= Solution
\b[Independent metric and <Hamiltonian> parametrization.] The Lorentzian <Polyakov action> is
$$
\boxed{I_{\mathrm P}[X;\gamma]=-\frac T2\int d^2\sigma\,\sqrt{-\gamma}\,
\gamma^{ab}\partial_aX\cdot\partial_bX.}
$$
The <worldsheet metric> is independent of the embedding. Its equation sets the traceless <worldsheet stress tensor> to zero, making the metric locally conformal to the <induced worldsheet metric>; substitution recovers the <Nambu–Goto action>. Eliminating momentum in the <Nambu-Goto phase-space action> instead gives $(\dot X-uX')^2/(2e)-eT^2X'^2/2$. On the positive-lapse branch, choose the independent metric, up to a positive <Weyl transformation>, as
$$
\gamma_{ab}=
\begin{pmatrix}u^2-e^2T^2&u\\u&1\end{pmatrix},\qquad
\sqrt{-\gamma}\,\gamma^{ab}=
\begin{pmatrix}
-1/(eT)&u/(eT)\\
u/(eT)&eT-u^2/(eT)
\end{pmatrix}.
$$
Its determinant is $-e^2T^2$. Multiplication by $-T/2$ recovers the reduced density, including the mixed term $-u\,\dot X\cdot X'/e$. This proves the <phase-space identification of the Polyakov metric>.
\b[Conformal gauge and residual coordinate freedom.] In the <Hamiltonian> formulation, <conformal gauge> is $e=1/T,u=0$. The equations become $\ddot X-X''=0$, while $\dot X^2+X'^2=0$ and $\dot X\cdot X'=0$ remain as constraints. In the metric formulation, <conformal gauge> is $\gamma_{ab}=e^{2\omega}\eta_{ab}$, or simply $\eta_{ab}$ after fixing <Weyl invariance>. The metric parametrization proves these choices equivalent. Gauge-fixing the metric does not discard the <Virasoro constraints>.
With $\sigma^\pm=t\pm\sigma$, independent reparameterizations $\sigma^\pm\mapsto f_\pm(\sigma^\pm)$ multiply the flat metric by a conformal factor. A compensating <Weyl transformation> restores its chosen representative. Infinitesimally the <Conformal Killing equation> is $\partial_-c^+=0,\partial_+c^-=0$. These are the <residual conformal transformations> of two-dimensional <Minkowski spacetime>, with the periodicity conditions appropriate to a closed string.
\b[Ghost determinant and anomaly.] For gauge conditions $G_A(\phi)=0$, the <Faddeev-Popov determinant> is the determinant of the linearized variation
$$
M_{AB}=\frac{\delta G_A(\phi^\epsilon)}{\delta\epsilon^B}\bigg|_{\epsilon=0}.
$$
Insert this determinant in the gauge-fixed <path integral> and exponentiate it with anticommuting ghosts and antighosts: the additional action is proportional to $\int b^AM_{AB}c^B$. In metric <gauge fixing>, the <trace> is removed by <Weyl invariance>, and a <worldsheet diffeomorphism> changes the <trace>-free metric through
$$
(P_1c)_{ab}=\nabla_ac_b+\nabla_bc_a-\gamma_{ab}\nabla_dc^d.
$$
Consequently $c^a$ is a vector ghost and $b^{ab}$ a symmetric <trace>-free antighost. A conventional normalization of the <worldsheet ghost action> is
$$
\boxed{I_{\mathrm{gh}}=\frac1{2\pi}\int d^2\sigma\,\sqrt{-\gamma}\,
b^{ab}(P_1c)_{ab}.}
$$
A rescaling of the antighost changes only this overall kinetic normalization. In <conformal gauge>, after normalizing the chiral fields, the kinetic terms are
$$
I_{\mathrm{gh}}\propto\int d^2\sigma\,
(b_{++}\partial_-c^++b_{--}\partial_+c^-).
$$
The closed-string ghosts are periodic, with independent left and right systems. Their <conformal weights> are $2$ for $b$ and $-1$ for $c$, so the supplied <bc ghost system> expression gives $c_{\mathrm{gh}}=-2(6\cdot2^2-6\cdot2+1)=-26$. Each free embedding boson has <central charge> one per chirality. Hence
$$
\boxed{c_{\mathrm{tot}}=D-26,\qquad D=26\text{ for anomaly cancellation}.}
$$
This <central charge of reparameterization ghosts> cancels the <string conformal anomaly> in each chirality separately. Having two sectors does not double the required target dimension.
\b[Channel variables, poles and the vector state.] With particles 1,2 incoming and 3,4 outgoing, the <Mandelstam variables> are $s=-(p_1+p_2)^2$ and $t=-(p_1-p_3)^2$. They measure squared center-of-mass <energy> and momentum transfer. In the specified units $\alpha'=1$, the external tachyon mass squared is $-1$, so $s+t+u=-4$.
The <Gamma function> <residue> at $-n$ is $(-1)^n/n!$. At $s=n-1$, its argument $-1-s$ varies with the opposite sign to $s$. The <Gamma function recurrence> gives
$$
\frac{\Gamma(-1-t)}{\Gamma(-1-n-t)}
=(-1)^n\prod_{j=2}^{n+1}(t+j).
$$
Thus the <Veneziano amplitude pole residues> are
$$
\boxed{s_n=n-1,\quad n=0,1,\ldots,\qquad
\operatorname*{Res}_{s=s_n}A=-\frac1{n!}\prod_{j=2}^{n+1}(t+j).}
$$
For generic fixed $t$ the <simple poles> are $-1,0,1,\ldots$. A zero of the <residue> product at an exceptional negative integer $t$ makes the corresponding pole removable. Fixed $t$ must be away from its own channel poles; fixing it at a channel singularity does not give an ordinary finite meromorphic function of $s$.
At $s=0$, the <residue> is $-(t+2)=-(t-u)/2$. This is proportional to $(p_1-p_2)\cdot(p_3-p_4)$, the contraction of conserved scalar-pair vector currents. Each current is orthogonal to the exchanged momentum because the external masses in its pair are equal. Alternatively, with scattering-angle cosine $z$, one has $t=-2(s/4+1)(1-z)$; at this pole $t+2=2z$, a pure spin-one angular dependence. Therefore
$$
\boxed{\text{The }s=0\text{ pole exhibits a massless spin-one intermediate string state}.}
$$
The <massless vector pole of the Veneziano amplitude> is the massless open-string vector seen in the two-tachyon exchange channel.
Back to article page
