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In a two-dimensional conformal field theory, a quasi-primary operator of conformal weights $(h,\tilde{h})$ obeysfor global conformal, or Möbius, transformations. A primary operator obeys this law for every locally invertible holomorphic and antiholomorphic conformal coordinate change. Quasi-primary is therefore weaker: the stress tensor at nonzero central charge is quasi-primary, but its Schwarzian transformation term prevents it from being primary under a general map.

A primary state is a simultaneous eigenstate of $L_0,{\tilde{L}}_0$ with eigenvalues $h,\tilde{h}$, satisfyingA quasi-primary state need only satisfy the corresponding $n=1$ conditions. Negative modes of the Virasoro algebra generate descendants.

In radial quantization, a local operator at the origin prepares a state on a surrounding circle; logarithmic radius is Euclidean time. The state–operator correspondence is $|\mathcal{O}⟩=\mathcal{O}(0)|0⟩$. Acting with a Virasoro generator means inserting its stress-tensor contour around that operator. For a primary operator, the operator product expansion isMultiplication by $z^{n+1}$ and the contour residue give $L_0|\mathcal{O}⟩=h|\mathcal{O}⟩$, $L_{-1}|\mathcal{O}⟩=|\partial \mathcal{O}⟩$, and $L_n|\mathcal{O}⟩=0$ for $n>0$. Conversely the contour modes are the coefficients of the singular expansion; vanishing positive modes removes every pole of order three and higher. The remaining double and simple poles generate precisely the primary transformation law. The antiholomorphic argument is identical. This proves the state-operator correspondence for Virasoro primaries in both directions.

An integrated string vertex operator is integrated with $d^{2}z$. The measure transforms with weights $(-1,-1)$, so conformal invariance requires its matter operator to have weights $(1,1)$. In old covariant string quantization the same requirement is $L_{n>0}={\tilde{L}}_{n>0}=0$ and $L_0={\tilde{L}}_0=1$. The ghost-dressed unintegrated vertex $c\bar{c}V$ has total weights $(0,0)$. Physical states are identified modulo null string states, or more systematically by BRST cohomology; raw counting of all primary operators includes gauge redundancy.

For a free embedding boson the holomorphic stress tensor is $T=-:\partial X\cdot \partial X:/{\alpha }^{\prime }$. Contracting it with $:e^{ik\cdot X}:$ gives weight ${\alpha }^{\prime }{k}^2/4$; every holomorphic oscillator of level $r$ adds $r$ to that weight. ThusWith Lorentzian target signature, $M^2=-k^2$, so the closed-bosonic-string mass shell from primary weights isThe ground state is tachyonic, level one is massless, and higher levels are massive. The primary conditions also constrain the polarization tensors; the two zero-mode conditions fix the mass and level matching.

For the commuting beta-gamma system, write $r=z-w$. The displayed exchange leaves the arguments attached to their respective fields. Exchanging the arguments instead givesThis radial-ordering sign in commuting beta-gamma contractions must be kept when differentiating. Expanding the total derivative in the proposed stress tensor givesThe contraction $\gamma (z)\beta (w)\sim 1/r$ and its derivative $\partial \gamma (z)\beta (w)\sim -1/r^2$ giveSimilarly $\partial \beta (z)\gamma (w)\sim 1/r^2$ and $\beta (z)\gamma (w)\sim -1/r$ giveThere are no higher poles. Both fields are holomorphic and $\bar{T}=0$, so

To compute the central charge, put $A=:\partial \beta \gamma :$ and $B=:\beta \partial \gamma :$. Only double Wick contractions can give the fourth-order pole in $TT$. Their coefficients arewhere only the double-contraction terms are shown. For example, the mixed contraction uses $\partial \beta (z)\partial \gamma (w)\sim 2/r^3$ and $\gamma (z)\beta (w)\sim 1/r$. All Wick pairings have bosonic signs. Therefore the fourth-order coefficient isThe stress-tensor operator product expansion has fourth-order coefficient $c/2$, yieldingThis is the stress tensor of a commuting beta-gamma system calculation. Single contractions supply the usual $2T/r^2+\partial T/r$ terms, with the potential third-order terms canceling.

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.

Insert the given closed-string mode expansion into the stress tensor and compare $T(z)={\sum }_n{L}_n{z}^{-n-2}$. Normal ordering gives

The inverse of the worldsheet Laplacian gives the free-field operator-product expansionHenceExtracting the Laurent modes by contour integration yields the string oscillator algebra

The vanishing of the worldsheet stress tensor becomes the Virasoro constraintsSince $L_0={\alpha }^{\prime }{k}^2/4+N$ and the bosonic-string normal-ordering constant is $a=1$, these zero-mode constraints are the target-space string mass-shell conditionThereforeThe positive-mode constraints further requireand null string states identify polarizations that differ by momentum-longitudinal terms. The symmetric trace-free, antisymmetric, and trace sectors of ${ϵ}_{\mu \nu }$ describe the graviton, Kalb–Ramond field, and dilaton, respectively.

Finally, the state–operator correspondence maps ${\alpha }_{-1}^{\mu }{\bar{\alpha }}_{-1}^{\nu }|k⟩$ toAdding its integrated massless closed-string vertex operator to the string nonlinear sigma model changes the background coupling byThus a Fourier mode $\delta G_{\mu \nu }(X)={ϵ}_{(\mu \nu )}{e}^{ik\cdot X}$ is precisely a linearized deformation of the target-space metric. Its transversality, mass-shell condition, and polarization gauge redundancy are the worldsheet statements that the deformation is marginal and is defined up to a linearized spacetime diffeomorphism.

Under the state–operator correspondence, let $|O_{ij}⟩$ be the state of the antisymmetric conformal primary operator $O_{ij}$. In radial quantization $P_i^{†}=K_i$, while primarity gives $K_i|O_{jk}⟩=0$. The norm of the level-one descendant obtained by taking a divergence is therefore determined by the conformal algebra commutatorUsing the two-form action of $M_{ab}$ gives, up to a positive normalization,Positivity of norm yields the two-form conformal unitarity boundAt saturation the descendant is null, and the operator obeys the conservation equation ${\partial }^i{O}_{ij}=0$.

In $d=3$, Hodge duality turns the two-form into the vector primary $V_k={ϵ}_{kij}{O}^{ij}/2$. The vector divergence descendant has norm proportional to $\Delta -d+1=\Delta -2$. Consequently the stronger bound israther than the formal two-form value $d-2=1$.

By the state–operator correspondence, a CFT operator of scaling dimension $\Delta$ creates a cylinder state of energy $E=\Delta /R$. The identity gives the vacuum; the first nontrivial low-energy single-trace operator is the conserved stress-energy tensor, with $\Delta =3$ in three dimensions; and one translation raises the dimension by one. Taking the scheme-dependent vacuum energy to vanish,All other single-trace primaries are heavy by assumption, while the first two-graviton state begins at $6/R$.

The state–operator correspondence maps $\mathcal{O}$ to the lowest one-particle state on $S^4\times \mathbb{R}$. On a sphere of radius $r$, cylinder energy equals scaling dimension divided by $r$. The primary has $\Delta =5$.

A translation generator raises the dimension by one and transforms as a vector of $SO(5)$. States with no angular momentum arise from scalar descendant pairs $P^2$, so the allowed descendants areTheir dimensions are $5+2n$, and hence the one-quantum, zero-angular-momentum energies areThese are the $l=0$ normal-mode energies of a massless scalar in global Anti-de Sitter spacetime.

A primary state is a simultaneous eigenvector of $L_0,{\tilde{L}}_0$ with eigenvalues $h,\tilde{h}$ that is annihilated by all positive modes of both Virasoro algebras. Negative modes generate descendants. The state–operator correspondence identifies it with a primary operator inserted at the origin.