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The Polyakov action introduces an independent worldsheet metric $h_{ab}$:Its redundancies are worldsheet diffeomorphisms and Weyl transformations. A Polyakov path integral over all metrics without dividing by this gauge volume would count physically equivalent configurations repeatedly. Fixing conformal gauge therefore needs the Faddeev-Popov determinant, not merely the substitution of a flat metric into the action.

After using the Weyl transformation to remove the trace of a metric variation, the infinitesimal worldsheet diffeomorphism ghost operator isThe Faddeev-Popov determinant of this operator is represented by anticommuting fields: a vector Faddeev-Popov ghost $c^a$ and a symmetric traceless Faddeev-Popov antighost field $b^{ab}$. In Euclidean coordinates their worldsheet ghost action is, up to a consistent overall normalization,The two chiral copies of this bc system have conformal weights $(2,-1)$ and anticommutators $\{b_m,c_n\}={\delta }_{m+n,0}$. They are auxiliary fields for the gauge determinant, not additional spacetime particles. Their use is different from the unwanted timelike negative-norm string states of covariant matter quantization.

The chiral holomorphic stress-energy tensor of the bc system isDouble contractions in its operator product expansion give the central charge of reparameterization ghosts $c_{\mathrm{gh}}=-26$. Equivalently, a fermionic bc system with antighost weight $\lambda$ has $c=1-3(2\lambda -1{)}^2$, and $\lambda =2$ gives $-26$. The free coordinate matter contributes $c_{\mathrm{m}}=D$. Thus the total worldsheet Weyl anomaly vanishes for the flat critical bosonic string theory precisely whenThis statement assumes no additional matter or compensating Liouville field theory.

BRST symmetry expresses the gauge symmetry after gauge fixing by replacing its infinitesimal parameter with the Grassmann variable $c$. In one chiral conformal patch, its graded action has the formthere is an analogous barred copy for a closed string. The Grassmann parity makes $s^{2}X=0$: the term from $sc$ cancels the term from applying $s$ to $\partial X$, while the term with $c^2$ vanishes. The gauge-fixed chiral transformations are on shell when their conservation equations are used. At the quantum level, BRST nilpotence also tests the anomaly.

For example, an open string BRST charge can be written using the matter Virasoro generators asUse the ghost oscillator vacuum $b_{n\ge 0}|0{⟩}_{mathrmgh}=c_{n>0}|0{⟩}_{mathrmgh}=0$ and the corresponding oscillator normal ordering. In this convention the last term is $-c_0$ in the critical theory. With a differently shifted ghost $L_0$, the displayed intercept must be shifted consistently; one must not count the same shift twice. Squaring $Q_B$ shows why both critical conditions are needed. Its anomalous terms, in this convention, have coefficientsThe independent cubic and linear terms vanish at $D=26$, $a=1$. The equivalent chiral BRST current is $j_B=c(T_{mathrmm}+T_{mathrmgh}/2)+(3/2){\partial }^{2}c$, with its contour integral defining the charge.

The graded identity $\{Q_B,b_n\}=L_n^{\mathrm{tot}}$ recovers the gauge constraints. In particular, on $|\psi ⟩\otimes |0{⟩}_{mathrmgh}$, BRST-closed imposes $L_{n>0}^{\mathrm{m}}|\psi ⟩=0$ and $(L_0^{\mathrm{m}}-1)|\psi ⟩=0$. Physical states are classes in BRST cohomology,at the appropriate ghost number: one for the usual open-string vertex and two for the closed-string vertex, with the closed-string zero-mode and closed-string level matching conditions also imposed. Since $Q_B^2=0$, an exact shift preserves closure. BRST Ward identities make exact insertions decouple from physical amplitudes and establish gauge independence. Together with the no-ghost theorem for the critical bosonic string, this explains how the auxiliary ghosts enforce gauge invariance while the physical spectrum retains positive norm.

Use a mostly-plus Minkowski metric and string tension $T=(2\pi {\alpha }^{\prime }{)}^{-1}$. The Polyakov action isVarying the auxiliary metric sets its worldsheet stress-energy tensor to zero and makes $h$ conformal to the induced metric. Substitution gives the equivalent Nambu–Goto action $-T\int \sqrt{-\det ({\partial }_{a}X\cdot {\partial }_{b}X)}$. The string coupling is separate from $T$. A constant dilaton contributes $S_{\Phi }={\Phi }_0\chi (\Sigma )$ to the Euclidean action, by the Gauss-Bonnet theorem, including its boundary term when necessary. With $g_s=e^{{\Phi }_0}$, a connected worldsheet has weight $g_s^{-\chi }$. For an oriented closed surface of genus $g$, this is $g_s^{2g-2}$; with $b$ boundaries it is $g_s^{2g+b-2}$. Normalizing each external closed string vertex with a factor $g_s$ gives $g_s^{2g+M-2}$ for $M$ external closed strings. This is the string genus expansion.

The local gauge symmetries are worldsheet diffeomorphisms, under which $X$ is a scalar field, and Weyl transformations $h_{ab}\mapsto e^{2\omega (\sigma )}{h}_{ab}$. Two coordinate functions and one Weyl function locally fix the three components of the two-dimensional metric, giving conformal gauge $h_{ab}=e^{2\omega }{\eta }_{ab}$ and then $\omega =0$. Globally one instead chooses a fiducial metric and integrates over the worldsheet moduli; conformal Killing transformations can remain. Target-space Poincaré transformations are global symmetries, not additional local gauge freedoms.

For an infinitesimal coordinate change $v$, the trace-free metric variation isAfter the Weyl trace is removed, gauge fixing produces the Faddeev-Popov determinant of this operator. An anticommuting vector $c^a$ and symmetric traceless tensor $b_{ab}$ represent that determinant through a Grassmann Gaussian integral. The resulting worldsheet ghost action is, up to field-normalization conventions,In complex coordinates this is a sum of chiral $b\bar{\partial }c$ and antichiral $\tilde{b}\partial \tilde{c}$ terms. The worldsheet ghost fields have conformal weights two and minus one. Their central charge is $-26$, while the $d$ free embedding bosons contribute $d$. In the critical dimension the total anomaly cancels, consistently with the stated nilpotence of the BRST charge.

Physical states are BRST cohomology classes at the appropriate ghost number: $\ker Q/\mathrm{im}Q$, with $Q|\varphi ⟩=0$ and $|\varphi ⟩\sim |\varphi ⟩+Q|\chi ⟩$. Nilpotence ensures that an exact state is closed and that the equivalence is well-defined. For the usual open string representatives the physical ghost number is one. BRST-exact excitations describe gauge or null directions rather than additional physical particles.

To calculate the ghost oscillator Virasoro generators, use the elementary fermionic identitieswhen $A,B,C$ are odd. In the first commutator only $\{c_{-k},b_n\}={\delta }_{k,n}$ survives, givingIn the second only $\{b_{m+k},c_n\}={\delta }_{m+k+n,0}$ survives, givingNormal-ordering constants commute with the string oscillators and do not change these results.

For the last request choose the ghost oscillator vacuum $|\downarrow ⟩$ with $b_{m>0}|\downarrow ⟩=c_{m>0}|\downarrow ⟩=b_0|\downarrow ⟩=0$. The given conditions restrict a general state to $|\varphi ⟩=|\Psi ⟩\otimes |\downarrow ⟩$ with no nonzero ghost excitations. Indeed, the positive modes annihilate both members of each creation–annihilation pair, and $b_0$ selects one member of the ghost zero-mode doublet. Use the ghost Virasoro zero-mode convention compatible with the displayed explicit $-c_0$ term. Reordering the charge givesThe normal-ordered cubic term annihilates $|\downarrow ⟩$. If both $c$ modes are creators, $m,n\ge 0$, the $b_{m+n}$ mode is an annihilator, including $b_0$. If either $c$ is an annihilator, normal ordering puts an annihilator on the right. ConsequentlyThese ghost states are independent. Acting with $b_0$ isolates $(L_0^{\alpha }-1)|\Psi ⟩$, while $b_m$, $m>0$, isolates $L_m^{\alpha }|\Psi ⟩$. Thus closure is equivalent toConversely these conditions make every term in $Q|\varphi ⟩$ vanish. They are exactly the physical-state Virasoro conditions for an open string; the cohomology quotient additionally identifies physically equivalent representatives.

Use Euclidean worldsheet conventions with ${ϵ}^{12}=1$. Including a target metric, a Kalb–Ramond field $B_{\mu \nu }$ and the dilaton $\Phi$ givesThe imaginary coefficient of the antisymmetric term comes from Euclidean continuation; the Lorentzian action has the corresponding real coupling. For constant ${\Phi }_0$, the Gauss-Bonnet theorem gives $S_{\Phi }={\Phi }_0\chi (\Sigma )$, with Euler characteristic $\chi =2-2g$ on a connected closed oriented surface. Thus dilaton Euler-characteristic weighting givesAdding one handle multiplies a contribution by $g_s^2$, identifying $g_s$ as the string coupling. With canonically normalized external closed-string states, $n$ vertices add $g_s^n$, giving the string genus expansion $g_s^{2g-2+n}$.

Before gauge fixing, the Polyakov path integral for inserted string vertex operators is schematicallyThe quotient removes descriptions related by worldsheet diffeomorphisms and Weyl transformations. Gauge fixing has a nontrivial Jacobian, the Faddeev-Popov determinant. Anticommuting vector ghosts $c^a$ and symmetric trace-free antighosts $b_{ab}$ exponentiate this determinant as a local worldsheet ghost action. These worldsheet ghost fields implement the gauge quotient; they are not additional spacetime particles. Their zero modes must be treated separately rather than included in an invertible determinant.

Three kinds of geometry enter the scattering calculation. The worldsheet moduli describe genuinely different conformal shapes that remain after local gauge fixing. Each such shape must be integrated over. The mapping class group consists of orientation-preserving diffeomorphisms modulo those continuously deformable to the identity; it identifies different markings of the same conformal surface, so only one representative of each class is counted. The worldsheet conformal Killing group is the continuous residual conformal symmetry of a chosen surface. It moves insertion positions without changing the geometry and must be divided out, usually by fixing enough vertex positions.

For a torus, write $z\sim z+1\sim z+\tau$, with $\mathrm{Im}\tau >0$. Its shape modulus is $\tau$, its mapping class group acts through the modular group, and its connected worldsheet conformal Killing group acts by translations. Thus the shape integral runs over the standard fundamental domain of the modular group, while one insertion position can be fixed by translation. On the sphere, three positions can instead be fixed by Möbius transformations; for genus at least two the continuous conformal Killing group is trivial.

Now use the half-normalized worldsheet diffeomorphism ghost operator in the question. A vector field $v^a$ can preserve the representative metric after a compensating Weyl transformation precisely when $P_{1}v=0$. The trace of the metric variation sets $\delta \omega =\frac{1}{2}{\nabla }_a{v}^a$, leaving the Conformal Killing equationThe kernel of this operator gives the infinitesimal worldsheet conformal Killing group.

For a surviving infinitesimal worldsheet modulus, choose its metric variation $t_{ab}$ orthogonal to all gauge variations. Orthogonality to Weyl transformations requires $h^{ab}{t}_{ab}=0$. On the closed surface, the formal adjoint of the worldsheet conformal Killing operator follows by integration by parts:Orthogonality to every diffeomorphism variation is therefore the adjoint-kernel condition, givingIn a local conformal coordinate these imply ${\partial }_{\bar{z}}{t}_{zz}=0$ and its conjugate: the moduli variations are represented by holomorphic quadratic differentials. Multiplying the definition of $P_1$ by two changes its adjoint by two but changes neither kernel.

For four tachyon vertex operators on the torus, there is one complex shape modulus, four complex insertion positions and one complex translation to remove. The worldsheet integration count after conformal gauge fixing is thereforeFix the fourth insertion at $z_4=0$. A schematic gauge-fixed expression isHere the antighost pair supplies the modulus measure, and the ghost pair at the fixed vertex absorbs the translation zero modes. The three unfixed positions and $\tau$ are the four complex integrations. The punctured Riemann surface moduli dimension gives the same result, $3g-3+n=4$. Target-spacetime zero-mode integrals producing momentum conservation are separate from this requested count of worldsheet integrations.

The Polyakov action is invariant under worldsheet diffeomorphisms and Weyl transformations, ${\gamma }_{ab}\mapsto e^{2\omega }{\gamma }_{ab}$. A two-dimensional metric has three local components: two coordinate transformations and one Weyl function remove these locally. In conformal gauge the factor $\sqrt{-\gamma }{\gamma }^{ab}$ is independent of the conformal scale, and on a flat coordinate patch the matter action isThus it is the action of $d$ free scalar fields, with their target-space contractions using the Minkowski metric. On a Euclidean sphere there are no unpunctured conformal moduli: a metric can be put into the conformal class of the round metric, and in stereographic patches the scalar action has this free-field form. This does not mean that the sphere has a globally flat metric. A spherical worldsheet here is understood after Wick rotation; it has no nonsingular global Lorentzian metric. The residual Möbius transformations and their zero modes still have to be divided out or fixed by vertex positions.

To derive the worldsheet ghost action, consider an infinitesimal diffeomorphism generated by $v^a$ and a Weyl variation $\omega$. The metric variation separates asIntegration over the Weyl parameter removes the trace gauge condition. The trace-free part produces the Faddeev-Popov determinant of an operator mapping vectors to symmetric traceless tensors. This is trace elimination in string Faddeev-Popov gauge fixing. The stated Grassmann Gaussian integral exponentiates this determinant with an odd vector $c^a$ and an odd symmetric trace-free antighost $b_{ab}$.

The normalization of the operator and antighost can be chosen to match the printed convention. For example, in a flat patch take $P=-P_1/2$. It differs from the orbit operator only by a field-independent determinant factor. Tracelessness and symmetry give $b_{ab}(Pc{)}^{ab}=-b_{ab}{\partial }^a{c}^b$. Integration by parts, followed by exchanging two odd factors, gives $\int b_{ab}{\partial }^a{c}^b=\int c^a{\partial }^b{b}_{ab}$. Consequently the determinant exponent is $+(2\pi {)}^{-1}\int c^a{\partial }^b{b}_{ab}=i{I}_{\mathrm{gh}}$ withOverall constant normalization does not affect the determinant's dependence on the fields. The Faddeev-Popov ghost fields are vectors, while the antighost fields are symmetric trace-free tensors; they are not extra target-space string excitations.

Now use ${\sigma }^{\pm }=\tau \pm \sigma$, ${\partial }_{\pm }=({\partial }_{\tau }\pm {\partial }_{\sigma })/2$ and $c^{\pm }=c^0\pm c^1$. In these coordinates ${\eta }_{+-}={\eta }_{-+}=-1/2$ and ${\partial }^{\pm }=-2{\partial }_{\mp }$. Tracelessness implies $b_{+-}=0$, so directlyThereforeFor a Cartesian check, $b_{11}=b_{00}$ and $b_{++}=(b_{00}+b_{01})/2$, $b_{--}=(b_{00}-b_{01})/2$ reproduce the same result.

The original PDF repeats ${\delta }_{\eta }{c}^{+}$ for its final ghost transformation. The second occurrence must be ${\delta }_{\eta }{c}^{-}$; otherwise the displayed rules give two incompatible transformations for one ghost and none for the other. Write the corrected transformation as ${\delta }_{\eta }\Phi =\eta s\Phi$, with $\eta$ on the left. Then the odd left-acting BRST differential satisfies the graded Leibniz rule $s(AB)=(sA)B+(-1{)}^{|A|}A(sB)$. Its local rules areThe last two expressions are $s{b}_{\pm \pm }=i{\theta }_{\pm \pm }/{\alpha }^{\prime }$ with the matter-plus-ghost worldsheet stress tensor in the question.

Here is an off-shell check of the action variation, including the Grassmann signs. Temporarily set $c=c^{+}$, $f=c^{-}$, $b=b_{++}$, $h=b_{--}$, $u={\partial }_{+}X$ and $v={\partial }_{-}X$. Apart from the common factor $1/\pi$, the total Lagrangian is $\mathcal{L}=u\cdot v/{\alpha }^{\prime }+i(c{\partial }_{-}b+f{\partial }_{+}h)$. Its matter variation isThe matter terms in $sb$ and $sh$ give the ghost-action contribution ${\alpha }^{\prime -1}[c{\partial }_{-}(u^2)+f{\partial }_{+}(v^2)]$. Combining these terms with the last two matter terms produces ${\alpha }^{\prime -1}[{\partial }_{-}(c{u}^2)+{\partial }_{+}(f{v}^2)]$.

The remaining contribution from the $c,b$ sector iswhere subscripts denote derivatives. In the last equality the terms $c_{-}{c}_{+}b$ and $c_{+}{c}_{-}b$ cancel by anticommutation. The $f,h$ sector has the same identity with $+\leftrightarrow -$. ThuswhereHence the total action is BRST invariant up to a boundary term, which vanishes on a closed worldsheet or under the usual compatible boundary and time-end conditions. This calculation requires no equations of motion.

Nilpotence needs a qualification for these particular chiral rules. For either ghost, $s(c\partial c)=(c\partial c)\partial c-c\partial (c\partial c)=0$, because $c^2=(\partial c{)}^2=0$. Thus $s^2{c}^{\pm }=0$ algebraically. For the embedding, however, the graded calculation givesMixed second derivatives of $X$ cancel because $c^{+}{c}^{-}=-c^{-}{c}^{+}$. The remaining terms vanish on the ghost equations of motion, ${\partial }_{-}{c}^{+}={\partial }_{+}{c}^{-}=0$, which follow by varying the antighosts in the chiral ghost action. They need not vanish off shell. An explicit counterexample uses independent Grassmann variables $\xi ,\rho$: take $c^{+}=\xi$, $c^{-}=\rho {\sigma }^{+}$ and one coordinate $X={\sigma }^{-}$. Then $s^{2}X=-\xi \rho \ne 0$.

For the antighosts, use the permitted stress-tensor identity $s{\theta }_{ab}=0$ on the classical solution space. Then $s^2{b}_{\pm \pm }=(i/{\alpha }^{\prime })s{\theta }_{\pm \pm }=0$. Indeed direct variation of the displayed stress tensors giveswhich vanish using the ghost equations and the free matter wave equation. Therefore two successive BRST transformations annihilate every field on shell. This is chiral conformal-gauge BRST nilpotence is on shell, not an unrestricted off-shell identity for arbitrary ghosts. Quantum BRST nilpotence additionally requires cancellation of the worldsheet anomaly, giving $d=26$ in the critical bosonic theory.

Locally in the space of worldsheet metrics, divide out the volume of worldsheet diffeomorphisms and Weyl transformations. With stabilizer zero modes omitted, the inverse Faddeev-Popov determinant at a metric $g$ isThis is the linearized orbit integral: the delta functional is the product of the metric-component delta functions, and its Jacobian is the inverse determinant of the gauge-orbit map. The omitted zero modes and any metric moduli must be handled separately; this local formula is not an integration over the physical moduli themselves.

The trace elimination in string Faddeev-Popov gauge fixing is essential. In two dimensions, split the variation asThe second term is trace-free. Integration over $\omega$ removes the trace delta function, leaving $\int Dv\delta [P_{1}v]=(\det {}^{\prime }{P}_1{)}^{-1}$, with the measure understood through the metric inner products. Equivalently, a Fourier representation of the metric delta functional introduces a commuting symmetric tensor ${\beta }^{ab}$. Integration over $\omega$ enforces $g_{ab}{\beta }^{ab}=0$, and the remaining exponent is proportional to $\int \sqrt{g}{\beta }^{ab}(P_{1}v{)}_{ab}$. This gives the same inverse Faddeev-Popov determinant and explains why its tensor variable must be trace-free.

For the Polyakov path integral, insert the gauge-fixing identitywhere $\hat{g}(m)$ is a representative depending on the worldsheet moduli $m$. Gauge invariance of the action, measure, and Faddeev-Popov determinant lets us change variables along the gauge orbit. Its volume cancels the original gauge-group denominator, leaving an integral over $X$, the moduli, and the determinant evaluated at $\hat{g}$. This is how the determinant enters the numerator of the gauge-fixed integral, despite the inverse determinant occurring in the orbit integral above.

A fermionic Gaussian integral exponentiates $\det {}^{\prime }{P}_1$ using an anticommuting vector $c^a$ and an anticommuting symmetric trace-free tensor $b^{ab}$:The constant $C$ can be absorbed into $b$. In local complex conformal gauge, trace-freeness sets $b_{z\bar{z}}=0$. The only surviving components of the worldsheet diffeomorphism ghost operator are proportional to $\bar{\partial }{c}^z$ and $\partial c^{\bar{z}}$. Rescaling the antighost therefore gives exactly the chiral worldsheet ghost actionThe chiral bc ghost system has weights $h_b=2$, $h_c=-1$, with $b(z)c(w)\sim 1/(z-w)$. Its worldsheet stress tensor is $T_{\mathrm{gh}}=-2:b\partial c:-:(\partial b)c:$. Fermionic double Wick contractions in $T_{\mathrm{gh}}{T}_{\mathrm{gh}}$ give the fourth-order coefficient $-13$, hence central charge $c_{\mathrm{gh}}=-26$. More generally this calculation gives $c=1-3(2h_b-1{)}^2$. Each embedding scalar contributes one, so the total chiral central charge is $D-26$. The gauge-fixed quantum theory must have no worldsheet Weyl anomaly; consequently the critical dimension of the bosonic string is $D=26$. Zero-mode insertions for Conformal Killing vector fields and moduli do not change this local anomaly calculation.

For the Veneziano amplitude, first examine an ordered beta-function term. A pole occurs when $-{\alpha }^{\prime }s-1=-n$, with $n=0,1,2,\dots$, so the exchanged masses follow the open bosonic string mass spectrumThere is a tachyon at level zero, a massless vector at level one, then an infinite tower with equally spaced squared masses. In oscillator quantization these levels are $N=n$; the poles alone do not determine all oscillator degeneracies or all states that might fail to couple to the chosen external particles.

The Veneziano amplitude pole residue gives considerably more information than a pole position. Near $s_n=(n-1)/{\alpha }^{\prime }$, use $\Gamma (-n+ϵ)\sim (-1{)}^n/(n!ϵ)$ and $ϵ=-{\alpha }^{\prime }(s-s_n)$. The finite Gamma function ratio is a product of $n$ linear factors. Thus, for generic $t$,The empty product at $n=0$ is one. This degree-$n$ angular polynomial decomposes into spin partial waves; it reveals the spins that couple, up to maximal spin $n$ in the ordered channel, and the corresponding products of three-point couplings. For example, the level-one linear residue encodes vector exchange. A single external-state amplitude need not separate all states with the same mass and spin.

There is a qualification specific to the sum printed in the question. For four identical on-shell tachyons, $s+t+u=-4/{\alpha }^{\prime }$. At ${\alpha }^{\prime }s=n-1$, put $P_n(t)={\prod }_{j=1}^n({\alpha }^{\prime }t+j+1)$. Then ${\alpha }^{\prime }u=-n-3-{\alpha }^{\prime }t$, and reversing the product gives $P_n(u)=(-1{)}^n{P}_n(t)$. The $t,u$ beta-function term is regular at a generic $s$-channel pole. The residue of the full symmetric amplitude is therefore proportional toThis is odd-level pole cancellation in the symmetric Veneziano amplitude. The fully symmetrized identical-tachyon amplitude has only even-level poles at generic angle, including the tachyon but not the massless vector. This is an external-state selection rule, not removal of odd-level states from the string theory. Ordered amplitudes, or suitable amplitudes with color factors and different external states, retain those exchanges. Thus interpreting every candidate Gamma function pole as a pole of the printed sum would give a false inference about that amplitude.

At high energy with $x=-t/s\in (0,1)$ fixed, the Gamma function reflection formula followed by the Stirling formula gives the fixed-angle softness of an open-string amplitude:up to oscillatory factors and possible additional cancellations. The bracket is positive at a genuinely nonforward fixed angle. This is the envelope away from resonance poles, or the corresponding analytically continued asymptotic regime; it is not a uniform bound through real-axis poles. String scattering is exponentially soft at fixed angle, unlike the typical power-law behavior of point-particle interactions. The finite string length distributes a hard collision over an extended object. This is evidence of improved ultraviolet behavior, not by itself a proof that every loop amplitude is finite. The fixed-angle limit also differs from the fixed-$t$ Regge limit.

Independent metric and Hamiltonian parametrization. The Lorentzian Polyakov action isThe 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 $(\dot{X}-u{X}^{\prime }{)}^2/(2e)-e{T}^2{X}^{\prime 2}/2$. On the positive-lapse branch, choose the independent metric, up to a positive Weyl transformation, asIts determinant is $-e^2{T}^2$. Multiplication by $-T/2$ recovers the reduced density, including the mixed term $-u\dot{X}\cdot X^{\prime }/e$. This proves the phase-space identification of the Polyakov metric.

Conformal gauge and residual coordinate freedom. In the Hamiltonian formulation, conformal gauge is $e=1/T,u=0$. The equations become $\ddot{X}-X^{\prime \prime }=0$, while ${\dot{X}}^2+X^{\prime 2}=0$ and $\dot{X}\cdot X^{\prime }=0$ 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.

Ghost determinant and anomaly. For gauge conditions $G_A(\varphi )=0$, the Faddeev-Popov determinant is the determinant of the linearized variationInsert this determinant in the gauge-fixed path integral and exponentiate it with anticommuting ghosts and antighosts: the additional action is proportional to $\int b^A{M}_{AB}{c}^B$. In metric gauge fixing, the trace is removed by Weyl invariance, and a worldsheet diffeomorphism changes the trace-free metric throughConsequently $c^a$ is a vector ghost and $b^{ab}$ a symmetric trace-free antighost. A conventional normalization of the worldsheet ghost action isA rescaling of the antighost changes only this overall kinetic normalization. In conformal gauge, after normalizing the chiral fields, the kinetic terms areThe 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 $c_{\mathrm{gh}}=-2(6\cdot 2^2-6\cdot 2+1)=-26$. Each free embedding boson has central charge one per chirality. HenceThis central charge of reparameterization ghosts cancels the string conformal anomaly in each chirality separately. Having two sectors does not double the required target dimension.

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 ${\alpha }^{\prime }=1$, 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 givesThus the Veneziano amplitude pole residues areFor generic fixed $t$ the simple poles are $-1,0,1,\dots$. 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. ThereforeThe massless vector pole of the Veneziano amplitude is the massless open-string vector seen in the two-tachyon exchange channel.

Use the classically equivalent Polyakov action, fix worldsheet diffeomorphisms and Weyl invariance, and perform a Wick rotation of the string worldsheet. In complex coordinates, the worldsheet ghost action for the closed string isThe worldsheet ghost fields are anticommuting. Their conformal weights are $b:(2,0)$, $c:(-1,0)$ and $\bar{b}:(0,2)$, $\bar{c}:(0,-1)$. The two chiral sectors are counted separately.

In either chiral sector, each embedding coordinate is a free boson conformal field theory with central charge one. For the bc system, putting $J=2$ in $c_{bc}=-2(6J^2-6J+1)$ gives $c_{bc}=-26$. Cancellation of the worldsheet Weyl anomaly requiresThe antiholomorphic sector gives the same equation, rather than doubling the required critical dimension of string theory. This dimension count assumes flat target spacetime with just the embedding fields; more general internal conformal field theories change the matter accounting.

For the spinning string, the worldsheet Majorana fermions have conformal weight $1/2$. At $J=1/2$, the complex anticommuting bc system has $c=1$; a real Majorana spinor contributes half of this. Hence $D$ real worldsheet Majorana fermions contribute $D/2$ per chiral sector. Ramond sector versus Neveu–Schwarz sector changes their mode numbers and vacuum energies, not this local central charge.

The additional local worldsheet supersymmetry has a fermionic gauge parameter, so its superconformal ghosts are commuting fields forming a beta-gamma system, with conformal weights $\beta :3/2$ and $\gamma :-1/2$ in the holomorphic sector. Commuting statistics reverse the sign of the bc system formula, givingTogether with the reparameterization worldsheet ghost fields, their total central charge is $-26+11=-15$. ThusThese cancellations make the string worldsheet gauge symmetries compatible with quantization; in the BRST symmetry formulation they enter the nilpotence condition for the BRST operator.

In Euclidean conformal gauge, complex coordinates reduce the worldsheet ghost action toup to a convention-dependent common normalization absorbed into the fields. Inverting $\bar{\partial }$ and $\partial$ giveswith mixed holomorphic-antiholomorphic correlators equal to zero. These are the standard local-plane propagators; compact worldsheets require the zero-mode qualifications described in part (a).