= Solution
Use target-space signature $(-,+,\ldots,+)$ and let a dot between target vectors denote contraction with the <Minkowski metric>. Varying the <string embedding map> in the <Polyakov action>, with periodic variations around the closed string, gives
$$
\delta_X I=\frac1{2\pi\alpha'}\int d^2\sigma\,\delta X_\mu\,\partial_a\big(\sqrt{-\gamma}\,\gamma^{ab}\partial_bX^\mu\big).
$$
Thus the embedding <equation of motion> is
$$
\boxed{\partial_a\big(\sqrt{-\gamma}\,\gamma^{ab}\partial_bX^\mu\big)=0.}
$$
The independent <worldsheet metric> variation uses $\delta\sqrt{-\gamma}=-\tfrac12\sqrt{-\gamma}\gamma_{ab}\delta\gamma^{ab}$. It sets the <worldsheet stress tensor> to zero:
$$
\boxed{\partial_aX\cdot\partial_bX-\frac12\gamma_{ab}\gamma^{cd}\partial_cX\cdot\partial_dX=0.}
$$
In <conformal gauge>, $\gamma_{ab}=e^{2\omega}\operatorname{diag}(-1,1)$, the Weyl factor cancels. Define worldsheet derivatives $\partial_\pm=(\partial_\tau\pm\partial_\sigma)/2$. The equations and <Virasoro constraints> reduce to
$$
\boxed{\partial_+\partial_-X^\mu=0,\qquad (\partial_+X)^2=(\partial_-X)^2=0.}
$$
Equivalently, $\ddot X-X''=0$, $\dot X\cdot X'=0$ and $\dot X^2+X'^2=0$. The free wave equation permits independent periodic left- and right-moving fields.
Take $0\le\sigma<2\pi$ and the uncompactified <closed-string mode expansion>
$$
X^\mu=x^\mu+\alpha'p^\mu\tau+i\sqrt{\frac{\alpha'}2}\sum_{n\ne0}\frac1n\left(\alpha_n^\mu e^{-in(\tau-\sigma)}+\widetilde\alpha_n^\mu e^{-in(\tau+\sigma)}\right),
\qquad \alpha_0^\mu=\widetilde\alpha_0^\mu=\sqrt{\frac{\alpha'}2}p^\mu.
$$
The <Virasoro generators> are the Fourier coefficients of the two quadratic <worldsheet stress tensors>. Their quantum expressions are
$$
L_n=\frac12\sum_m:\!\alpha_{n-m}\cdot\alpha_m\!:,
\qquad \widetilde L_n=\frac12\sum_m:\!\widetilde\alpha_{n-m}\cdot\widetilde\alpha_m\!:.
$$
The <canonical commutation relations> are $[\alpha_m^\mu,\alpha_n^\nu]=m\eta^{\mu\nu}\delta_{m+n,0}$, with an identical tilded algebra and commuting chiral sectors. <Normal ordering> places annihilation modes to the right. It gives $L_0=\alpha'p^2/4+N$, where $N=\sum_{n>0}\alpha_{-n}\cdot\alpha_n$, and similarly on the other side. In <old covariant string quantization>, <physical string states> obey
$$
L_{n>0}|\Psi\rangle=\widetilde L_{n>0}|\Psi\rangle=0,
\qquad (L_0-a)|\Psi\rangle=(\widetilde L_0-a)|\Psi\rangle=0,
$$
modulo physical <null string states>. Negative generators are adjoint constraints on bras, not further annihilation conditions on the same ket. In critical <bosonic string theory>, $d=26$ and the <string intercept> is $a=1$. These <closed-string physical-state Virasoro conditions> imply
$$
N=\widetilde N,\qquad M^2=-p^2=\frac4{\alpha'}(N-1).
$$
Thus the <massless first closed-string level> has $N=\widetilde N=1$, and its general state is
$$
|\zeta,p\rangle=\zeta_{\mu\nu}\alpha_{-1}^\mu\widetilde\alpha_{-1}^\nu|0,p\rangle.
$$
To derive the <polarization tensor> constraints, the oscillator algebra gives $[L_n,\alpha_m^\mu]=-m\alpha_{n+m}^\mu$. Applying $L_1$ to this state replaces $\alpha_{-1}^\mu$ by $\alpha_0^\mu$; the independent tilded condition replaces the other oscillator by $\widetilde\alpha_0^\nu$. All higher positive generators annihilate the state automatically. Therefore
$$
\boxed{p^2=0,\qquad p^\mu\zeta_{\mu\nu}=0,\qquad p^\nu\zeta_{\mu\nu}=0.}
$$
There is no condition that the tensor trace vanish: each chiral side has only one oscillator, so neither $L_2$ nor $\widetilde L_2$ imposes a trace condition. The <null-state quotient of a string> also identifies
$$
\zeta_{\mu\nu}\sim\zeta_{\mu\nu}+p_\mu\xi_\nu+p_\nu\widetilde\xi_\mu,
\qquad p\cdot\xi=p\cdot\widetilde\xi=0.
$$
For example, $L_{-1}(\xi_\nu\widetilde\alpha_{-1}^\nu|0,p\rangle)$ is a momentum-longitudinal state with zero inner product against every transverse physical polarization. This explains the <string-state gauge redundancy>, rather than merely counting covariant components. For an ordinary nonzero null momentum, choose a frame adapted to $p$. The constraints remove one null direction in each index, and these null-state identifications remove the remaining longitudinal directions. The representatives are then $\zeta_{ij}$, $i,j=1,\ldots,24$.
Decompose that transverse matrix into its symmetric trace-free part, antisymmetric part and trace. \b[They describe the <graviton>, <Kalb–Ramond field> and <dilaton>, respectively.] Their <particle polarization> counts are
$$
\boxed{\left(\frac{24\cdot25}{2}-1\right)+\frac{24\cdot23}{2}+1=299+276+1=576.}
$$
These are representations of the rotational part $SO(24)$ of the massless <little group>. The dilaton corresponds to the transverse trace; a covariant scalar polarization can be written proportional to $\eta_{\mu\nu}-p_\mu\ell_\nu-\ell_\mu p_\nu$, with $\ell^2=0$ and $p\cdot\ell=1$. It is transverse in both indices and has trace 24. The nonzero-momentum assumption is appropriate for propagating particle states; zero momentum requires separate treatment of gauge and background zero modes.
In a general background, these fields replace the flat-space free action by the <string nonlinear sigma model>. A convenient Euclidean convention is
$$
\begin{aligned}
S_E[X,\gamma;G,B,\Phi]
={}&\frac1{4\pi\alpha'}\int d^2\sigma\,\sqrt\gamma\,\gamma^{ab}G_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu\\
&+\frac{i}{4\pi\alpha'}\int d^2\sigma\,\epsilon^{ab}B_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu
+\frac1{4\pi}\int d^2\sigma\,\sqrt\gamma\,\Phi(X)R^{(2)}.
\end{aligned}
$$
Here $\epsilon^{ab}$ is the orientation density; reversing its convention changes the displayed sign of the two-form coupling. The <metric tensor> is $G_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}+\cdots$, $B$ is the antisymmetric <Kalb–Ramond field>, and $\Phi$ is the <dilaton>. Their worldsheet couplings reproduce the corresponding <massless closed-string vertex operators> when expanded about flat space. The <Polyakov path integral> integrates $e^{-S_E}$ over embedding fields and metrics modulo <worldsheet diffeomorphisms> and <Weyl transformations>; <gauge fixing> supplies ghost determinants and moduli integrations. Quantum consistency of a general background additionally requires vanishing <sigma-model beta functions>, not just insertion of arbitrary backgrounds into a free theory.
Finally, separate the constant dilaton mode, $\Phi=\Phi_0+\widehat\Phi$. By the <Gauss-Bonnet theorem>, its action is
$$
S_{\Phi_0}=\frac{\Phi_0}{4\pi}\int\sqrt\gamma R^{(2)}=\Phi_0\chi(\Sigma).
$$
For a connected oriented surface of <genus> $g$, its <Euler characteristic> is $\chi=2-2g$. Defining the <string coupling> by $g_s=e^{\Phi_0}$ gives
$$
\boxed{e^{-S_{\Phi_0}}=g_s^{-\chi}=g_s^{2g-2}.}
$$
Thus the <string genus expansion> is a topology sum with two extra powers of $g_s$ per handle. An amplitude with $n$ canonically normalized external closed-string vertices has the additional vertex normalization $g_s^n$, so its overall factor is $g_s^{2g-2+n}$. The curvature term, rather than a guessed coupling inserted at each internal interaction, supplies this dependence directly from the functional integral.
Back to article page
