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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.

The closed string already contains the particle required for quantum gravity. For bosonic string theory in its critical dimension of string theory, the mass-shell condition and closed-string level matching areAt $N=\tilde{N}=1$, the states ${ϵ}_{mn}{\alpha }_{-1}^m{\tilde{\alpha }}_{-1}^n|0;k⟩$ are massless. After imposing the Virasoro constraints and quotienting null string states, the transverse polarization tensor decomposes into symmetric trace-free, antisymmetric and scalar pieces under $SO(D-2)$. They describe a graviton, the Kalb–Ramond field and the dilaton. In particular, the symmetric trace-free sector is a massless spin-two graviton. Its number of degrees of freedom is $D(D-3)/2$.

The state–operator correspondence associates each physical string state with a string vertex operator. For example, the matter part of a massless closed-string vertex operator isIts conformal weights are $(1,1)$, so its integrated form $\int d^{2}z{V}_{ϵ,k}$ is invariant under changes of string worldsheet coordinates. At fixed insertion positions it is accompanied by $c\bar{c}$ from the worldsheet ghost fields. Physical vertices represent BRST cohomology classes; longitudinal changes of the polarization tensor are BRST-exact operators and decouple from physical scattering amplitudes. For the graviton, this string-state gauge redundancy becomes the linearized target-space diffeomorphism $h_{mn}\mapsto h_{mn}+{\partial }_m{\xi }_n+{\partial }_n{\xi }_m$.

The Polyakov path integral computes a closed-string scattering amplitude by inserting the external string vertex operators, integrating their unfixed positions and the inequivalent worldsheet moduli, and including the worldsheet ghost fields. On a Riemann sphere three positions are fixed by the residual conformal group. For example, contractions of tachyon vertex operators produce the Koba-Nielsen factor, whose position integral gives the Virasoro–Shapiro amplitude.

The string dual resonance property means that one crossing-symmetric scattering amplitude has equivalent expansions in the different channels: its poles exhibit intermediate string states in each channel, rather than separate channel contributions being added again. The scattering-amplitude factorization at a pole identifies the intermediate particle and its couplings. Set $8\pi T=1$, so ${\alpha }^{\prime }=4$ and the closed-string tachyon has $M^2=-1$. For four such external states the Mandelstam variables obey $s+t+u=-4$. The Virasoro–Shapiro amplitude has generic $s$-channel poles at $s=-1,0,1,\dots$, exactly the bosonic string mass spectrum in these units.

The massless pole gives an especially direct check. With overall normalization $C$ and $u=-4-s-t$, the Gamma function recurrence givesThusThe quadratic residue in $t$ contains a spin-two exchange. A scalar exchange alone could not give this angular dependence. Combined with the known massless spectrum and scattering-amplitude factorization, it identifies exchange of the graviton, with possible scalar contributions from the dilaton; the antisymmetric Kalb–Ramond field does not couple to two identical scalar tachyons here. Consequently the graviton participates in interactions, rather than being an isolated free state.

Decoupling longitudinal graviton polarizations forces a universal coupling to the conserved stress-energy tensor. Consistency of this massless spin-two gauge invariance extends the linearized coupling to the nonlinear dynamics of general relativity. At distances large compared with $\sqrt{{\alpha }^{\prime }}$, the gravitational sector of the effective action begins with the Einstein-Hilbert action, alongside dilaton and Kalb–Ramond field terms and higher-derivative string corrections. The infinite tower of string states supplies the short-distance completion of this quantum gravity expansion.

Interactions are organized by string worldsheet topology. A constant dilaton gives the string coupling $g_s=e^{{\Phi }_0}$, and a connected oriented surface of genus $g$ has Euler characteristic $\chi =2-2g$. Its weight is $g_s^{-\chi }$; with $n$ normalized external vertices,The sphere is tree order, the torus is one loop, and each additional handle adds a factor $g_s^2$. Each coefficient is an integral over worldsheet moduli, so this is string perturbation theory in $g_s$, with an independent low-energy expansion in ${\alpha }^{\prime }$.

For the one-loop vacuum contribution, torus modular invariance identifies $\tau$ with $(a\tau +b)/(c\tau +d)$, $ad-bc=1$. Integration is over the standard fundamental domain of the modular group,The potential short-proper-time region ${\tau }_2\to 0$, responsible for a point-particle ultraviolet divergence, is absent. It would count metrics already represented elsewhere in $\mathcal{F}$. Torus modular invariance is therefore the geometric reason for one-loop ultraviolet finiteness. More explicitly, in $D=26$ the vacuum integrand is proportional to $d^2\tau {\tau }_2^{-14}|\eta (\tau ){|}^{-48}$, with $\eta$ the Dedekind eta function; the complete expression is invariant under the modular group.

Ultraviolet finiteness does not make the bosonic one-loop vacuum energy finite. The remaining long-tube region ${\tau }_2\to \infty$ is an infrared divergence from the tachyon: $|\eta (\tau ){|}^{-48}\sim e^{4\pi {\tau }_2}$. It signals instability of the bosonic vacuum. Tachyon-free consistent backgrounds remove this particular obstruction, though other infrared effects must still be treated. The ultraviolet improvement comes from the extended string and the complete spectrum together with the worldsheet gauge identifications; the genus expansion remains a perturbative description of quantum gravity.

The BRST operator $Q_B$ is Grassmann odd and represents the gauge symmetry on the gauge-fixed state space. Requiring two successive BRST transformations to vanish meansThis nilpotence makes physical states a BRST cohomology. If $[Q_B,S_0]=0$ and $\Psi$ is the gauge-fixing fermion, the graded Jacobi identity givesso $S=S_0+\{Q_B,\Psi \}$ is BRST invariant.

A holomorphic field of conformal weight $h$ has the Laurent expansionUnder the state–operator correspondence, $\varphi (z)|0⟩$ must be regular at the origin. Terms with $n>-h$ have negative powers, soFor the anticommuting bc system,Separating creation and annihilation modes and summing the geometric series for $|z|>|w|$ gives the bc ghost operator-product expansion

The BRST current built from the matter and ghost stress tensors has an operator-product expansion with $c$ whose residue givesFor a matter Virasoro primary operator $\Phi (z,\bar{z})$ of weights $(1,1)$, the two standard string vertex operators areThe first is the local, unintegrated vertex. Using $Q_{B}c=c\partial c$, $Q_B\bar{c}=\bar{c}\bar{\partial }\bar{c}$, and the weight-$(1,1)$ transformation of $\Phi$, the terms cancel pairwise and $Q_{B}U=0$. For the integrated vertex,so $\{Q_B,V\}=0$ on a closed worldsheet because the variation is a total derivative. Both vertices therefore represent the same BRST cohomology class.

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 $1-h$.

A local operator $\mathcal{O}(w,\bar{w})$ is a primary operator of conformal weights $(h,\tilde{h})$ when its operator product expansions with the holomorphic and antiholomorphic stress tensors arewith no more singular poles. Equivalently, under a local conformal transformation it transforms covariantly with holomorphic exponent $h$ and antiholomorphic exponent $\tilde{h}$.

LetThe exponential is a primary operator with conformal weights $({\alpha }^{\prime }{k}^2/4,{\alpha }^{\prime }{k}^2/4)$. In the $T(z)V(w,\bar{w})$ OPE, contracting one derivative in $T$ with $\partial X^{\mu }$ and the other with the exponential produces a third-order pole proportional to $k^{\mu }{ϵ}_{\mu \nu }$. Its antiholomorphic counterpart is proportional to ${ϵ}_{\mu \nu }{k}^{\nu }$. Therefore $V$ is primary exactly when its polarization tensor is transverse in both indices,The remaining second-order poles give

The matter part of the massless closed-string vertex operator isIts symmetric trace-free polarization is the graviton, while its antisymmetric polarization is the Kalb–Ramond field, or B-field. The mass-shell condition $k^2=0$ and transversality make $V_{ϵ}$ a primary operator of conformal weights $(1,1)$, as required for an integrated string vertex operator.

The graviton polarization has the linearized gauge redundancywhile the antisymmetric polarization obeysIn either case the change in the integrated vertex is a worldsheet total derivative, hence vanishes on a closed worldsheet; in covariant language it is BRST-exact. This string-state gauge redundancy is the vertex-operator form of linearized target-space diffeomorphism or two-form gauge invariance.

For $E_{\mu \nu }={\eta }_{\mu \nu }$, transversality would require $k^{\mu }{\eta }_{\mu \nu }=k_{\nu }=0$. Hence the operator has a third-order stress-tensor pole and is not a primary operator for any nonzero $k$.

Forthe two transversality conditions coincide and reduce toIf $k^2=0$, this says $k\cdot \xi =-1$, with arbitrary additional component transverse to $k$. If $k^2\ne 0$, it forcesso $E_{\mu \nu }={\eta }_{\mu \nu }-k_{\mu }{k}_{\nu }/k^2$ is the transverse projection operator. Once this condition removes the third-order poles, the conformal weights are again $(1+{\alpha }^{\prime }{k}^2/4,1+{\alpha }^{\prime }{k}^2/4)$; in the null case they are $(1,1)$.

An operator $\mathcal{O}(w,\bar{w})$ is a primary operator of conformal weights $(h,\tilde{h})$ when its OPEs with the stress tensors arewith no more singular terms. Its scaling dimension and two-dimensional spin are

A primary operator of conformal weight $(h,\tilde{h})$ transforms under $z\mapsto w(z)$ asThe derivative $\partial X$ is a primary of weight $(1,0)$, and $\bar{\partial }X$ is a primary of weight $(0,1)$.

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 $\tilde{h}$.

Commutation allows exchanging the factors at their existing arguments; it does not allow exchanging their arguments while retaining the same pole sign. With the displayed convention, differentiation produces $\partial \beta (z)\gamma (w)\sim (z-w{)}^{-2}$ and $\partial \gamma (z)\beta (w)\sim -(z-w{)}^{-2}$. These signs are essential for the conformal weights and central charge of a commuting beta-gamma system.

At $R=\sqrt{{\alpha }^{\prime }}$, a chiral compact boson has operator product expansion $Y_L(z)Y_L(w)\sim -{\alpha }^{\prime }\log (z-w)/2$. The currents $J^3=i\partial Y_L/\sqrt{{\alpha }^{\prime }}$ and $J^{\pm }=:e^{\pm 2i{Y}_L/\sqrt{{\alpha }^{\prime }}}:$ have conformal weight one. Contractions give $J^3(z)J^3(w)\sim 1/[2(z-w{)}^2]$, $J^3(z)J^{\pm }(w)\sim \pm J^{\pm }(w)/(z-w)$ and $J^{+}(z)J^{-}(w)\sim 1/(z-w{)}^2+2J^3(w)/(z-w)$. These are the level-one SU(2) Lie algebra currents. The right-moving copy supplies a second factor. Combining a current with a noncompact coordinate derivative gives a massless string vertex operator for each of the six vectors; four are additional to the generic two Cartan vectors.

The commuting beta-gamma system produced by worldsheet supersymmetry gauge fixing has conformal weights $(3/2,-1/2)$ and central charge $11$ per chiral sector.

For free embedding bosons, the operator product expansion of the stress tensor with ${\zeta }_{\mu }:\partial X^{\mu }{e}^{ipX}:$ has a cubic pole $-i{\alpha }^{\prime }(p\cdot \zeta ):e^{ipX}:/2$. Its vanishing is precisely the primary condition, with weight $1+{\alpha }^{\prime }{p}^2/4$. The two-derivative closed-string vertex has the two displayed transversality conditions from its holomorphic and antiholomorphic cubic poles. Being primary does not require $p^2=0$; that extra condition follows when a physical integrated vertex must have conformal weights $(1,1)$.

Each real chiral fermion has conformal weight $1/2$ and contributes $1/2$ to the central charge. The Ramond sector and Neveu–Schwarz sector differ by periodic versus antiperiodic boundary conditions.