m+n,0}$. The oscillator <commutator> gives
$$
[L_m,\alpha_n^\mu]=-n\alpha_{m+n}^\mu,\qquad [L_m,L_n]=(m-n)L_{m+n}+\frac D{12}(m^3-m)\delta_{m+n,0}.
$$
For example, for $m>0$ the extra vacuum term is $(D/2)\sum_{r=1}^{m-1}r(m-r)=D(m^3-m)/12$. This exhibits the <Virasoro central extension> rather than concealing it in a classical constraint equation. The matter <central charge> is $D$; the <worldsheet ghost fields> have charge $-26$. Quantum consistency sets $D=26$ and the <string intercept> to one.
In <old covariant string quantization> the <physical-state Virasoro conditions for an open string> are
$$
L_m|\Psi\rangle=0\quad(m>0),\qquad (L_0-1)|\Psi\rangle=0,\qquad L_0=\alpha'k^2+N.
$$
<Physical string states> are then identified modulo physical <null string states>. One does not impose both positive and negative generators as annihilation conditions: their central extension would obstruct that prescription. The resulting <null-state quotient of a string> agrees with the transverse light-cone spectrum at nonzero <momentum>.
At $N=0$, the <oscillator vacuum> $|k\rangle$ is a single <spacetime> <scalar field> with $\alpha'M^2=-1$: the bosonic <tachyon>. At $N=1$, write $|\Psi\rangle=\epsilon_\mu\alpha_{-1}^\mu|k\rangle$. The zero-mode equation gives $k^2=0$, and $L_1$ gives $k\cdot\epsilon=0$. The null descendant $L_{-1}|k\rangle=\sqrt{2\alpha'}k\cdot\alpha_{-1}|k\rangle$ identifies $\epsilon\sim\epsilon+\lambda k$. \b[This leaves 24 physical polarizations of a massless gauge vector.]
At $N=2$, use the <level-two open-string polarization decomposition>
$$
|\Psi\rangle=\left(\frac12h_{\mu\nu}\alpha_{-1}^\mu\alpha_{-1}^\nu+b_\mu\alpha_{-2}^\mu\right)|k\rangle,\qquad h_{\mu\nu}=h_{\nu\mu}.
$$
The mass is $M^2=1/\alpha'$. The oscillator <commutators> give all the nontrivial positive-mode conditions:
$$
\sqrt{2\alpha'}\,k^\mu h_{\mu\nu}+2b_\nu=0,\qquad \frac12h^\mu{}_{\mu}+2\sqrt{2\alpha'}\,k\cdot b=0.
$$
Higher positive modes annihilate this level. To see the quotient explicitly, the 25 transverse-parameter null states $L_{-1}(\xi\cdot\alpha_{-1}|k\rangle)$, with $k\cdot\xi=0$, shift
$$
\Delta h_{\mu\nu}=\sqrt{2\alpha'}(k_\mu\xi_\nu+k_\nu\xi_\mu),\qquad \Delta b_\mu=\xi_\mu.
$$
The additional <level-two scalar Virasoro null state> $(L_{-2}+\tfrac32L_{-1}^2)|k\rangle$ shifts
$$
\Delta h_{\mu\nu}=\eta_{\mu\nu}+6\alpha'k_\mu k_\nu,\qquad \Delta b_\mu=\frac52\sqrt{2\alpha'}\,k_\mu.
$$
Substitution into the two physical equations verifies the null-state conditions; the second residual is $(D-26)/2$. Because $k$ is timelike, these null shifts span all components of $b$, so choose a representative with $b=0$. The physical conditions then say $k^\mu h_{\mu\nu}=0$ and $h^\mu{}_{\mu}=0$. In the <rest frame> this is a <traceless second-rank tensor> in 25 spatial dimensions, with $25\cdot26/2-1=324$ polarizations. The light-cone basis independently gives $24$ states $\alpha_{-2}^i|k\rangle$ and $24\cdot25/2=300$ symmetric states $\alpha_{-1}^i\alpha_{-1}^j|k\rangle$. Hence
$$
\boxed{\alpha'M^2=-1,0,1:\qquad1\ \text{scalar},\quad24\ \text{vector polarizations},\quad324\ \text{massive spin-two polarizations}.}
$$
For the fermionic extension, an explicit convention prevents ambiguity in the signs. Keep <worldsheet> signature $(-,+)$ and use
$$
\rho^0=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\qquad \rho^1=\begin{pmatrix}0&i\\i&0\end{pmatrix},\qquad \{\rho^a,\rho^b\}=-2\eta^{ab}.
$$
Take $\psi=(\psi_1,\psi_2)^T$ and the constant parameter $\epsilon=(\epsilon_1,\epsilon_2)^T$ to be real Grassmann-odd Majorana components, with $\bar\psi=\psi^T\rho^0$. Put $\partial_\pm=\partial_\tau\pm\partial_\sigma$. Apart from the common factor $1/(4\pi\alpha')$, the combined Lagrangian is
$$
\mathcal L=\partial_+X\cdot\partial_-X+i\psi_1\cdot\partial_+\psi_1+i\psi_2\cdot\partial_-\psi_2.
$$
The rigid transformations can be written
$$
\delta X=\bar\epsilon\psi,\qquad \delta\psi=-i\rho^a\partial_aX\,\epsilon,
$$
or, in components,
$$
\delta X=i(\epsilon_2\psi_1-\epsilon_1\psi_2),\qquad \delta\psi_1=-\epsilon_2\partial_-X,\qquad \delta\psi_2=\epsilon_1\partial_+X.
$$
For the $\epsilon_2$ part, the bosonic variation is $i\epsilon_2(\partial_+\psi_1\cdot\partial_-X+\partial_+X\cdot\partial_-\psi_1)$, while the fermionic variation is $-i\epsilon_2\partial_-X\cdot\partial_+\psi_1+i\epsilon_2\psi_1\cdot\partial_+\partial_-X$. Anticommuting the constant parameter past the <fermion> is essential in the second term. The first terms cancel. The other chiral component works in the same way, giving the <chiral proof of rigid worldsheet supersymmetry>
$$
\boxed{\delta\mathcal L=i\epsilon_2\partial_-(\psi_1\cdot\partial_+X)-i\epsilon_1\partial_+(\psi_2\cdot\partial_-X).}
$$
Thus the action is invariant off shell up to a <boundary term>. The transformation is a genuine <supersymmetry>: on $X$ its <commutator> for parameters $\epsilon,\zeta$ is the translation
$$
[\delta_\epsilon,\delta_\zeta]X=2i(\epsilon_2\zeta_2\partial_-+\epsilon_1\zeta_1\partial_+)X.
$$
On $\psi$ the same translation follows using $\partial_+\psi_1=0$ and $\partial_-\psi_2=0$; the remaining cross terms are proportional to these <Dirac equations>. This is <on-shell closure of rigid worldsheet supersymmetry>. On a <surface with boundary>, compatible endpoint conditions must make the boundary flux vanish. On a compact <worldsheet>, a constant parameter must preserve the <spin structure>; the local identity alone does not make a constant transformation compatible with every antiperiodic sector.
The <fermions> contribute to the <worldsheet stress-energy tensor> and produce a fermionic <superconformal current>. In holomorphic normalization with $\psi^\mu(z)\psi^\nu(w)\sim\eta^{\mu\nu}/(z-w)$,
$$
T=-\frac1{\alpha'}:\partial X\cdot\partial X:-\frac12:\psi\cdot\partial\psi:,qquad G=i\sqrt{\frac2{\alpha'}}:\psi\cdot\partial X:.
$$
The free-field contractions give $G(z)G(w)\sim(2c/3)/(z-w)^3+2T(w)/(z-w)$ and <conformal weight> $3/2$ for $G$, where $c=D+D/2=3D/2$. Consequently the modes obey the <N=1 super-Virasoro algebra>
$$
\begin{aligned}
[L_m,L_n]&=(m-n)L_{m+n}+\frac c{12}(m^3-m)\delta_{m+n,0},\\
[L_m,G_r]&=\left(\frac m2-r\right)G_{m+r},\\
\{G_r,G_s\}&=2L_{r+s}+\frac c3\left(r^2-\frac14\right)\delta_{r+s,0}.
\end{aligned}
$$
Dropping the central terms gives the classical graded <constraint algebra>. <Integer> $r$ gives the <Ramond sector>; half-integer $r$ gives the <Neveu–Schwarz sector>. In the displayed conformal-plane convention $G_0^2=L_0-c/24$ in the <Ramond sector>. Shifting to the Ramond vacuum-normal-ordered <zero mode> gives the usual <Ramond supercurrent zero-mode square> $G_0^2=L_0^{(R)}$.
Rigid symmetry of matter already supplies these currents, but promoting their vanishing to new gauge constraints requires coupling to <worldsheet> <supergravity> and varying its <worldsheet gravitino>. In <superconformal gauge> one retains $T=G=0$. The fermionic current constraints and their physical-state conditions extend the bosonic Virasoro conditions. The ghost charges are $-26+11=-15$, so cancellation with $3D/2$ gives the <critical dimension of the RNS superstring> $D=10$; the NS and Ramond intercepts are respectively $1/2$ and zero. This distinguishes the global <supersymmetry> established above from its locally gauged string completion.
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