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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.

The stress-energy tensor generates translations. One may improve it by adding a divergence without changing the translation charges under suitable boundary conditions, and choose it symmetric in a relativistic theory. If ${\Theta }^{\mu \nu }$ is symmetric and conserved, a spacetime vector field $\omega$ defines the current $j^{\mu }={\Theta }^{\mu \nu }{\omega }_{\nu }$. For a Conformal Killing vector field,Thus a traceless conserved stress-energy tensor produces all the conformal charges. The dilation current is $x_{\nu }{\Theta }^{\mu \nu }$, and the special conformal currents are $(2x_{\lambda }{x}_{\nu }-x^2{\eta }_{\lambda \nu }){\Theta }^{\mu \nu }$. A trace that is merely the divergence of a virial current gives scale invariance; conformal invariance requires the additional improvement that removes the trace. These statements distinguish an actual traceless tensor from a trace discarded without justification.

For a free massless scalar field, the canonical stress-energy tensor has trace $(1-d/2)(\partial \varphi {)}^2$. For $d>1$, the improved stress-energy tensor of a free massless scalar isIts improvement is identically conserved. On $\square \varphi =0$, the identity $\square {\varphi }^2=2(\partial \varphi {)}^2$ shows that its trace vanishes. In $d=2$, no improvement is needed: the canonical stress-energy tensor is already traceless. In Lorentzian null coordinates, conservation and tracelessness imply that its two surviving components depend separately on $x^{+}$ and $x^{-}$. After Euclidean continuation, these become the holomorphic stress-energy tensor $T(z)$ and its antiholomorphic counterpart $\bar{T}(\bar{z})$, withLocal conformal changes are arbitrary holomorphic and antiholomorphic coordinate changes, not just the three global Möbius transformation generators in either sector.

To make the free-field quantum normalization explicit, rescale the field to $\phi =\sqrt{4\pi }\varphi$, so that its plane contraction is $\phi (z,\bar{z})\phi (w,\bar{w})\sim -\log |z-w{|}^2$. Define $J=i\partial \phi$, so $J(z)J(w)\sim (z-w{)}^{-2}$. Normal ordering givesA single Wick contraction shows that $J$ is a weight-one primary operator. The two double contractions in $T(z)T(w)$ produce $\frac{1}{2}(z-w{)}^{-4}$; single contractions, followed by Taylor expansion at $w$, produce $2T(w)/(z-w{)}^2+\partial T(w)/(z-w)$. Thus the operator product expansion isThe same holds independently for $\bar{T}$, with $\bar{c}=1$. The undifferentiated noncompact scalar has a logarithmic contraction and a zero-mode subtlety; its derivative current has the ordinary primary transformation law.

The holomorphic stress-energy tensor implements infinitesimal conformal transformations through contour charges $Q_v=\frac{1}{2\pi i}\oint dzv(z)T(z)$. A primary field $\mathcal{O}$ of weight $h$ obeysFor example the free-field vertex operator $:e^{i\alpha \phi }:$ has holomorphic weight $h={\alpha }^2/2$, as follows by contracting $\partial \phi$ with the exponential twice. In correlators the contour may encircle each insertion, giving the conformal Ward identity $⟨T(z){\prod }_i{\mathcal{O}}_i(z_i)⟩={\sum }_i[h_i/(z-z_i{)}^2+{\partial }_{z_i}/(z-z_i)]⟨{\prod }_i{\mathcal{O}}_i(z_i)⟩$. This also yields the global conformal Ward identities for chiral correlators.

Writing $T(z)={\sum }_n{L}_n{z}^{-n-2}$ and integrating the stress-tensor operator product expansion gives the Virasoro algebraIts central charge is the quantum correction to the classical conformal algebra; the correction vanishes on $L_{-1},L_0,L_1$. Equivalently, in the pullback convention a finite holomorphic change $z\mapsto f(z)$ sends $T$ to $(f^{\prime }{)}^{2}T(f)+\frac{c}{12}\{f,z\}$, where $\{f,z\}=f^{\prime \prime \prime }/f^{\prime }-\frac{3}{2}(f^{\prime \prime }/f^{\prime }{)}^2$ is the Schwarzian derivative. This anomalous term vanishes for Möbius transformations. On a curved background the related trace anomaly is proportional to its curvature; it does not destroy the flat-space conformal symmetry of this massless theory.

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.

Use a mostly-plus target Minkowski spacetime metric and restore the string tension by $T=(2\pi {\alpha }^{\prime }{)}^{-1}$. After Wick rotation, the Polyakov action on a Riemann surface isThe Polyakov path integral integrates over embeddings and worldsheet metrics, dividing by worldsheet diffeomorphisms and Weyl transformations. For a consistent flat bosonic string theory, take the critical dimension of the bosonic string $d=26$. Conformal gauge turns the embedding fields into free scalars; its Faddeev-Popov determinant is represented by the bc system. The remaining integrations are over worldsheet moduli and external insertion points.

A tachyon vertex operator isThe worldsheet Green function is $⟨X^a(z)X^b(w)⟩=-({\alpha }^{\prime }/2){\eta }^{ab}\log |z-w{|}^2$. The operator product expansion with the holomorphic stress-energy tensor gives conformal weights $({\alpha }^{\prime }{k}^2/4,{\alpha }^{\prime }{k}^2/4)$. An integrated string vertex operator must have weights $(1,1)$, so the mass-shell condition is $k^2=4/{\alpha }^{\prime }$, or tachyonic mass parameter $m^2=-4/{\alpha }^{\prime }$.

At lowest order in the string coupling, the worldsheet is a Riemann sphere. For four external states, three insertion points can be fixed by a Möbius transformation. In the worldsheet ghost field description these three vertices carry $c\bar{c}$, while the fourth is integrated. The three-point worldsheet ghost correlator gives the squared product of the three fixed-point separations, precisely the sphere gauge fixing for four string vertices factor. Thus fixing three points is a gauge choice, rather than deleting three integrations without their Jacobian.

The integral over the constant embedding worldsheet zero mode gives $(2\pi {)}^d{\delta }^{(d)}({\sum }_i{k}_i)$. The remaining Gaussian integral, using normal ordering to remove self-contractions, gives the Koba-Nielsen factorSet $(z_1,z_2,z_3,z_4)=(0,z,1,\infty )$; the vertex at infinity is defined with its conformal-weight factor. Let all momenta be incoming and defineThe mass-shell condition and momentum conservation imply $s+t+u=-16/{\alpha }^{\prime }$. Since ${\alpha }^{\prime }{k}_1\cdot k_2=-4-{\alpha }^{\prime }s/2$ and similarly for the pair $(2,3)$, the sphere tachyon position integral becomesHere $\mathcal{N}$ depends on the normalization of external string vertex operators; it does not affect the kinematic dependence.

To evaluate the remaining worldsheet modulus, put $a=-1-{\alpha }^{\prime }s/4$ and $b=-1-{\alpha }^{\prime }t/4$. The complex beta integral isFor example, this identity follows by writing the two powers as Schwinger-parameter Gamma integrals, doing the two-dimensional Gaussian integral, and changing the two positive parameters to their sum and ratio. The complex beta integral first holds for $\mathrm{Re}a,\mathrm{Re}b>0$ and $\mathrm{Re}(a+b)<1$; the scattering result is its analytic continuation. The third gamma-function argument is $1-a-b=-1-{\alpha }^{\prime }u/4$. Absorbing $\pi$ into $\mathcal{N}$, the result is the tree-level four-tachyon amplitude, the Virasoro–Shapiro amplitude:It is symmetric in the three Mandelstam variables and has generic poles at $q=4(N-1)/{\alpha }^{\prime }$, matching the bosonic string mass spectrum. The formula is the leading term of string perturbation theory. Higher orders are constructed by the same Polyakov path integral prescription on higher-genus worldsheets, with four marked points and the corresponding ghost and worldsheet moduli measure; genus $g$ has coupling power $g_s^{2g+2}$.

The holomorphic part of the free-boson worldsheet propagator givesApply Wick theorem to $T(z)T(w)$. The two double contractions give $D/[2(z-w{)}^4]$, while the single contractions reconstruct $T$ and its derivative. Thus the stress-tensor operator-product expansion isand the $D$ embedding coordinates have central charge $\boxed{c=D}$.

The holomorphic stress-energy tensor generates an infinitesimal conformal transformation throughTaking the three residues givesThe third derivative is the anomalous term that prevents $T$ from transforming as an ordinary weight-two Virasoro primary operator.

With Virasoro algebra modes $L_n=(2\pi i{)}^{-1}\oint dz{z}^{n+1}T(z)$, a second contour calculation givesso $A(m)=c(m^3-m)/12=D(m^3-m)/12$. This Virasoro central extension is the quantum conformal anomaly. In string theory the matter and ghost contributions must cancel; $c_{\mathrm{matter}}+c_{\mathrm{ghost}}=D-26=0$ gives the critical dimension of string theory $D=26$ and makes the BRST operator nilpotent.

Applying Wick theorem to the stated holomorphic stress-energy tensor givesandThese are exactly the stress-tensor OPEs of primary operators. Hence $\gamma$ has holomorphic conformal weight $h$ and $\beta$ has weight $1-h$.

Under a finite conformal coordinate change $z^{\prime }=f(z)$ and ${\bar{z}}^{\prime }=\bar{f}(\bar{z})$, a primary operator of conformal weight $(h,\tilde{h})$ obeysEquivalently, its operator product expansion with the holomorphic stress-energy tensor has singular partThe antiholomorphic stress tensor gives the analogous formula with $\tilde{h}$.