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The worldsheet ghost fields form a bc system with conformal weights $2$ and $-1$. Setting $J=2$ in $c=-2(6J^2-6J+1)$ gives central charge $-26$ in each chirality. The worldsheet Weyl anomaly of $D$ free embedding bosons cancels at $D=26$.

Each of $D$ embedding bosons contributes one to the chiral central charge, and each real worldsheet Majorana fermion contributes one half. The reparameterization bc system contributes $-26$, and the superconformal ghosts contribute $+11$. Vanishing total worldsheet Weyl anomaly therefore requires $3D/2-26+11=0$, or $D=10$. This differs from the critical dimension 26 of the bosonic string, which has no worldsheet fermions or superconformal ghosts.

For the anticommuting bc system, $\{b_m,c_n\}={\delta }_{m+n,0}$. A useful oscillator vacuum obeys $b_n|0⟩=c_n|0⟩=0$ for $n>0$ and $b_0|0⟩=0$. Its partner $c_0|0⟩$ is nonzero. This specifies a ghost zero-mode doublet and differs from imposing annihilation by both zero modes.

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.

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 RNS string adds worldsheet Majorana fermions ${\psi }^{\mu }$ to the coordinate fields $X^{\mu }$. The new fields are spinors on the worldsheet but vectors under the target Lorentz group; this distinction matters when interpreting their oscillator states. In a flat target, a convenient Lorentzian convention for the free gauge-fixed action iswhere ${\rho }^a$ are two-dimensional gamma matrices. The bosonic and fermionic kinetic terms are related by worldsheet supersymmetry. Before fixing local worldsheet supersymmetry, one introduces a worldsheet gravitino along with the worldsheet metric. After superconformal gauge fixing, the residual matter constraints are the Virasoro constraints and the constraints generated by the superconformal current $G\propto \psi \cdot \partial X$.

Each real worldsheet Majorana fermion contributes $1/2$ to the chiral central charge, while each coordinate contributes one. The reparameterization bc system contributes $-26$, and fixing local worldsheet supersymmetry also produces commuting superconformal ghosts of weights $(3/2,-1/2)$, with central charge $11$. ThusIn light-cone gauge in string theory there are eight physical transverse coordinate fields and eight transverse worldsheet Majorana fermions.

On the doubled spatial interval of an open string, the two allowed fermionic periodicities areThey arise from the relative signs of the left-right fermion gluing conditions at the two ends. Thus the Ramond sector has integer-mode oscillators $d_n$, including $d_0$, and the Neveu–Schwarz sector has half-integer oscillators $b_r$, $r\in \mathbb{Z}+1/2$, without zero modes. In either sector,A closed string has independent left- and right-moving periodicities, giving four sector choices rather than two.

The Ramond zero-mode Clifford algebra isHence its ground states carry a Spinor representation of the Lorentz group. The zero-mode supercurrent constraint reduces on a ground state to the Dirac equation $p_{\mu }{\Gamma }^{\mu }u=0$. In the Neveu–Schwarz sector the oscillator vacuum instead carries a scalar representation. Consequently the open string NS sector describes spacetime bosons, while the R sector describes spacetime fermions. Exciting a target-vector oscillator does not change this statistics assignment.

The string intercepts follow from the transverse zero-point energy. Per boson the integer-mode contribution is $\frac{1}{2}\zeta (-1)=-1/24$. Per fermion it is the negative of the corresponding half-sum. Since $\zeta (-1,1/2)=1/24$, the eight transverse pairs giveThus $a_{\mathrm{NS}}=1/2$, $a_{\mathrm{R}}=0$, and the open string mass formula is ${\alpha }^{\prime }{M}^2=N-a$. This distinguishes the lowest states explicitly:

The supersymmetric GSO projection keeps odd oscillator fermion number in the NS sector. Defining $F_{\mathrm{osc}}$ so that the NS vacuum has $F_{\mathrm{osc}}=0$, its projector isIt removes the scalar tachyon and keeps the massless vector. In the R sector, fermion parity contains the zero-mode chirality operator, so one may writeHere $F_{\mathrm{osc}}$ counts nonzero-mode fermionic excitations. On the ground states, this keeps one Majorana-Weyl spinor chirality: 16 real covariant components, or eight on-shell polarizations after the Dirac equation. The projected massless open-string states are an eight-polarization gauge vector and an eight-polarization gaugino. The compatible projected tower exhibits spacetime supersymmetry; unprojected worldsheet supersymmetry by itself did not eliminate the tachyon or equalize these massless counts.

For a closed string, both chiral zero-mode constraints must hold:This shifted closed-string level matching is particularly important in the mixed sectors. The lowest states and their spacetime nature are as follows.

For type IIA superstring theory, the two R sector ground-state chiralities are opposite. The Ramond–Ramond potentials can be represented by $C_1,C_3$, with transverse counts $8+56=64$. The two mixed-sector gravitinos have opposite chiralities, and the ten-dimensional theory is nonchiral. For type IIB superstring theory, the two R sector chiralities agree; the Ramond–Ramond potentials are $C_0,C_2,C_4$, with self-dual five-form field strength, giving $1+28+35=64$. Its two gravitinos have the same chirality. Thus both projected type II theories have 128 massless bosonic and 128 massless fermionic polarizations and no tachyon, while their chiralities and form spectra differ.

If unoriented open strings are included in the complete theory, the supersymmetric example is type I string theory. Worldsheet orientation reversal projects the closed type IIB superstring theory spectrum and identifies the two mixed sectors; the surviving closed fields include the graviton, dilaton, a Ramond–Ramond potential $C_2$, and one gravitino and dilatino. The open states form vector and gaugino multiplets with Chan-Paton factors; tadpole and anomaly cancellation select $SO(32)$. This is an additional orientation projection, not part of the GSO projection itself. The heterotic string instead pairs one supersymmetric chiral side with a bosonic chiral side carrying an internal gauge sector; it does not have a Ramond choice on both sides.

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 $g_s^{2g+2}$.

A Majorana spinor equals its transform under charge conjugation, so its spinor components are not independent of their conjugates. In a real two-dimensional gamma matrix representation it can be taken real, with anticommuting Grassmann variables as components. Each target index labels a separate worldsheet Majorana fermion; it is not itself a target-space spinor.

Use worldsheet signature $(-,+)$ and the real gamma matricesTake $\bar{\psi }={\psi }^{T}C$ with $C={\gamma }^0$. Define the chirality matrix with the chosen worldsheet orientation byIt squares to one and anticommutes with each gamma matrix. Chirality is its eigenvalue, selected by the chiral projectors. Its overall sign is conventional.

Writing $\psi =({\psi }_{+},{\psi }_{-}{)}^T$, the fermion equation of motion ${\gamma }^{\mu }{\partial }_{\mu }\psi =0$ becomesHenceThe first profile travels toward increasing $\sigma$, and the second toward decreasing $\sigma$. Thus the choice of chirality orientation realizes the requested worldsheet chirality and propagation direction correspondence; reversing the orientation interchanges the labels.

For the rigid worldsheet supersymmetry variation, let $ϵ$ be a constant Grassmann-odd Majorana spinor. With the displayed conventions,The minus sign follows from $({\gamma }^{\mu }{)}^{T}C=-C{\gamma }^{\mu }$. A second identity from the Majorana Grassmann bilinear interchange is $\bar{\chi }{\gamma }^{\nu }{\gamma }^{\mu }ϵ=\overline{ϵ}{\gamma }^{\mu }{\gamma }^{\nu }\chi$. Keeping these Grassmann signs is essential.

The bosonic kinetic variation is $2i{\partial }_{\mu }{X}_a\overline{ϵ}{\partial }^{\mu }{\psi }^a$. The two fermionic kinetic variations, using the two identities above, give the full resultFor the second equality use ${\gamma }^{\nu }{\gamma }^{\mu }=2{\eta }^{\mu \nu }-{\gamma }^{\mu }{\gamma }^{\nu }$ and symmetry of the second embedding derivative. No field equation has been used. Thus the variation is an off-shell total derivative:This is the rigid worldsheet supersymmetry boundary term. It vanishes when the boundary flux vanishes, for example on a closed worldsheet with compatible fields and variations. On a spatial circle, a constant supersymmetry parameter must also respect the chosen spin structure. Periodic fermions allow this rigid transformation; antiperiodic fermions and a constant parameter would give an antiperiodic $\delta X$ and fail to preserve a periodic bosonic embedding. The local action identity remains valid, but global symmetry requires compatible boundary data. This is the spin-structure obstruction to constant worldsheet supersymmetry.

For the spectrum, assume the usual critical RNS string, physical-state constraints, a nonzero null target momentum, and the supersymmetric GSO projection. Analyze just one chiral sector. Its matter central charge is $d+d/2$; the bc system and superconformal ghosts contribute $-26+11$. Therefore $3d/2-15=0$ gives the critical dimension of the RNS superstring $d=10$. Light-cone gauge in string theory leaves eight transverse bosonic and eight transverse fermionic string oscillator directions.

In the Neveu–Schwarz sector, the normal-ordering intercept is $a_{\mathrm{NS}}=1/2$. The massless level is $N=1/2$ and has statesThe GSO projection retains these eight transverse vector polarizations while removing the tachyonic ground state. They are spacetime bosons, although the creating string oscillator is a worldsheet fermion. In covariant language transversality and the longitudinal null-state quotient leave $10-2=8$ polarizations.

In the Ramond sector, bosonic and fermionic zero-point energies cancel, so $a_{\mathrm{R}}=0$ and the ground states are massless. The eight transverse fermion zero modes obeyThis Ramond zero-mode Clifford algebra acts on a sixteen-dimensional ground-state space. Its two chirality subspaces each have dimension eight. The GSO projection keeps one, an $8_s$ or $8_c$ spinor of $\mathrm{SO}(8)$, the rotation subgroup of the massless little group. These are spacetime fermions. Equivalently a ten-dimensional Majorana-Weyl spinor has sixteen real components before the massless Dirac equation reduces the physical polarization count to eight.

Consequently the massless chiral RNS spectrum after GSO projection satisfiesThe GSO projection is essential: without it, the transverse Ramond sector would retain both eight-dimensional spinor chiralities. These are counts for one chiral sector, not the tensor-product counts of the full closed-string spectrum.

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 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.

The worldsheet ghosts form an anticommuting bc system of weights $(2,-1)$. Its double contractions give central chargeThereforeand hence