Polyakov action

The Polyakov action introduces an independent worldsheet metric $g_{\alpha \beta }$:Its metric equation makes $g_{\alpha \beta }$ conformal to the induced worldsheet metric, reducing it classically to the Nambu–Goto action.

Replacing the flat target metric in the Polyakov action by a spacetime metric $g_{\mu \nu }(X)$ gives the displayed string sigma model. The embedding fields $X^{\mu }$ map the worldsheet into target spacetime, while $h_{ab}$ is an independent worldsheet metric. Classical worldsheet diffeomorphisms and Weyl transformations permit conformal gauge; quantum consistency additionally requires vanishing sigma-model beta functions.

In conformal gauge and light-cone gauge in string theory $X^{+}=\kappa \tau$, the transverse part of the string sigma model in a spacetime metric reduces toThe omitted longitudinal term is a total time derivative. Each transverse coordinate obeys a massive wave equation. If the spatial period is $L$, the longitudinal momentum is $p^{+}=T\kappa L$ in this normalization. Keeping the gauge slope explicit prevents missing a factor of $p^{+}$ when changing conventions.

Expand every transverse closed string coordinate in a real orthonormal Fourier basis of the spatial circle. The canonical momenta are $p_{Ia}=T{\dot{q}}_{Ia}$, and spatial orthogonality reduces the field Hamiltonian to the displayed independent harmonic oscillators. Canonical quantization gives ${\omega }_a(a_{Ia}^{†}{a}_{Ia}+1/2)$ for each nonzero-frequency mode; a zero-frequency mode retains the free particle term $p_{Ia}^2/(2T)$. The formal zero-point energy requires a regulator; it is not automatically the massless flat-background intercept. Opposite travelling-wave oscillator combinations diagonalize the translation generator and satisfy closed-string level matching.

For a closed string in a string sigma model in an isotropic plane wave, insert $e^{-i{\omega }_n\tau +2\pi in\sigma /L}$ into the transverse wave equation. Periodicity requires integer $n$, giving the displayed dispersion relation. The zero spatial mode is an oscillator of frequency $|m\kappa |$; when this mass vanishes it becomes a free particle mode and needs the position-plus-linear-time expansion.

Varying the independent metric in the Polyakov action sets the worldsheet stress tensor to zero. For a nondegenerate timelike embedding in two worldsheet dimensions, the metric is proportional to the induced worldsheet metric. The conformal factor cancels from the action, which becomes the Nambu–Goto action. This elimination is classical; the quantum measure and worldsheet Weyl anomaly require separate treatment.

Varying the independent worldsheet metric gives $T_{ab}=T[{\partial }_{a}X\cdot {\partial }_{b}X-\frac{1}{2}{\gamma }_{ab}{\gamma }^{cd}{\partial }_{c}X\cdot {\partial }_{d}X]$. Its classical vanishing is the metric equation and produces the Virasoro constraints.

For positive lapse, eliminating momentum gives $(\dot{X}-u{X}^{\prime }{)}^2/(2e)-e{T}^2{X}^{\prime 2}/2$. This is the Polyakov action for ${\gamma }_{ab}=\left(\begin{array}{cc}{u}^2-e^2{T}^2& u\\ u& 1\end{array}\right)$, up to a Weyl transformation. Thus $e=1/T,u=0$ is conformal gauge.

The Polyakov path integral integrates over string embeddings and worldsheet metrics, divided by worldsheet diffeomorphisms and Weyl transformations. Gauge fixing introduces worldsheet ghosts and moduli integrals.

The Polyakov path integral integrates over string worldsheet metrics as well as embedding fields, modulo worldsheet diffeomorphisms and Weyl transformations. Summing surface topologies gives perturbative two-dimensional quantum gravity coupled to the coordinate matter fields. Gauge fixing retains worldsheet moduli and introduces ghosts; a conformal gauge is locally flat, not a global flat metric on every genus. Cancellation of the worldsheet Weyl anomaly makes the critical bosonic version consistent.

Worldsheet moduli parameterize inequivalent conformal structures and marked insertion points after worldsheet diffeomorphisms and Weyl invariance are fixed. At each genus, string perturbation theory integrates over the corresponding moduli space.

The worldsheet conformal Killing group is the connected residual group of conformal automorphisms of the chosen compact Riemann surface. Its infinitesimal vectors satisfy the Conformal Killing equation, or $P_{1}v=0$ for the worldsheet diffeomorphism ghost operator. Its complex dimensions are three on a sphere, one on a torus, and zero for genus at least two. The sphere group consists of Möbius transformations, while the torus's connected group is translation. Additional discrete automorphisms are handled by the moduli quotient rather than counted as continuous zero modes.

After quotienting infinitesimal worldsheet diffeomorphisms and Weyl transformations, choose a metric variation orthogonal to the gauge directions. Orthogonality to Weyl scaling makes its trace zero; the formal adjoint of the worldsheet conformal Killing operator makes its divergence zero. These are transverse-traceless variations of the two-dimensional metric. In a local conformal complex coordinate, their nonzero components satisfy ${\partial }_{\bar{z}}{t}_{zz}=0$ and its conjugate, linking them to holomorphic quadratic differentials.

The torus worldsheet path integral is invariant under the modular group acting on its complex parameter $\tau$. Integrating over the standard fundamental domain of the modular group counts each inequivalent torus once and excludes the short-proper-time ultraviolet region.

A closed-string scattering amplitude is a worldsheet correlation function of external string vertex operators, integrated over their unfixed positions and worldsheet moduli, with the necessary worldsheet ghost fields.

For a stable $n$-point closed-string scattering amplitude on a genus-$g$ surface, let $d_g$ count complex unmarked worldsheet moduli and $k_g$ count complex dimensions of the worldsheet conformal Killing group. Integrating the $n$ vertex positions and moduli, then fixing $k_g$ positions, leaves $d_g+n-k_g$ complex integrations. The punctured Riemann surface moduli dimension identifies this as $3g-3+n$. Faddeev-Popov ghost fields and antighost insertions implement the removed zero modes and the surviving modulus measure; they do not add independently integrated worldsheet coordinates.

The tree-level Polyakov path integral for closed-string tachyons reduces in conformal gauge to their vertex-position integral on the Riemann sphere. Three positions are fixed by the residual Möbius group; the other $N-3$ complex positions are integrated. The embedding-field Gaussian integral produces momentum conservation and the Koba-Nielsen factor.

At genus zero the residual Möbius transformation group fixes three marked insertion points, leaving one complex position to integrate. The corresponding worldsheet ghost fields give $|z_{ij}{z}_{jk}{z}_{ki}{|}^2$ for the three fixed points. For on-shell weight-$(1,1)$ string vertex operators, these ghost factors cancel the coordinate dependence of the fixed-point normalization, including the limit at infinity. The remaining Koba-Nielsen factor for four tachyons is the integrand of the complex beta integral, producing the Virasoro–Shapiro amplitude.

A single crossing-symmetric scattering amplitude has equivalent pole expansions in different exchange channels. Its scattering-amplitude factorization reproduces the string spectrum and couplings in each channel without adding the channel expansions again.

A worldsheet Green function inverts the scalar kinetic operator after its zero modes have been removed. In a locally flat complex coordinate it satisfies $-(\pi {\alpha }^{\prime }{)}^{-1}\partial \bar{\partial }G(z,w)={\delta }^{(2)}(z-w)$ and is $G(z,w)=-({\alpha }^{\prime }/2)\log |z-w{|}^2$ up to harmonic and constant terms.

A string vertex operator inserts an asymptotic string state at a point of the worldsheet. Its conformal weights encode the physical-state and mass-shell conditions.

These insertions are described by the state-field correspondence of a vertex operator algebra; an individual insertion is not the whole algebra.

An integrated closed-string vertex is inserted as $\int d^{2}zV(z,\bar{z})$. In conformal gauge its matter operator must have conformal weights $(1,1)$ so that the insertion is invariant under worldsheet coordinate changes.

At the first massless closed-string level, the matter vertex has the form ${ϵ}_{\mu \nu }:\partial X^{\mu }\bar{\partial }{X}^{\nu }{e}^{ik\cdot X}:$, with null momentum and a polarization tensor transverse in both indices. Its symmetric trace-free, antisymmetric and trace sectors describe the graviton, Kalb–Ramond field and dilaton.

A closed-string polarization tensor specifies the target-spacetime tensor state multiplying $\partial X^{\mu }\bar{\partial }{X}^{\nu }{e}^{ik\cdot X}$. Its decomposition into symmetric trace-free, antisymmetric and trace parts separates the graviton, Kalb–Ramond field and dilaton polarizations.

Changing a massless closed-string polarization by a momentum-longitudinal tensor changes the integrated vertex by a worldsheet total derivative, equivalently by a BRST-exact state. The corresponding target-space transformations are linearized diffeomorphisms and two-form gauge transformations.

A string mass-shell condition relates the target momentum of a vertex operator to its oscillator level. Together with polarization constraints, it makes the matter vertex have the conformal weights required for an integrated insertion.

The bosonic closed-string tachyon vertex is the normal-ordered exponential $:e^{ip\cdot X}:$ integrated over its worldsheet position. Pairwise contractions of such exponentials generate the Koba-Nielsen factor.

A worldsheet diffeomorphism is a smooth reparameterization of the worldsheet coordinates under which $X^{\mu }$ is a scalar and $g_{\alpha \beta }$ is a rank-two tensor.

On a future-directed patch, choose target time as worldsheet time, $X^0=t$. The momentum equation then gives $P^0=1/e$ and $X^{\prime 0}=0$. Freedom to vary target time at an open-string endpoint sets its shift to zero, simplifying the endpoint momentum-flux condition.

Static gauge identifies selected target-space coordinates with worldvolume coordinates, such as $X^0=\tau$ for a string.

For a string, this local static gauge identifies two embedding coordinates with the worldsheet coordinates, $X^0=t$ and $X^1=\sigma$. The remaining coordinates describe a graph over that coordinate plane. Folded embeddings may require more than one patch.

For $X=(t,\sigma ,k(\sigma +t))$, the vertical coordinate velocity is $k$, but its normal projection has magnitude $|k|/\sqrt{1+k^2}<1$. Tangential motion merely changes string labels. The induced worldsheet metric has determinant $-1$ even when constant-label curves become spacelike for $|k|>1$.

With unit tension and $X^0=t$, $X^1=\sigma$, the Virasoro constraints give $P_1=-\Pi {\Phi }^{\prime }$ and $P_0=-\sqrt{(1+{\Phi }^{\prime 2})(1+{\Pi }^2)}$. The reduced energy density is $\mathcal{H}=-P_0$. On the chiral branch $\Pi ={\Phi }^{\prime }$, the Hamiltonian field equation becomes $\dot{\Phi }={\Phi }^{\prime }$.

A Weyl transformation locally rescales a metric tensor, $g\mapsto e^{2\omega }g$, while retaining the same points and coordinates. Applied to a worldsheet metric, it leaves the embedding $X^{\mu }$ unchanged, and the classical Polyakov action is invariant under these Weyl transformations. Applied to a spacetime metric, a Weyl rescaling can relate different frame conventions, such as a string metric and an Einstein metric; the accompanying field normalizations must be transformed as well. It is not itself a coordinate transformation.

Conformal gauge uses worldsheet diffeomorphisms and Weyl transformations to set $g_{\alpha \beta }=e^{2\varphi }{\eta }_{\alpha \beta }$ locally.

After conformal gauge fixing, independent reparameterizations ${\sigma }^{+}\mapsto f({\sigma }^{+})$ and ${\sigma }^{-}\mapsto \tilde{f}({\sigma }^{-})$ remain, accompanied by a compensating Weyl transformation.

Light-cone gauge uses residual conformal transformations to make the target-space coordinate $X^{+}$ linear in worldsheet time. The Virasoro constraints then determine $X^{-}$ from the transverse fields.

This is the string-worldsheet realization of light cone gauge.

On a patch with ${\alpha }_0^{+}\ne 0$, use worldsheet diffeomorphisms and worldsheet supersymmetry to fix ${\alpha }_{n\ne 0}^{+}=0$ and ${\psi }_r^{+}=0$. The Virasoro constraints and superconformal current constraints then determine the longitudinal modes by the displayed transverse bilinears. In the Ramond sector, reduction of the fermionic zero modes includes the zero-mode Dirac constraint; fixing only nonzero modes does not complete the reduction. Eight transverse bosons and eight transverse fermions remain in the critical ten-dimensional RNS string.

Fixing nonzero longitudinal oscillators leaves the center-of-mass symplectic term, transverse oscillator terms, and a zero-mode multiplier imposing the mass-shell condition. On a patch with $p^{+}\ne 0$, further choosing $x^{+}$ as time solves that constraint for $p^{-}=[p_{\perp }^2+(N-a)/{\alpha }^{\prime }]/(2p^{+})$. This distinguishes partial oscillator gauge fixing from the subsequent time choice.

The worldsheet metric equation in conformal gauge sets the stress tensor to zero, $T_{++}=T_{--}=0$. In oscillator language these are the Virasoro constraints.

Critical closed bosonic strings have independent left- and right-moving Virasoro constraints. Positive modes in both sectors annihilate physical kets, while both zero-mode conditions impose the mass shell and closed-string level matching. Quotient by physical null string states, equivalently use the appropriate BRST cohomology. With $L_0={\alpha }^{\prime }{k}^2/4+N$, the result is $N=\tilde{N}$ and $M^2=4(N-1)/{\alpha }^{\prime }$. Negative matter modes are adjoints rather than additional ket-annihilation constraints.

The covariant physical-state conditions impose positive-mode Virasoro constraints on kets and the adjoint conditions on bras, with intercept $a$ in the zero mode. They are a string version of Gupta-Bleuler quantization. Imposing every positive and negative matter mode on the same state would conflict with the Virasoro central extension. In critical bosonic string theory, $a=1$ and the matter central charge is $26$, canceled by the reparameterization ghosts.

With the oscillator symplectic form of an open string, the quadratic generators $L_n={\sum }_k{\alpha }_k\cdot {\alpha }_{n-k}/2$ obey $\{{\alpha }_k,L_n\}=-ik{\alpha }_{k+n}$. Their Poisson brackets close without a central term, so they are first-class constraints. Quantization requires normal ordering and can produce the Virasoro central extension.

The string nonlinear sigma model couples the worldsheet embedding fields to target-space backgrounds such as a metric, a Kalb–Ramond field and a dilaton. Quantum Weyl invariance requires its beta functions to vanish.

The leading massless-sector spacetime action is proportional to $\int \sqrt{-G}{e}^{-2\Phi }[R+4(\nabla \Phi {)}^2-H^2/12]$, with $H=dB$. The backgrounds are the string-frame metric, dilaton, and Kalb–Ramond field. Massive string exchange generates higher-derivative corrections ordered by ${\alpha }^{\prime }$. This perturbative bosonic massless-sector truncation does not remove the tachyon instability.

The string-frame metric is the target metric appearing directly in the kinetic term of the string nonlinear sigma model. With conventional Euclidean dilaton $\phi$, its leading spacetime gravitational action contains $\sqrt{-g^{(s)}}{e}^{-2\phi }(R+4|\nabla \phi {|}^2)$. The dilaton-dependent Ricci coefficient distinguishes it from the Einstein-frame metric. Frame changes are local field redefinitions where the conformal factor is finite and nonzero; the Lorentzian dilaton coupling sign convention fixes which named field is $\phi$.

For $D>2$, rescale the string-frame metric by $g^{(E)}=e^{-4\phi /(D-2)}{g}^{(s)}$. The Weyl transformation of the Ricci scalar makes the coefficient of the Einstein-frame Ricci term independent of the dilaton. Indeed $g^{(s)}=e^{4\phi /(D-2)}{g}^{(E)}$ contributes factors $e^{2D\phi /(D-2)}$ from the volume, $e^{-2\phi }$ from the original coupling and $e^{-4\phi /(D-2)}$ from the Ricci term; their product is one. Derivative terms supply the transformed dilaton kinetic term. The two-dimensional case is excluded from this particular rescaling.

Expand the string embedding map along target geodesics as $X={\exp }_{\bar{X}}Y$. The quadratic Euclidean fluctuation action has a covariant derivative term and a curvature term $-R_{acbd}{Y}^a{Y}^b\partial {\bar{X}}^c\partial {\bar{X}}^d$. The logarithmically divergent coincident contraction is proportional to ${\alpha }^{\prime }{g}^{ab}$; its curvature contraction is the Ricci tensor. A compensating metric counterterm gives ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}+O({\alpha }^{\prime 2})$. This is why the target gravitational equation arises from quantum worldsheet Weyl anomaly cancellation, rather than from the classical worldsheet metric variation. Infrared regulation and removal of equation-of-motion terms by field redefinitions specify the usual one-loop scheme.

The Kalb–Ramond field is an antisymmetric two-form background coupled to the oriented worldsheet area element. Its gauge-invariant field strength is the three-form $H=dB$.

When ${\partial }_{-}$ is invertible, choose $B_{-m}=0$. Residual gauge parameters are exact one-forms and act trivially on $B$. The remaining independent modes are antisymmetric transverse components $B_{IJ}$, satisfying $\square B_{IJ}=0$; longitudinal components obey $B_{+I}={\partial }_{-}^{-1}{\partial }^J{B}_{IJ}$. There are $(D-2)(D-3)/2$ physical polarizations.

The three-form field strength of the Kalb–Ramond field is $H=dB$. Its components are $H_{mnp}={\partial }_m{B}_{np}+{\partial }_n{B}_{pm}+{\partial }_p{B}_{mn}$. In flat spacetime the free equation is ${\partial }^m{H}_{mnp}=0$.

The H-field is the three-form field strength of a Kalb–Ramond field. It obeys the Bianchi identity $dH=0$ and contributes quadratically to the leading string background equations.

A sigma-model beta function measures the failure of a worldsheet background coupling to remain scale invariant. Vanishing metric, B-field and dilaton beta functions gives the spacetime field equations order by order in ${\alpha }^{\prime }$.

For the Euclidean kinetic normalization $(4\pi {\alpha }^{\prime }{)}^{-1}\int g_{ab}(X)\partial X^a\partial X^b$, vanishing two-form and constant dilaton, the leading beta function is ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}$. The covariant Gaussian fluctuation operator contains a curvature vertex whose one-loop trace contracts to the Ricci tensor. Its logarithmic divergence renormalizes the metric. The scale convention is $\beta =\mu dg/d\mu$; Weyl consistency requires vanishing of this beta function as well as the other anomaly coefficients.

For the conventional positive Euclidean dilaton curvature coupling, with no antisymmetric background field, the leading scalar Weyl coefficient is the displayed expression. The matter/ghost central charge gives $(D-26)/6$. In a geodesic background field expansion of a string sigma model, contracting $\frac{1}{2}{\nabla }_a{\nabla }_b\phi Y^a{Y}^b$ in the curvature coupling produces the counterterm $\delta \phi =-{\alpha }^{\prime }\square \phi /(2\epsilon )$ and hence the Laplacian term. The full embedding equation ${\square }_h{X}^a+{\Gamma }^a{}_{bc}\partial X^b\partial X^c=({\alpha }^{\prime }/2)R^{(2)}{\nabla }^a\phi$ supplies the gradient-square improvement in the dilaton Weyl variation. Setting this coefficient to zero fixes the constant in the dilaton equation from contracted Bianchi identity; the metric equation alone does not. The Lorentzian dilaton coupling sign convention determines the named-field translation.

The noncritical coefficient is also given in Tong's string theory notes, Section 7.4.4.

For the conventional Euclidean dilaton coupling $+(4\pi {)}^{-1}\int \sqrt{h}R\phi$ and no antisymmetric background, the metric worldsheet Weyl anomaly vanishes at first order in ${\alpha }^{\prime }$ when $R_{ab}+2{\nabla }_a{\nabla }_b\phi =0$. The one-loop Ricci term follows from the background field expansion of a string sigma model. Under a Weyl transformation, the dilaton coupling varies as $-(2\pi {)}^{-1}\int \sqrt{h}\omega {\square }_h\phi$; the embedding equations replace this by the target Hessian contribution. Full Weyl invariance also requires the dilaton curvature coefficient, including the central-charge deficit, to vanish. See the Lorentzian dilaton coupling sign convention when translating an action with a specified Lorentzian sign.

The conventional coefficient agrees with the background equations in Tong's string theory notes, Section 7.2.3.

Diverging $R_{ab}+2{\nabla }_a{\nabla }_b\phi =0$ and using the contracted Bianchi identity and Ricci identity gives ${\nabla }_b(R+4{\square }_g\phi -4|\nabla \phi {|}^2)=0$. The trace equation $R+2{\square }_g\phi =0$ then gives ${\square }_g\phi -2|\nabla \phi {|}^2=C$, with $C$ constant on each connected component. Zero does not follow from the tensor equation alone. The additional dilaton anomaly coefficient fixes $C=(D-26)/(3{\alpha }^{\prime })$ at leading order for the bosonic string without an antisymmetric background. In critical dimension it is zero; noncritical backgrounds have a central-charge-deficit term.

In flat target space, $\phi =q_a{X}^a$ has zero Hessian and therefore solves the leading metric-dilaton Weyl condition for every constant covector $q$. But $\square \phi -2|\nabla \phi {|}^2=-2q^2$, which need not vanish. This demonstrates the integration constant left by the contracted Bianchi identity. Full quantum Weyl invariance fixes the allowed $q^2$ through the central charge, as in a linear dilaton conformal field theory; the metric equation alone does not.

For the dilaton equation from contracted Bianchi identity, put $F=e^{-2\phi }$. The chain rule gives ${\square }_{g}F=F(-2{\square }_g\phi +4|\nabla \phi {|}^2)=-2CF$. Thus the exponential obeys a linear Klein-Gordon equation on the given background, although the original dilaton equation is nonlinear and the string-frame metric remains coupled to the dilaton. With the opposite named-field convention $\phi =-\Phi$, this variable is $F=e^{2\Phi }$.

A worldsheet Weyl anomaly is the quantum failure of the gauge-fixed string path integral to remain invariant under a Weyl transformation. Its local coefficients are the sigma-model beta functions, while its constant part is controlled by the total central charge.

To first order in ${\alpha }^{\prime }$, the conventional Euclidean dilaton coupling gives ${\overline{\beta }}_{\mu \nu }^G={\alpha }^{\prime }[R_{\mu \nu }-H_{\mu \rho \sigma }{H}_{\nu }{}^{\rho \sigma }/4+2{\nabla }_{\mu }{\nabla }_{\nu }\Phi ]$ and ${\overline{\beta }}_{\mu \nu }^B={\alpha }^{\prime }[-{\nabla }^{\rho }{H}_{\rho \mu \nu }/2+({\nabla }^{\rho }\Phi )H_{\rho \mu \nu }]$. The scalar coefficient is $(d-26)/6+{\alpha }^{\prime }[(\nabla \Phi {)}^2-{\nabla }^2\Phi /2-H^2/24]$. Here $H$ is the Kalb-Ramond field strength. Vanishing of these coefficients, including higher-order corrections, is the background-field condition for quantum Weyl transformations to remain a gauge symmetry. The scalar coefficient includes the central charge deficit and is not implied by the metric equation alone.

Gauge fixing worldsheet diffeomorphism and Weyl symmetry introduces an anticommuting vector ghost $c^{\alpha }$ and a symmetric traceless antighost $b_{\alpha \beta }$. In complex coordinates they form holomorphic and antiholomorphic $bc$ systems.

On the Riemann sphere, three holomorphic reparameterization ghost zero modes correspond to the three infinitesimal Möbius transformations. Three $c$ insertions saturate their Grassmann integral. In a basis $1,z,z^2$, their integral is the Vandermonde determinant, giving $z_{12}{z}_{13}{z}_{23}$ up to an ordering-dependent sign. Including the antiholomorphic factor gives $|z_{12}{z}_{13}{z}_{23}{|}^2$, the Jacobian in sphere gauge fixing for four string vertices.

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.

With the ghost oscillator vacuum, one can define $L_0^{\mathrm{gh}}={\sum }_{k>0}k(b_{-k}{c}_k+c_{-k}{b}_k)$ with vacuum eigenvalue zero. The corresponding Virasoro algebra contains a linear shift: $[L_m^{\mathrm{gh}},L_n^{\mathrm{gh}}]=(m-n)(L_{m+n}^{\mathrm{gh}}-{\delta }_{m+n,0})-\frac{26}{12}(m^3-m){\delta }_{m+n,0}$. Shifting the zero mode by minus one removes the linear term but changes the stated vacuum eigenvalue. A BRST charge must use a consistent choice throughout.

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.

With Hermitian $b_0,c_0$ and $\{b_0,c_0\}=1$, a ket annihilated by $b_0$ has zero bare full-ghost norm. Nonzero amplitudes need FP ghost zero-mode saturation. Low-level Virasoro descendant calculations can instead normalize the highest-weight coefficient and compute $L_n{L}_{-n}|\Omega ⟩$ algebraically. Such normalized contravariant coefficients are not positive-definite Hilbert norms. The pairing convention must be stated before using FP ghost descendants in a norm argument.

The BRST operator is the Grassmann-odd conserved charge that combines a gauge transformation with its ghost field. Quantum gauge consistency requires $Q_B^2=0$, and physical states are its cohomology classes.

If $L_m=\{b_m,Q\}$, nilpotence gives $[L_m,Q]=0$. The graded Jacobi identity and $[b_m,L_n]=(m-n)b_{m+n}$ then imply the displayed algebra. Thus a quantum central extension in the total generators obstructs BRST nilpotence.

For a matter state tensored with the ghost oscillator vacuum, the BRST charge produces the independent ghost states $c_0|0⟩$ and $c_{-m}|0⟩$ with coefficients $(L_0-1)|\Psi ⟩$ and $L_m|\Psi ⟩$. Thus BRST closure is equivalent to $(L_0-1)|\Psi ⟩=0$ and $L_{m>0}|\Psi ⟩=0$ in this sector. It reproduces old covariant string quantization and the open bosonic string mass spectrum.

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.

For the traceless diffeomorphism operatorthe ghost action is

In conformal gauge, let the odd left-acting BRST differential act by $sX=c^{+}{\partial }_{+}X+c^{-}{\partial }_{-}X$ and $s{c}^{\pm }=c^{\pm }{\partial }_{\pm }{c}^{\pm }$. The graded Leibniz rule gives the displayed expression for $s^{2}X$. It vanishes using the ghost equations of motion ${\partial }_{-}{c}^{+}={\partial }_{+}{c}^{-}=0$, but it is not zero for arbitrary off-shell fields. For independent Grassmann variables $\xi ,\rho$, take $c^{+}=\xi$, $c^{-}=\rho {\sigma }^{+}$ and $X={\sigma }^{-}$; then $s^{2}X=-\xi \rho \ne 0$. Each ghost itself satisfies $s^2{c}^{\pm }=0$ algebraically. With the eliminated-antighost transformation $s{b}_{\pm \pm }=i{\theta }_{\pm \pm }/{\alpha }^{\prime }$ and $s{\theta }_{\pm \pm }=0$ on the solution space, the antighosts are nilpotent too. This on-shell closure is compatible with an off-shell total-derivative variation of the gauge-fixed action. A full off-shell BRST quantization uses the complete gauge transformation and, where required, auxiliary fields rather than silently dropping mixed ghost derivatives.

In two dimensions the metric variation splits into a scalar trace and the trace-free diffeomorphism operator $(P_{1}v{)}_{ab}={\nabla }_a{v}_b+{\nabla }_b{v}_a-g_{ab}\nabla \cdot v$. Integrating the Weyl parameter in the inverse Faddeev-Popov determinant removes the trace delta function. The remaining determinant is exponentiated by an anticommuting vector ghost and symmetric trace-free antighost, yielding the two chiral $b\bar{\partial }c$ terms. Conformal Killing vectors and moduli are zero modes treated separately from this local determinant.

The trace-free metric variation is $(P_{1}c{)}_{ab}={\nabla }_a{c}_b+{\nabla }_b{c}_a-{\gamma }_{ab}{\nabla }_d{c}^d$. A symmetric trace-free antighost exponentiates the Faddeev-Popov determinant. In light-cone worldsheet coordinates its two chiral kinetic terms are $b_{++}{\partial }_{-}{c}^{+}$ and $b_{--}{\partial }_{+}{c}^{-}$.

Use the half-normalized operator $(P_{1}v{)}_{ab}=({\nabla }_a{v}_b+{\nabla }_b{v}_a-h_{ab}{\nabla }_c{v}^c)/2$. On a closed worldsheet, a symmetric trace-free tensor obeysThis integration by parts identifies the displayed formal adjoint operator. A convention with twice $P_1$ has twice this adjoint, with identical kernels. The worldsheet conformal Killing group corresponds to the kernel of $P_1$, while infinitesimal worldsheet moduli correspond to the trace-free kernel of its adjoint.