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The BRST current is the conserved worldsheet current whose contour integral is the BRST operator. For the bosonic string it is built from the reparameterization ghost and the matter and ghost stress tensors; its self-OPE is nonsingular only when the total central charge vanishes.

Contracting an antighost with the cubic FP ghost term of the BRST operator produces the bilinear generators. They give $[L_m,b_n]=(m-n)b_{m+n}$ and $[L_m,c_n]=-(2m+n)c_{m+n}$. Their zero-mode additive constant depends on the vacuum and normal ordering convention; it must agree with the corresponding $c_0$ term in the charge.

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 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 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.

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.