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Expand the string embedding map along target geodesics as $X={\exp }_{\bar{X}}Y$. The quadratic Euclidean fluctuation action has a covariant derivative term and a curvature term $-R_{acbd}{Y}^a{Y}^b\partial {\bar{X}}^c\partial {\bar{X}}^d$. The logarithmically divergent coincident contraction is proportional to ${\alpha }^{\prime }{g}^{ab}$; its curvature contraction is the Ricci tensor. A compensating metric counterterm gives ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}+O({\alpha }^{\prime 2})$. This is why the target gravitational equation arises from quantum worldsheet Weyl anomaly cancellation, rather than from the classical worldsheet metric variation. Infrared regulation and removal of equation-of-motion terms by field redefinitions specify the usual one-loop scheme.

Use flat target spacetime with metric ${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,\dots ,1)$, and let $h_{ab}$ be an independent Lorentzian worldsheet metric. The Polyakov action isHere $X$ is the string embedding map and $T$ the string tension. Variation of $X$, followed by integration by parts, givesThus the embedding equation isThe boundary term vanishes by periodicity for a closed string. At an open-string boundary it requires, for example, vanishing normal derivative for a Neumann boundary condition, or $\delta X=0$ for a Dirichlet boundary condition. Mixed choices give corresponding endpoint conditions.

Define the induced worldsheet metric ${\gamma }_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$. Using $\delta \sqrt{-h}=-\frac{1}{2}\sqrt{-h}{h}_{ab}\delta h^{ab}$, metric variation givesThe second set of equations is therefore the vanishing worldsheet stress tensor:For a nondegenerate induced metric this says $h_{ab}=e^{2\omega }{\gamma }_{ab}$. Substituting into the action gives $S_P=-T\int \sqrt{-\det \gamma }{d}^2\sigma$, proving the classical equivalence of Polyakov and Nambu–Goto actions. The auxiliary metric has not added physical metric degrees of freedom.

There are three important symmetry statements. Worldsheet diffeomorphisms change the parameterization, with $X$ transforming as a scalar and $h$ as a tensor; the integral is invariant. Weyl transformations $h_{ab}\mapsto e^{2\omega (\sigma )}{h}_{ab}$ leave $\sqrt{-h}{h}^{ab}$ invariant specifically in two dimensions. These are gauge redundancies. Locally, the two diffeomorphism functions and one Weyl function fix the three independent components of $h$ to conformal gauge, $h_{ab}=e^{2\omega }{\eta }_{ab}$. Global worldsheet moduli may remain and cannot be gauged away. Separately the flat-target Poincaré group acts globally on $X$ and supplies conserved target momentum and angular momentum, rather than another worldsheet gauge constraint.

In conformal gauge the embedding equations and metric constraints becomeor equivalently ${\dot{X}}^2+X^{\prime 2}=0$ and $\dot{X}\cdot X^{\prime }=0$. The Virasoro constraints must still be imposed: fixing the metric in the action before varying it would miss them. Residual conformal changes of the two null worldsheet coordinates are generated by these constraints. Classically these statements hold in any target dimension; cancellation of the quantum worldsheet Weyl anomaly is a further condition, not part of the classical variation.

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.

Use Minkowski spacetime with signature $(-,+,\dots ,+)$, and write the string tension as $T=1/(2\pi {\alpha }^{\prime })$. If $X^{\mu }(\tau ,\sigma )$ is the string embedding map, its induced worldsheet metric is ${\gamma }_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$. The Nambu–Goto action isThe first variation uses $\delta {\gamma }_{ab}=2{\partial }_{(a}{X}_{\mu }{\partial }_{b)}\delta X^{\mu }$ and therefore givesAn integration by parts yields the bulk Nambu–Goto equations of motion and the endpoint term:For a closed string, periodicity cancels the endpoint term. For a freely moving open string, arbitrary endpoint variations require zero momentum flux, which is the free-end string boundary condition.

Choose orthogonal conformal coordinates on the nondegenerate part of the worldsheet, so that ${\gamma }_{\tau \sigma }=0$ and ${\gamma }_{\tau \tau }+{\gamma }_{\sigma \sigma }=0$. The Nambu–Goto equations of motion then become a wave equation, with the accompanying Virasoro constraints:Here a dot and a prime mean ${\partial }_{\tau }$ and ${\partial }_{\sigma }$. The last two equations are essential: a solution of the wave equation alone need not be a relativistic fundamental string motion.

For a free-ended open string, take $0\le \sigma \le \pi$ and $a>0$, and setwith the other coordinates constant. Every component satisfies the wave equation, andMoreover $X^{\prime }=0$ at both endpoints, giving the Neumann boundary condition. At fixed target time $t=a\tau$, the spatial image is the straight segmentThus its angular velocity is $1/a$. The speed at coordinate $r$ is $|r|/a$, so the endpoints have the null motion of a free string endpoint. The induced worldsheet metric degenerates there; the conserved densities below have finite limits, so the endpoint degeneracy causes no divergent charge.

In these coordinates ${\Pi }^{\tau \mu }=T{\dot{X}}^{\mu }$. The target-space Noether charges of a string give the energy, momentum and angular momentum:Since this is the centre-of-momentum frame, the rest mass is $M=E$. Eliminating $a$ proves the free-ended straight-string Regge relation:A spacetime Lorentz transformation of the solution changes its frame, not this rest-frame relation.

There is also a folded closed string version: use the same formula over $0\le \sigma \le 2\pi$. The spatial segment is then covered twice, with folds at $\sigma =0,\pi$, and the closed-string mode expansion is periodic. Each charge is twice its open string value, soThe folds also have degenerate induced metric and are understood through the finite canonical densities or a limiting smooth motion. The two classical Regge trajectories have different slopes because the closed string has two coincident branches. No quantum string intercept enters this classical calculation.

Let $Y$ be the compact coordinate, identified modulo $2\pi R$. The closed string may wrap the circle:Single-valued momentum eigenfunctions $e^{ipY}$ require $p=n/R$, $n\in \mathbb{Z}$. These are the momentum and winding modes. Write the chiral zero-mode momenta asThe left-moving string oscillators will be tilded: $N_L=\tilde{N}$, $N_R=N$. This fixes the sign in compact-circle closed-string level matching.

Define the lower-dimensional mass by $M^2=-p_{\mathrm{noncompact}}^2$. The two closed-string physical-state Virasoro conditions, with bosonic string intercept one, areTheir average and difference derive the complete free bosonic string mass spectrum on the circle:The allowed states have nonnegative integer oscillator levels and obey the Virasoro constraints within each chiral sector. Compactification has not removed oscillator polarizations; it has added quantized compact momentum and allowed winding while changing the lower-dimensional interpretation of those polarizations.

At large $R$, the momentum and winding modes with $w=0$ have closely spaced momentum energies $|n|/R$, whereas nonzero winding costs $|w|R/{\alpha }^{\prime }$. At small $R$ their roles reverse. The precise equality is T-duality:It leaves $p_L$ unchanged and reverses $p_R$, and leaves the mass formula and closed-string level matching unchanged. On the string embedding map it reverses the right-moving part of $Y$. Therefore the small-radius spectrum is the large-radius spectrum of the dual theory, rather than a spectrum in which all charged states simply become heavy.

At a generic radius, the neutral level $(N_L,N_R)=(1,1)$ contains the lower-dimensional graviton, Kalb–Ramond field and dilaton, two gauge bosons from the mixed components of the metric and two-form, and the radius modulus of a string compactification. The two vector charges are the Cartan charges of $U(1{)}_L\times U(1{)}_R$.

Consider instead the following charged states:Every state satisfies compact-circle closed-string level matching. The single noncompact string oscillator supplies a vector polarization. At the massless point, the level-one Virasoro constraint imposes transverse polarization and its physical null string state identifies polarizations differing by the noncompact momentum. Thus these are genuine lower-dimensional vector states. Their squared mass isThus four additional charged massless spin-one particles appear atFor $(N_L,N_R)=(0,1)$ the charges have $p_R=0$, $p_L=\pm 2/\sqrt{{\alpha }^{\prime }}$; for $(1,0)$ they have $p_L=0$, $p_R=\pm 2/\sqrt{{\alpha }^{\prime }}$. This identifies the two pairs of charged roots.

The enhanced gauge group can be derived rather than just inferred from counting. At the self-dual circle, the chiral compact boson satisfies $Y_L(z)Y_L(u)\sim -({\alpha }^{\prime }/2)\log (z-u)$. The self-dual circle current algebra has currentsThe exponentials have conformal weight ${\alpha }^{\prime }{p}_L^2/4=1$. Free-boson contractions give the operator product expansionsThey form the level-one SU(2) Lie algebra, and there is an independent right-moving copy. The string vertex operators $J_L^a\bar{\partial }{X}^{\mu }$ and $\partial X^{\mu }{J}_R^a$ give the six massless vectors. ConsequentlyChanging $R$ gives the four charged vectors the mass found above, consistent with the radius deformation as a Higgs mechanism.

For completeness, the enhanced spectrum also has scalar fields. Using a compact rather than noncompact oscillator in the four singly excited charged families gives four scalars. At $(N_L,N_R)=(0,0)$, closed-string level matching requires $nw=0$; the mass equation then gives the four charges $(\pm 2,0),(0,\pm 2)$. These exceptional massless ground states of a bosonic circle give another four scalars. Together with the radius scalar they form nine scalars in the $(3,3)$ representation, in addition to the neutral dilaton. The ever-present neutral bosonic tachyon still has $M^2=-4/{\alpha }^{\prime }$. The enhancement establishes the additional massless vectors, not stability of the entire bosonic vacuum.

Use target-space signature $(-,+,\dots ,+)$ 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, givesThus the embedding equation of motion isThe independent worldsheet metric variation uses $\delta \sqrt{-\gamma }=-\frac{1}{2}\sqrt{-\gamma }{\gamma }_{ab}\delta {\gamma }^{ab}$. It sets the worldsheet stress tensor to zero:In conformal gauge, ${\gamma }_{ab}=e^{2\omega }\mathrm{diag}(-1,1)$, the Weyl factor cancels. Define worldsheet derivatives ${\partial }_{\pm }=({\partial }_{\tau }\pm {\partial }_{\sigma })/2$. The equations and Virasoro constraints reduce toEquivalently, $\ddot{X}-X^{\prime \prime }=0$, $\dot{X}\cdot X^{\prime }=0$ and ${\dot{X}}^2+X^{\prime 2}=0$. The free wave equation permits independent periodic left- and right-moving fields.

Take $0\le \sigma <2\pi$ and the uncompactified closed-string mode expansionThe Virasoro generators are the Fourier coefficients of the two quadratic worldsheet stress tensors. Their quantum expressions areThe 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 $L_0={\alpha }^{\prime }{p}^2/4+N$, where $N={\sum }_{n>0}{\alpha }_{-n}\cdot {\alpha }_n$, and similarly on the other side. In old covariant string quantization, physical string states obeymodulo 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 implyThus the massless first closed-string level has $N=\tilde{N}=1$, and its general state isTo 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 ${\tilde{\alpha }}_0^{\nu }$. All higher positive generators annihilate the state automatically. ThereforeThere is no condition that the tensor trace vanish: each chiral side has only one oscillator, so neither $L_2$ nor ${\tilde{L}}_2$ imposes a trace condition. The null-state quotient of a string also identifiesFor example, $L_{-1}({\xi }_{\nu }{\tilde{\alpha }}_{-1}^{\nu }|0,p⟩)$ 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 ${\zeta }_{ij}$, $i,j=1,\dots ,24$.

Decompose that transverse matrix into its symmetric trace-free part, antisymmetric part and trace. They describe the graviton, Kalb–Ramond field and dilaton, respectively. Their particle polarization counts areThese 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 isHere ${ϵ}^{ab}$ 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, $\Phi ={\Phi }_0+\hat{\Phi }$. By the Gauss-Bonnet theorem, its action isFor a connected oriented surface of genus $g$, its Euler characteristic is $\chi =2-2g$. Defining the string coupling by $g_s=e^{{\Phi }_0}$ givesThus 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.

Use target-space signature $(-,+,+,+)$ and conformal gauge $g_{ab}=e^{2\omega }\mathrm{diag}(-1,1)$. Dots and primes denote differentiation with respect to $\tau$ and $\sigma$. Varying the Polyakov action with respect to the string embedding map gives the wave equation; varying the worldsheet metric sets the worldsheet stress tensor to zero. Explicitly,With ${\sigma }^{\pm }=\tau \pm \sigma$ and ${\partial }_{\pm }=\frac{1}{2}({\partial }_{\tau }\pm {\partial }_{\sigma })$, the two Virasoro constraints are $({\partial }_{+}X{)}^2=({\partial }_{-}X{)}^2=0$.

A fixed spatial circle would sweep out a cylindrical string worldsheet. In static coordinates its string embedding map is $X=(t,R\cos \theta ,R\sin \theta ,0)$ and its induced worldsheet metric is $\mathrm{diag}(-1,R^2)$. The radial part of the Nambu–Goto equations of motion is proportional to $R^{-2}{\partial }_{\theta }^2(R\cos \theta ,R\sin \theta )=-(\cos \theta ,\sin \theta )/R$, which is nonzero. Thus a nonzero-radius cylinder cannot solve the equations. Motion of labels around this same cylinder is a worldsheet diffeomorphism and cannot alter this conclusion.

The familiar pulsating circular string makes the obstruction particularly explicit. Put $X^0=\kappa \tau$ and $X^1+i{X}^2=r(\tau )e^{i\sigma }$. The wave equation and Virasoro constraints reduce toA constant $r=R\ne 0$ violates the first equation. Even rotating the labels, $X^1+i{X}^2=R{e}^{i(\sigma +\Omega \tau )}$, cannot help: the wave equation requires ${\Omega }^2=1$, whereas $\dot{X}\cdot X^{\prime }=R^2\Omega$ requires $\Omega =0$.

Nor does a rigid rotation of the circular plane rescue the usual circular ansatz in three spatial dimensions. Write its spatial part as $A(\tau )\cos \sigma +B(\tau )\sin \sigma$, with $A^2=B^2=R^2$ and $A\cdot B=0$. The wave equation gives $\ddot{A}=-A$ and $\ddot{B}=-B$. The constant norms give $A\cdot \dot{A}=B\cdot \dot{B}=0$ and ${\dot{A}}^2={\dot{B}}^2=R^2$. The mixed Virasoro constraint, together with the derivative of $A\cdot B=0$, gives $\dot{A}\cdot B=A\cdot \dot{B}=0$. Differentiating $A\cdot B$ twice gives $\dot{A}\cdot \dot{B}=0$. This would require four mutually orthogonal nonzero spatial vectors in ${\mathbb{R}}^3$. This obstruction is specific to the stated spatial dimension; a rigid circular string in four spatial dimensions has more room.

To express the Virasoro constraints in modes, extend the string oscillators byDifferentiating the two chiral expansions gives ${\partial }_{-}{X}^{\mu }=\sqrt{{\alpha }^{\prime }/2}{\sum }_n{\alpha }_n^{\mu }{e}^{-in{\sigma }^{-}}$ and the corresponding expression with tildes for ${\partial }_{+}X$. Equating every Fourier coefficient of their squares to zero yieldsReality of the string embedding map means ${\alpha }_{-n}={\alpha }_n^{\ast }$ and ${\tilde{\alpha }}_{-n}={\tilde{\alpha }}_n^{\ast }$. In particular, define the classical string level operators by $N={\sum }_{n>0}{\alpha }_{-n}\cdot {\alpha }_n$ and $\tilde{N}={\sum }_{n>0}{\tilde{\alpha }}_{-n}\cdot {\tilde{\alpha }}_n$. The zero-mode Virasoro constraints becomeConsequently the classical bosonic string mass spectrum isThe equality of the two levels is closed-string level matching. There is no string intercept here: that shift comes from quantum normal ordering, not from these classical equations.

For the compact boson, closed-string periodicity permits a winding number $w\in \mathbb{Z}$, so $X^3(\sigma +2\pi )-X^3(\sigma )=2\pi wR$. Let $q$ be the momentum in this direction. Its zero-mode expansion and the corresponding chiral momenta areThus replace the equal left and right zero modes by ${\tilde{\alpha }}_0^3=\sqrt{{\alpha }^{\prime }/2}{p}_L^3$ and ${\alpha }_0^3=\sqrt{{\alpha }^{\prime }/2}{p}_R^3$; the nonzero string oscillators remain integer-moded. Classically $q$ is continuous. Quantizing the compact boson makes momentum wavefunctions single-valued around its circle, giving $q=n/R$ with $n\in \mathbb{Z}$.

If $k^a$, $a=0,1,2$, is the noncompact momentum and $M^2=-k^a{k}_a$, the two zero-mode Virasoro constraints giveAveraging and subtracting gives the momentum and winding modes and compact-circle closed-string level matching formulasThe latter becomes $\tilde{N}-N+nw=0$ on imposing quantum momentum quantization. The sign follows from assigning tildes to the left mover ${\sigma }^{+}$.

An explicit winding-supported circular string, valid classically for every $R>0$, isThe noncompact projection is a circle of constant radius. Every coordinate obeys the wave equation. Its compact winding is $w=1$, and a direct calculation givesThus both Virasoro constraints hold and the induced worldsheet metric is $2R^2\mathrm{diag}(-1,1)$, not a degenerate cylinder. The compact motion cancels the mixed stress that spoiled rotation of labels in flat space. Here $q=-R/{\alpha }^{\prime }$, $N=0$ and $\tilde{N}=R^2/{\alpha }^{\prime }$ satisfy the classical compact-circle closed-string level matching condition. Requiring this particular classical momentum to equal an individual quantum eigenvalue would impose an additional radius-dependent condition; that is not required to construct the classical solution.

The conformal condition is a quantum worldsheet Weyl anomaly cancellation, not the classical metric equation of the Polyakov action. Assume a smooth target metric, no antisymmetric background field, and curvature/gradient scales large compared with $\sqrt{{\alpha }^{\prime }}$. Use $[{\nabla }_a,{\nabla }_b]v^c=R^c{}_{dab}{v}^d$ and $R_{ab}=R^c{}_{acb}$, consistent with the Ricci-scalar Weyl transformation in the hint.

A sign convention in the printed action must be made explicit. With worldsheet signature $(-,+)$, continuation $\tau =-i{\tau }_E$ and $e^{i{S}_L}=e^{-S_E}$ sends its negative kinetic term to a positive Euclidean kinetic term, while its positive Lorentzian curvature coupling becomes a negative Euclidean curvature coupling. Define the conventional Euclidean dilaton by $\phi =-\Phi$ for this literal action. Then $S_{E,\mathrm{dil}}=(4\pi {)}^{-1}\int \sqrt{h}{R}^{(2)}\phi$. The Lorentzian dilaton coupling sign convention is therefore important: a convention with a negative Lorentzian curvature coupling would instead use $\phi =\Phi$ and reverse every term linear in the printed $\Phi$ below.

For completeness, the metric part of the one-loop calculation can be obtained by a geodesic background field expansion of a string sigma model. Write $X={\exp }_{\bar{X}}Y$. Its quadratic Euclidean action containsHere $D_{\mu }{Y}^a={\partial }_{\mu }{Y}^a+{\Gamma }^a{}_{bc}{\partial }_{\mu }{\bar{X}}^b{Y}^c$. In dimension $2-\epsilon$, the ultraviolet coincident contraction has pole $⟨Y^a{Y}^b{⟩}_{\mathrm{div}}={\alpha }^{\prime }{g}^{ab}/\epsilon$ with an infrared regulator. Contracting the curvature term gives a divergence $-(4\pi \epsilon {)}^{-1}\int \sqrt{h}{R}_{cd}\partial {\bar{X}}^c\partial {\bar{X}}^d$. It is cancelled by the metric counterterm $\delta g_{ab}={\alpha }^{\prime }{R}_{ab}/\epsilon$, giving the one-loop sigma-model beta function ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}+O({\alpha }^{\prime 2})$ before the dilaton improvement. Equivalently its local Euclidean Weyl variation is $-(4\pi {)}^{-1}\int \sqrt{h}\omega R_{ab}\partial X^a\partial X^b$, with terms proportional to the embedding equations understood as field redefinitions.

Set $\Omega =e^{\omega }$ in the supplied curvature transformation. On a closed string worldsheet, integration by parts givesThe chain rule and leading string embedding map equation implyThe kinetic embedding equation suffices for extracting the leading metric coefficient. More precisely, the full Euclidean embedding equation makes the parenthesis $({\alpha }^{\prime }/2)R^{(2)}{\nabla }^a\phi$, giving a curvature contribution $-({\alpha }^{\prime }/4\pi )\int \sqrt{h}\omega R^{(2)}|\nabla \phi {|}^2$ to the Weyl variation. This contributes to the scalar coefficient below rather than to the leading metric tensor coefficient. Combining the curvature counterterm anomaly with this variation gives the leading metric-dilaton Weyl condition:This is the gravitational equation in the string frame, rather than an ordinary Einstein equation with a minimally coupled scalar. In terms of the field and signs literally printed in this Lorentzian action it readsThe frequently used $R_{ab}+2{\nabla }_a{\nabla }_b\Phi =0$ is obtained if the printed $\Phi$ is identified with the conventional Euclidean dilaton, which requires the opposite Lorentzian curvature-coupling sign. Both conventions describe the same mathematics after $\Phi \mapsto -\Phi$, but one cannot change only the field equation silently. Also, vanishing of this tensor coefficient is the metric part of Weyl invariance; the dilaton curvature coefficient and central-charge condition remain to be checked.

Now derive the scalar consequence without presupposing its integration constant. The contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. The Ricci identity applied to the gradient of a scalar givesDiverging the metric equation and then substituting $R_{ab}=-2{\nabla }_a{\nabla }_b\phi$ yieldsThus the bracket is constant on each connected target component. Its trace equation is $R+2{\square }_g\phi =0$, so the dilaton equation from contracted Bianchi identity becomesThe second equation uses the literal printed sign convention. These are nonlinear scalar wave equations analogous to the Klein-Gordon equation. For the exponentiated dilaton wave equation, put $F=e^{-2\phi }=e^{2\Phi }$; differentiating twice givesHere ${\square }_g={\nabla }^a{\nabla }_a$ is the Lorentzian Laplace-Beltrami operator, and $|\nabla \phi {|}^2=g^{ab}{\partial }_a\phi {\partial }_b\phi$ is a Lorentzian contraction, not necessarily nonnegative.

The Bianchi and Ricci identities alone do not imply $C=0$. The linear dilaton counterexample to zero integration constant is flat target space with $\phi =q_a{X}^a$: the tensor equation holds for every constant $q$, while $C=-2q^2$. A non-null $q$ disproves any deduction of the zero-constant scalar equation from that tensor equation alone.