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The worldsheet ghost fields form a bc system with conformal weights $2$ and $-1$. Setting $J=2$ in $c=-2(6J^2-6J+1)$ gives central charge $-26$ in each chirality. The worldsheet Weyl anomaly of $D$ free embedding bosons cancels at $D=26$.

A closed-string scattering amplitude is a worldsheet correlation function of external string vertex operators, integrated over their unfixed positions and worldsheet moduli, with the necessary worldsheet ghost fields.

Liouville field theory is a two-dimensional scalar theory with a curvature background charge and an exponential interaction. In the usual spacelike normalization, quantum conformal invariance gives $Q=b+b^{-1}$ and central charge $c_L=1+6Q^2$. When a noncritical string worldsheet conformal factor is retained as a dynamical field, its effective action has this form and its central charge combines with the matter and worldsheet ghost field charges. It is closely related to a background-charge scalar field, but its exponential potential supplies an additional interaction.

Time-coordinate oscillators produce negative norms in the covariant indefinite Fock space. A physical negative-norm state would violate probabilistic unitarity. Such ghosts differ from the anticommuting worldsheet ghost fields introduced by gauge fixing.

The Polyakov path integral integrates over both the string embedding map and the worldsheet metric. After the usual analytic continuation, a formal closed-string vacuum functional has the schematic formThe genus weights organize string perturbation theory; the coupling $g_s$ is related to the constant dilaton. External states are inserted as string vertex operators, with the usual additional external coupling normalization. Division by the gauge group prevents counting different parameterizations and Weyl representatives as distinct configurations.

Gauge-fix $h=e^{2\varphi }\hat{h}(m)$, where $m$ labels inequivalent conformal structures. Conformal gauge fixes the local metric redundancies, not the worldsheet moduli. Unmarked compact surfaces have no complex moduli at genus zero, one at genus one and $3g-3$ at genus $g\ge 2$. Integrating over the appropriate moduli quotient includes large diffeomorphisms; at genus one it requires a modular fundamental domain, as in torus modular invariance.

The infinitesimal traceless diffeomorphism operator isIts Faddeev-Popov determinant is represented by anticommuting worldsheet ghost fields $c^a$ and traceless symmetric $b^{ab}$, with action proportional to $\int \sqrt{h}{b}^{ab}(P_{1}c{)}_{ab}$. In conformal complex coordinates this is the pair of chiral bc systems of weights $(2,-1)$. Ghost zero modes must be handled rather than included in an ordinary nonzero determinant: conformal Killing vectors correspond to residual gauge transformations, while antighost zero modes pair with surviving moduli variations. At genus zero three insertion positions can be fixed by the Möbius transformations; corresponding $c\tilde{c}$ factors supply the ghost zero modes. At higher genus the measure includes antighost insertions paired with the moduli.

In a flat target, integrating the free coordinate fields gives a Gaussian determinant $[{\det }^{\prime }(-{\Delta }_{\hat{h}}){]}^{-d/2}$, with the zero-mode integral supplying target volume or momentum conservation. Vertex insertions give the corresponding free-field correlators. For example, on a sphere, exponential operators have the factorThe remaining unfixed insertion positions and moduli are integrated with the ghost measure. Physical closed-string operators have matter conformal weights $(1,1)$; an unintegrated insertion is $c\tilde{c}V$, while an integrated insertion uses $\int d^{2}zV$. This explains how on-shell scattering amplitudes arise from the metric-and-embedding functional integral.

The quantum measure can break the classical Weyl symmetry. Each free coordinate contributes central charge one, while each chiral reparameterization ghost system contributes $-26$. Hence $c_{\mathrm{total}}=d-26$, and the worldsheet Weyl anomaly vanishes for the ordinary flat bosonic string exactly at $d=26$. Only then can the conformal factor be removed as a gauge degree of freedom without adding further dynamics. For a subcritical matter system it generally produces a Liouville-type theory, unless an appropriate additional conformal sector restores the central charge balance.

Equivalently the gauge-fixed construction uses a nilpotent string BRST operator. Physical states are its closed states modulo exact states, expressing the BRST cohomology of the gauge constraints. In the critical theory the ghost and matter contributions make this operator nilpotent and the measure gauge independent. Degenerations of moduli describe long-tube propagation and factorization onto intermediate string states. The bosonic tachyon can still cause infrared divergences; anomaly cancellation is not a claim that every vacuum integral is finite or that the bosonic vacuum is stable. The path integral sums embeddings and conformal structures, with ghosts removing local gauge overcounting and moduli retaining the genuine geometry.

The Nambu–Goto action is invariant under worldsheet diffeomorphisms. Its canonical Nambu–Goto phase-space constraints areIn conformal gauge, $\Pi =T\dot{X}$, so these become $(\dot{X}\pm X^{\prime }{)}^2=0$. Their smeared generators implement the residual conformal transformations rather than independent propagating degrees of freedom. Classically their Fourier modes form the classical Virasoro constraint algebra. Quantization promotes them to the Virasoro algebra, and its constraints select the physical string states from the indefinite covariant Fock space.

For clarity, start with a Neumann boundary condition at both ends of an open string, $0\le \sigma \le \pi$. Its open-string mode expansion isCanonical quantization gives the string oscillator relationsWith ${\partial }_{\pm }=({\partial }_{\tau }\pm {\partial }_{\sigma })/2$, the worldsheet stress-energy tensor obeysSubstituting the open-string mode expansion into this quadratic expression derives the Virasoro generators:The colons mean normal ordering, placing the negative-mode creation operators to the left. The string level operator $N$ increases by $r$ when an oscillator ${\alpha }_{-r}$ is added.

Using the commutator once givesApplying this to the quadratic expression for $L_n$ produces $(m-n)L_{m+n}$, together with the double-contraction Virasoro central extension. To determine its coefficient, take $m>0$ and evaluate $[L_m,L_{-m}]$ on a zero-momentum oscillator vacuum. Only the creation pairs with $1\le r\le m-1$ contribute, givingHere $D$ is the target spacetime dimension; the timelike free boson also contributes one to the central charge, since a double contraction contains two metric factors. ThusThis derives both the Virasoro generators and their quantum anomaly.

In old covariant string quantization, the physical-state Virasoro conditions for an open string arewhere $a$ is the normal-ordering constant of a string. Requiring all positive and negative modes to annihilate a state would contradict the nonzero Virasoro central extension; the positive-mode prescription is the appropriate covariant subsidiary condition. The zero-mode equation gives the open bosonic string mass spectrum,For a closed string, there are two commuting Virasoro algebras. With uncompactified zero modes, $L_0={\alpha }^{\prime }{p}^2/4+N_R$ and ${\tilde{L}}_0={\alpha }^{\prime }{p}^2/4+N_L$. Imposing both positive-mode conditions and both zero-mode conditions gives the closed-string level matching $N_L=N_R$ and $M^2=4(N_L-a)/{\alpha }^{\prime }$.

Covariance initially retains the timelike string oscillators. For example, ${\alpha }_{-1}^0|0;p⟩$ has negative norm because ${\eta }^{00}=-1$. Its existence in the unconstrained space is not an inconsistency if it is absent from the physical quotient. The no-ghost theorem for the critical bosonic string states that, at $D=26$ and $a=1$, the inner product on physical string states at ordinary nonzero momentum is positive semidefinite. After quotienting physical null string states, this physical space is represented by the 24 positive-norm transverse string oscillators.

A spurious string state has the form ${\sum }_{n>0}{L}_{-n}|{\chi }_n⟩$. Its inner product with every physical string state vanishes, by $L_n|\psi ⟩=0$ and $L_n^{†}=L_{-n}$. Whenever such a state is itself physical, it is a null string state and represents a gauge redundancy, not an extra polarization. Consequently the physical quotient has no negative-norm propagating states, and agrees with light-cone gauge in string theory. This is the significance of the theorem; a proof is not needed here. It does not eliminate the bosonic tachyon, whose negative mass squared is distinct from negative norm, nor does it remove the auxiliary worldsheet ghost fields used in gauge fixing.

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 has independent worldsheet metric ${\gamma }_{ab}$ and embedding fields $X^{\mu }$. Varying the metric before imposing a gauge givesWorldsheet 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, leavingThese are free two-dimensional scalar fields, but the surviving metric equations requireEquivalently, ${\dot{X}}^2+X^{\prime 2}=0$ and $\dot{X}\cdot X^{\prime }=0$. Gauge fixing does not license dropping these Virasoro constraints.

For an ordinary open string, the boundary conditions relate the two chiral sectors. With ${\alpha }_0^{\mu }=\sqrt{2{\alpha }^{\prime }}{k}^{\mu }$, their independent Fourier generators areThe classical oscillator Poisson brackets $\{{\alpha }_m^{\mu },{\alpha }_n^{\nu }\}=-im{\eta }^{\mu \nu }{\delta }_{m+n,0}$ give the classical Virasoro constraint algebraThus 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 givesFor 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 arePhysical 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 $|k⟩$ is a single spacetime scalar field with ${\alpha }^{\prime }{M}^2=-1$: the bosonic tachyon. At $N=1$, write $|\Psi ⟩={ϵ}_{\mu }{\alpha }_{-1}^{\mu }|k⟩$. The zero-mode equation gives $k^2=0$, and $L_1$ gives $k\cdot ϵ=0$. The null descendant $L_{-1}|k⟩=\sqrt{2{\alpha }^{\prime }}k\cdot {\alpha }_{-1}|k⟩$ identifies $ϵ\sim ϵ+\lambda k$. This leaves 24 physical polarizations of a massless gauge vector.

At $N=2$, use the level-two open-string polarization decompositionThe mass is $M^2=1/{\alpha }^{\prime }$. The oscillator commutators give all the nontrivial positive-mode conditions:Higher positive modes annihilate this level. To see the quotient explicitly, the 25 transverse-parameter null states $L_{-1}(\xi \cdot {\alpha }_{-1}|k⟩)$, with $k\cdot \xi =0$, shiftThe additional level-two scalar Virasoro null state $(L_{-2}+\frac{3}{2}{L}_{-1}^2)|k⟩$ shiftsSubstitution 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 $25\cdot 26/2-1=324$ polarizations. The light-cone basis independently gives $24$ states ${\alpha }_{-2}^i|k⟩$ and $24\cdot 25/2=300$ symmetric states ${\alpha }_{-1}^i{\alpha }_{-1}^j|k⟩$. Hence

For the fermionic extension, an explicit convention prevents ambiguity in the signs. Keep worldsheet signature $(-,+)$ and useTake $\psi =({\psi }_1,{\psi }_2{)}^T$ and the constant parameter $ϵ=({ϵ}_1,{ϵ}_2{)}^T$ 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 $1/(4\pi {\alpha }^{\prime })$, the combined Lagrangian isThe rigid transformations can be writtenor, in components,For the ${ϵ}_2$ part, the bosonic variation is $i{ϵ}_2({\partial }_{+}{\psi }_1\cdot {\partial }_{-}X+{\partial }_{+}X\cdot {\partial }_{-}{\psi }_1)$, while the fermionic variation is $-i{ϵ}_2{\partial }_{-}X\cdot {\partial }_{+}{\psi }_1+i{ϵ}_2{\psi }_1\cdot {\partial }_{+}{\partial }_{-}X$. 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 supersymmetryThus the action is invariant off shell up to a boundary term. The transformation is a genuine supersymmetry: on $X$ its commutator for parameters $ϵ,\zeta$ is the translationOn $\psi$ 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 ${\psi }^{\mu }(z){\psi }^{\nu }(w)\sim {\eta }^{\mu \nu }/(z-w)$,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 algebraDropping 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.

For an oriented Euclidean string worldsheet with metric $h_{ab}$, use the string nonlinear sigma model actionHere ${ϵ}^{ab}$ is an antisymmetric density with ${ϵ}^{12}=1$. The factor $i$ in the Euclidean Kalb–Ramond field term follows from continuing the real Lorentzian two-form coupling; reversing orientation or the convention for $B$ reverses its sign. The conventional positive Euclidean dilaton coupling is being used. With worldsheet boundaries its curvature coupling also needs ${\int }_{\partial \Sigma }\Phi kds/(2\pi )$, where $k$ is the boundary geodesic curvature, to have the proper topological normalization for constant $\Phi$.

All three terms are invariant under worldsheet diffeomorphisms. Target-coordinate changes act on $X,G,B,\Phi$ by their ordinary scalar and tensor transformation rules, making the target description covariant as well. For the two-form there is also the target gauge symmetry $B\mapsto B+d\Lambda$. Its change in the closed-worldsheet action is an integral of an exact two-form and hence zero. With boundaries it is instead $i{\int }_{\partial \Sigma }{X}^{\ast }\Lambda /(2\pi {\alpha }^{\prime })$; an endpoint gauge field with $\delta A=-\Lambda /(2\pi {\alpha }^{\prime })$ cancels it.

Under a local Weyl transformation $h_{ab}\mapsto e^{2\omega }{h}_{ab}$, the metric term is invariant because the two-dimensional factors $\sqrt{h}$ and $h^{ab}$ cancel. The $B$ term is independent of $h$ and is separately invariant. The curvature transforms asOn a closed surface the dilaton variation is consequentlyThus a general nonconstant dilaton term is not separately invariant under Weyl transformations. Its variation is the Weyl variation of a nonconstant dilaton coupling; it vanishes for constant $\Phi$ after the boundary completion where necessary.

For $\Phi ={\Phi }_0$, the Gauss-Bonnet theorem gives $S_{\Phi }={\Phi }_0\chi (\Sigma )$. In the Euclidean path integral the resulting weight isFor a connected orientable genus-$g$ surface with $b$ boundaries, $\chi =2-2g-b$, so the weight is $g_s^{2g+b-2}$. A closed-string handle therefore adds two powers of string coupling. External vertex normalizations add their own coupling factors; they are distinct from this dilaton Euler-characteristic weighting. The constant dilaton controls the string genus expansion, rather than adding a force to the classical embedding equation.

Quantization treats the target fields as position-dependent couplings of a two-dimensional interacting quantum field theory. Expanding $X$ around a background embedding gives fluctuation vertices involving target curvature and derivatives of $B$ and $\Phi$. Short-distance contractions require local counterterms, so the target fields become renormalized couplings. In particular a curvature vertex contracted once yields a Ricci tensor counterterm proportional to ${\alpha }^{\prime }{R}_{\mu \nu }$. Each additional sigma-model loop brings another power of ${\alpha }^{\prime }$ and more target derivatives. The expansion is useful when the target variation scale is large compared with $\sqrt{{\alpha }^{\prime }}$; it is not the separate expansion in powers of $g_s$.

After gauge fixing, a worldsheet Weyl anomaly would make amplitudes depend on the discarded local conformal factor. Quantum consistency therefore requires the trace anomaly to vanish, including the reparameterization worldsheet ghost field contribution. In a conventional renormalization scheme the resulting bosonic sigma-model Weyl anomaly coefficients start asThe barred quantities include the dilaton stress-energy tensor improvement; ordinary coupling sigma-model beta functions can differ by field redefinitions or target diffeomorphisms. Setting these coefficients to zero gives differential equations for the background fields order by order, including the metric gravitational equation and the antisymmetric-field equation. These conventions for the leading coefficients are described in the background-field lecture notes. Higher orders supply higher-curvature and higher-derivative corrections. This explains how spacetime dynamics follows from quantum gauge consistency of the worldsheet, not from asking only for a classical stationary worldsheet in an arbitrarily chosen background.

The apparently noninvariant dilaton term is essential outside the critical dimension because its stress-energy tensor improvement can cancel the quantum central charge mismatch. A flat constant-dilaton background has matter central charge $d$ and worldsheet ghost field central charge $-26$. If $d\ne 26$, changing the constant value of $\Phi$ only changes $g_s$ and cannot cure that mismatch.

A concrete solution is the linear dilaton conformal field theory with $G=\eta$, $B=0$, and $\Phi ={\Phi }_0+V_{\mu }{X}^{\mu }$. Its metric and antisymmetric-field anomaly coefficients vanish, while the scalar condition isOne can also see this cancellation directly from the improved holomorphic stress-energy tensor. With the positive Euclidean curvature convention above,The free-field contraction is $X^{\mu }(z)X^{\nu }(w)\sim -{\alpha }^{\prime }{\eta }^{\mu \nu }\log (z-w)/2$. Differentiating it twice at each point gives a fourth-order pole $3{\alpha }^{\prime }{V}^2/(z-w{)}^4$ from the improvement. Since the fourth-order stress-tensor pole is $c/2$, the matter central charge becomes $c=d+6{\alpha }^{\prime }{V}^2=26$. The worldsheet ghost field contribution then cancels it. This is why a classical Weyl variation need not mean inconsistency: it can cancel the quantum variation of the other terms. For $d<26$ the required gradient is spacelike; for $d>26$ it is timelike in the Lorentzian target theory. Such backgrounds are not flat constant-dilaton Minkowski vacua with full target Lorentz symmetry, nor does a central charge cancellation by itself prove stability or unitarity of every noncritical background. If the conformal factor is instead retained dynamically, its Liouville field theory supplies the corresponding noncritical description.

Take ${\partial }_{\pm }=({\partial }_{\tau }\pm {\partial }_{\sigma })/2$ on the strip $0\le \sigma \le \pi$. Varying the Grassmann-odd worldsheet Majorana fermions gives ${\partial }_{-}{\psi }_1=0$, ${\partial }_{+}{\psi }_2=0$ and, up to the orientation of the strip boundary, the surface termAt each endpoint it vanishes if ${\psi }_1=\eta {\psi }_2$ with $\eta =\pm 1$, with the same relation on variations. A common sign can be absorbed by redefining one chiral field, so set ${\eta }_0=+1$. Since ${\psi }_1=f(\tau +\sigma )$ and ${\psi }_2=f(\tau -\sigma )$, the second endpoint imposes $f(u+2\pi )={\eta }_{\pi }f(u)$. Equal endpoint signs give the periodic Ramond sector, and opposite endpoint signs give the antiperiodic Neveu–Schwarz sector:These gluing choices preserve the Neumann-endpoint worldsheet supersymmetry conditions and give integer or half-integer worldsheet Majorana fermion modes, respectively.

First fix a normalization issue in the original PDF. With its stated canonical string oscillator algebra, the standard superconformal current mode iswithout the printed factor $1/2$. To check the Ramond supercurrent zero-mode square directly, pair positive and negative integer modes:The zero-mode Clifford relations give $({\alpha }_0\cdot {\psi }_0{)}^2={\alpha }_0^2/2$. Different $Q_m$ anticommute, and so do $Q_m$ with the zero-mode term. The two same-sign terms in $Q_m^2$ vanish by antisymmetry of the fermions and symmetry of the bosonic product. The cross terms giveThe second term comes from commuting ${\alpha }_m$ through ${\alpha }_{-m}$, so the frequency factor $m$ is essential. ThereforeHere $L_0^R$ uses vacuum-annihilating Ramond normal ordering; its string intercept is zero. A plane-to-cylinder central shift must be included consistently if a different zero-mode convention is used.

For the mode $F_r=G_r/2$ actually printed, the algebra instead gives $\{F_0,F_0\}=L_0^R/2$. Already the oscillator-free term gives $\{F_0,F_0\}={\alpha }_0^2/4$, whereas $2L_0^R={\alpha }_0^2$. Thus the requested identity is not valid with both that prefactor and the supplied string oscillator algebra. Removing the prefactor gives the identity just derived. Renaming $L_0^R/4$ as $L_0$ would fix this isolated equality but would spoil the usual Virasoro normalization and the subsequent mass formula. We use the standard normalized modes $G_r$ in the remaining physical constraints; their vanishing is unaffected by a common nonzero rescaling.