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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 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 $D=26$.

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.

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.

Keeping the ordinary reparameterization bc ghost system, four embedding bosons require 44 additional real chiral fermions in each chirality to cancel the worldsheet Weyl anomaly. These are internal matter fields, not an assumed locally supersymmetric RNS construction. Their bilinear currents generate the chiral $\mathfrak{so}(44)$ current algebra, with $\left(\genfrac{}{}{0ex}{}{44}{2}\right)=946$ currents. A full closed fermionic model also requires consistent spin-structure and modular projections; the central-charge count alone does not determine these choices.

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.

For integer target dimension $2\le d\le 26$, intercept $a=1$ and ordinary nonzero momentum, the old covariant string quantization conditions leave a positive-semidefinite physical inner product. Quotienting its physical null string states removes gauge degeneracy. A DDF operator basis displays $d-2$ positive transverse oscillators; below dimension 26 the remaining physical longitudinal sector can be organized as a unitary Virasoro algebra sector with central charge $26-d$ and highest weight zero. At $d=26$ this extra sector becomes null and the quotient reduces to the 24 transverse oscillators. The result concerns negative-norm states, not the removal of the bosonic tachyon, and it does not itself cancel the worldsheet Weyl anomaly of a subcritical matter theory.

Use flat target spacetime with metric ${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,\dots ,1)$, and let $h_{ab}$ be an independent Lorentzian worldsheet metric. The Polyakov action isHere $X$ is the string embedding map and $T$ the string tension. Variation of $X$, followed by integration by parts, givesThus the embedding equation isThe boundary term vanishes by periodicity for a closed string. At an open-string boundary it requires, for example, vanishing normal derivative for a Neumann boundary condition, or $\delta X=0$ for a Dirichlet boundary condition. Mixed choices give corresponding endpoint conditions.

Define the induced worldsheet metric ${\gamma }_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$. Using $\delta \sqrt{-h}=-\frac{1}{2}\sqrt{-h}{h}_{ab}\delta h^{ab}$, metric variation givesThe second set of equations is therefore the vanishing worldsheet stress tensor:For a nondegenerate induced metric this says $h_{ab}=e^{2\omega }{\gamma }_{ab}$. Substituting into the action gives $S_P=-T\int \sqrt{-\det \gamma }{d}^2\sigma$, proving the classical equivalence of Polyakov and Nambu–Goto actions. The auxiliary metric has not added physical metric degrees of freedom.

There are three important symmetry statements. Worldsheet diffeomorphisms change the parameterization, with $X$ transforming as a scalar and $h$ as a tensor; the integral is invariant. Weyl transformations $h_{ab}\mapsto e^{2\omega (\sigma )}{h}_{ab}$ leave $\sqrt{-h}{h}^{ab}$ invariant specifically in two dimensions. These are gauge redundancies. Locally, the two diffeomorphism functions and one Weyl function fix the three independent components of $h$ to conformal gauge, $h_{ab}=e^{2\omega }{\eta }_{ab}$. Global worldsheet moduli may remain and cannot be gauged away. Separately the flat-target Poincaré group acts globally on $X$ and supplies conserved target momentum and angular momentum, rather than another worldsheet gauge constraint.

In conformal gauge the embedding equations and metric constraints becomeor equivalently ${\dot{X}}^2+X^{\prime 2}=0$ and $\dot{X}\cdot X^{\prime }=0$. The Virasoro constraints must still be imposed: fixing the metric in the action before varying it would miss them. Residual conformal changes of the two null worldsheet coordinates are generated by these constraints. Classically these statements hold in any target dimension; cancellation of the quantum worldsheet Weyl anomaly is a further condition, not part of the classical variation.

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 first massless level of a closed string contains a spin-two field, an antisymmetric two-form and a scalar: the graviton, Kalb–Ramond field and dilaton. The polarization redundancies found in Question 2 become, in position space, the linearized transformations $\delta G_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$ and $\delta B_{\mu \nu }={\partial }_{\mu }{\Lambda }_{\nu }-{\partial }_{\nu }{\Lambda }_{\mu }$. Thus the massless spin-two field already carries the gauge symmetry of linearized general relativity.

To describe slowly varying backgrounds, promote these polarizations to fields of the embedding coordinates. The Euclidean string nonlinear sigma model isHere ${ϵ}^{ab}$ is the antisymmetric coordinate density. Quantum consistency requires worldsheet Weyl anomaly cancellation, not merely the classical background equations. At first order in ${\alpha }^{\prime }$ the bosonic sigma-model Weyl anomaly coefficients givewhere $H=dB$ is the Kalb-Ramond field strength. The scalar anomaly condition, combined with the trace of the metric condition in the critical dimension, givesThese are the Euler–Lagrange equations of the leading string-frame massless effective actionFor example, variation of $B$ gives the divergence equation above. Varying $\Phi$ and integrating its kinetic variation by parts gives $R+4{\nabla }^2\Phi -4(\nabla \Phi {)}^2-H^2/12=0$. Combining this with the metric variation yields the Ricci equation. Thus the string's quantum conformal consistency reproduces target-space gravity coupled to matter. Equivalently, expanding string scattering amplitudes at small momenta gives the same action: massless exchange poles are reproduced by these fields, while analytic terms from massive exchange become local higher-derivative interactions.

The apparent dilaton-dependent coefficient of $R$ is a frame choice. Put $\varphi =\Phi -{\Phi }_0$ and, in dimension $D>2$, define the Einstein-frame metricTo see the cancellation explicitly, write $G=e^{2\omega }{g}^{(E)}$, $\omega =2\varphi /(D-2)$. ThenThe Laplacian term is a boundary term. Combining the last term with the original positive dilaton kinetic term produces $-4({\nabla }_E\varphi {)}^2/(D-2)$. The three inverse metrics in $H^2$ give its remaining exponential factor. HenceFor the uncompactified bosonic theory $D=26$, these coefficients are $-1/6$ and $e^{-\varphi /3}$. The Ricci term is now the Einstein-Hilbert action; its metric variation gives the Einstein field equations with stress-energy tensor from the scalar and two-form. The leading low-energy theory is general relativity coupled to the massless string fields.

The approximation requires characteristic momenta $E\sqrt{{\alpha }^{\prime }}\ll 1$ and small curvature in string units. Massive string modes have masses of order $1/\sqrt{{\alpha }^{\prime }}$ and can be integrated out, generating corrections with extra powers of ${\alpha }^{\prime }{E}^2$ or ${\alpha }^{\prime }R$. The string genus expansion supplies independent loop corrections controlled by $g_s=e^{{\Phi }_0}$; at weak coupling tree-level gravity dominates. Dimensional analysis gives ${\kappa }_{26}^2\propto g_s^2({\alpha }^{\prime }{)}^{12}$.

Compactification supplies lower-dimensional gravity and additional matter: metric and two-form components give gauge fields, internal geometry gives scalar moduli, and light modes of Kaluza-Klein theory must be kept if their masses lie below the chosen cutoff. Open strings add gauge fields, and supersymmetric string theories add spacetime fermions. Their particular matter spectrum depends on the compactification and projection; universal massless spin-two interactions retain the gravitational form. Finally, the bosonic massless-sector action is a formal truncation around a tachyonically unstable vacuum. It does not remove that instability. Tachyon-free superstring backgrounds provide stable applications of this low-energy gravitational interpretation.

Use a mostly-plus target metric tensor and the standard open-string mode expansion convention ${\alpha }_0^{\mu }=\sqrt{2{\alpha }^{\prime }}{p}^{\mu }$. The integer string oscillator level is $N={\sum }_{n\ge 1}{\alpha }_{-n}\cdot {\alpha }_n$, whose eigenvalues are nonnegative integers. Then $L_0=N+{\alpha }^{\prime }{p}^2$, so the string mass-shell condition givesThe ground state is a tachyon, and the level-one state is massless. The physical string state conditions in old covariant string quantization areOnly the positive-mode Virasoro constraints annihilate a ket; their negative-mode partners annihilate the corresponding bra. Imposing both signs on every ket would contradict the nonzero Virasoro central extension.

The central charge comes from normal ordering the quadratic oscillator expressions. Classically their Poisson brackets have no central term. Quantum mechanically, moving annihilation operators through creation operators produces double contractions. From the oscillator commutator one obtains $[L_m,{\alpha }_n^{\mu }]=-n{\alpha }_{m+n}^{\mu }$, which gives the noncentral part $(m-n)L_{m+n}$.

The coefficient is fixed by the free-boson Virasoro central term. On a formal zero-momentum vacuum, for $m>0$,The two contraction pairings giveHere ${\eta }_{\mu \nu }{\eta }^{\mu \nu }=D$; the time coordinate contributes one, just as each spatial coordinate does. Comparing with the Virasoro algebra provesThis counts covariant target coordinates. It is distinct from the $D-2$ physical transverse polarizations. Including the central charge of reparameterization ghosts gives $c_{\mathrm{total}}=D-26$; cancellation of the worldsheet Weyl anomaly gives the usual critical dimension of the bosonic string $D=26$.

A general level-one state isThe zero-mode constraint requires $k^2=0$. Since $[L_1,{\alpha }_{-1}^{\mu }]={\alpha }_0^{\mu }$, the remaining nontrivial positive-mode constraint is $k\cdot ϵ=0$; all $L_m$ with $m\ge 2$ act trivially at this level. Thus the polarization vector is transverse to a nonzero null momentum.

There is also a null string state $L_{-1}|0;k⟩=\sqrt{2{\alpha }^{\prime }}k\cdot {\alpha }_{-1}|0;k⟩$. It is physical when $k^2=0$ and orthogonal to every physical polarization. The null-state quotient of a string therefore identifiesChoose $k=(E,E,0,\dots ,0)$. Transversality sets ${ϵ}^1={ϵ}^0$, and the null-state identification removes this common component. The remaining $D-2$ spatial components have positive norm. Hence the physical level-one states form a massless vector with $D-2$ polarizations, or 24 at the critical dimension. The removed component is gauge redundancy. States violating the constraints, including independent timelike negative-norm polarizations, are unphysical; physical null states are removed by the quotient rather than counted as additional particles.

For an oriented Euclidean string worldsheet with metric $h_{ab}$, use the string nonlinear sigma model actionHere ${ϵ}^{ab}$ is an antisymmetric density with ${ϵ}^{12}=1$. The factor $i$ in the Euclidean Kalb–Ramond field term follows from continuing the real Lorentzian two-form coupling; reversing orientation or the convention for $B$ reverses its sign. The conventional positive Euclidean dilaton coupling is being used. With worldsheet boundaries its curvature coupling also needs ${\int }_{\partial \Sigma }\Phi kds/(2\pi )$, where $k$ is the boundary geodesic curvature, to have the proper topological normalization for constant $\Phi$.

All three terms are invariant under worldsheet diffeomorphisms. Target-coordinate changes act on $X,G,B,\Phi$ by their ordinary scalar and tensor transformation rules, making the target description covariant as well. For the two-form there is also the target gauge symmetry $B\mapsto B+d\Lambda$. Its change in the closed-worldsheet action is an integral of an exact two-form and hence zero. With boundaries it is instead $i{\int }_{\partial \Sigma }{X}^{\ast }\Lambda /(2\pi {\alpha }^{\prime })$; an endpoint gauge field with $\delta A=-\Lambda /(2\pi {\alpha }^{\prime })$ cancels it.

Under a local Weyl transformation $h_{ab}\mapsto e^{2\omega }{h}_{ab}$, the metric term is invariant because the two-dimensional factors $\sqrt{h}$ and $h^{ab}$ cancel. The $B$ term is independent of $h$ and is separately invariant. The curvature transforms asOn a closed surface the dilaton variation is consequentlyThus a general nonconstant dilaton term is not separately invariant under Weyl transformations. Its variation is the Weyl variation of a nonconstant dilaton coupling; it vanishes for constant $\Phi$ after the boundary completion where necessary.

For $\Phi ={\Phi }_0$, the Gauss-Bonnet theorem gives $S_{\Phi }={\Phi }_0\chi (\Sigma )$. In the Euclidean path integral the resulting weight isFor a connected orientable genus-$g$ surface with $b$ boundaries, $\chi =2-2g-b$, so the weight is $g_s^{2g+b-2}$. A closed-string handle therefore adds two powers of string coupling. External vertex normalizations add their own coupling factors; they are distinct from this dilaton Euler-characteristic weighting. The constant dilaton controls the string genus expansion, rather than adding a force to the classical embedding equation.

Quantization treats the target fields as position-dependent couplings of a two-dimensional interacting quantum field theory. Expanding $X$ around a background embedding gives fluctuation vertices involving target curvature and derivatives of $B$ and $\Phi$. Short-distance contractions require local counterterms, so the target fields become renormalized couplings. In particular a curvature vertex contracted once yields a Ricci tensor counterterm proportional to ${\alpha }^{\prime }{R}_{\mu \nu }$. Each additional sigma-model loop brings another power of ${\alpha }^{\prime }$ and more target derivatives. The expansion is useful when the target variation scale is large compared with $\sqrt{{\alpha }^{\prime }}$; it is not the separate expansion in powers of $g_s$.

After gauge fixing, a worldsheet Weyl anomaly would make amplitudes depend on the discarded local conformal factor. Quantum consistency therefore requires the trace anomaly to vanish, including the reparameterization worldsheet ghost field contribution. In a conventional renormalization scheme the resulting bosonic sigma-model Weyl anomaly coefficients start asThe barred quantities include the dilaton stress-energy tensor improvement; ordinary coupling sigma-model beta functions can differ by field redefinitions or target diffeomorphisms. Setting these coefficients to zero gives differential equations for the background fields order by order, including the metric gravitational equation and the antisymmetric-field equation. These conventions for the leading coefficients are described in the background-field lecture notes. Higher orders supply higher-curvature and higher-derivative corrections. This explains how spacetime dynamics follows from quantum gauge consistency of the worldsheet, not from asking only for a classical stationary worldsheet in an arbitrarily chosen background.

The apparently noninvariant dilaton term is essential outside the critical dimension because its stress-energy tensor improvement can cancel the quantum central charge mismatch. A flat constant-dilaton background has matter central charge $d$ and worldsheet ghost field central charge $-26$. If $d\ne 26$, changing the constant value of $\Phi$ only changes $g_s$ and cannot cure that mismatch.

A concrete solution is the linear dilaton conformal field theory with $G=\eta$, $B=0$, and $\Phi ={\Phi }_0+V_{\mu }{X}^{\mu }$. Its metric and antisymmetric-field anomaly coefficients vanish, while the scalar condition isOne can also see this cancellation directly from the improved holomorphic stress-energy tensor. With the positive Euclidean curvature convention above,The free-field contraction is $X^{\mu }(z)X^{\nu }(w)\sim -{\alpha }^{\prime }{\eta }^{\mu \nu }\log (z-w)/2$. Differentiating it twice at each point gives a fourth-order pole $3{\alpha }^{\prime }{V}^2/(z-w{)}^4$ from the improvement. Since the fourth-order stress-tensor pole is $c/2$, the matter central charge becomes $c=d+6{\alpha }^{\prime }{V}^2=26$. The worldsheet ghost field contribution then cancels it. This is why a classical Weyl variation need not mean inconsistency: it can cancel the quantum variation of the other terms. For $d<26$ the required gradient is spacelike; for $d>26$ it is timelike in the Lorentzian target theory. Such backgrounds are not flat constant-dilaton Minkowski vacua with full target Lorentz symmetry, nor does a central charge cancellation by itself prove stability or unitarity of every noncritical background. If the conformal factor is instead retained dynamically, its Liouville field theory supplies the corresponding noncritical description.

The string nonlinear sigma model regards the target-space metric, the Kalb–Ramond field and the dilaton as couplings of the two-dimensional worldsheet theory. Schematically its Euclidean terms areQuantization produces running background couplings and a possible worldsheet Weyl anomaly. Gauge-fixed matter bosons contribute central charge $D$, while the reparameterization bc ghost system contributes $-26$ in each chirality. In the critical flat background cancellation therefore requires $D=26$. In a curved background, vanishing improved sigma-model beta functions further imposes field equations for $G,B,\Phi$. For example, the leading metric condition isThese are spacetime conditions because the couplings are functions of the embedding coordinates, not additional worldsheet coordinates.

The leading equations arise by varying the string-frame massless effective actionThe factor $e^{-2\Phi }$ is the sphere contribution in the string genus expansion. Higher sigma-model orders generate higher-derivative terms in the ${\alpha }^{\prime }$ expansion; higher genera give string-loop corrections. Thus conformal consistency of the quantum worldsheet determines the low-energy target equations and their action. This is a massless-sector truncation: the bosonic tachyon is not removed by writing an action only for $G,B,\Phi$.

For a brane embedding $X^{\mu }(\xi )$, the D-brane worldvolume pullbacks areThe dilaton is evaluated at the embedding. When $T_p$ denotes the physical tension in a background with constant mode ${\Phi }_0$, the convention is $\tilde{\Phi }=\Phi (X)-{\Phi }_0$, so $e^{-\tilde{\Phi }}=1$ in that vacuum. Equivalently one can use $\tilde{\Phi }=\Phi (X)$ and a coefficient with the constant $e^{-{\Phi }_0}$ not already included. The physics is the same; the two conventions must not count that constant factor twice.

Several features motivate the Dirac-Born-Infeld action. Without gauge or two-form fields, it is the relativistic induced-volume action for a tensionful extended object. An open-string boundary couples to its gauge potential $A$, so a bulk transformation $B\mapsto B+d\Lambda$ is canceled by $A\mapsto A-P[\Lambda ]/(2\pi {\alpha }^{\prime })$. Thus only $P[B]+2\pi {\alpha }^{\prime }F$ can enter. For slowly varying constant field strength, the open-string disk partition function resums its dependence into the determinant; expansion gives the Maxwell kinetic term. The disk has Euler characteristic one, giving $e^{-\Phi }$ rather than the closed-string sphere's $e^{-2\Phi }$. Worldvolume covariance and the induced metric then combine these ingredients into the determinant expression.

Explicitly, in flat background and static gauge $X^a={\xi }^a$ with transverse fluctuations $X^I$, the weak-field expansion starts asThe linear term in the antisymmetric matrix $F$ vanishes; the quadratic term follows from expanding $\frac{1}{2}\mathrm{Tr}\log (1+2\pi {\alpha }^{\prime }{\eta }^{-1}F)$. Thus the DBI action contains both the scalar and gauge kinetic energies, while higher powers and derivatives are string-scale corrections.

For two coincident D-branes, an oriented open string has independent labels $r,s\in \{1,2\}$ at its ends. Its Chan-Paton factors are the matrix units $E_{rs}$. The four massless vector modes span the Lie algebra of $U(2)$: diagonal strings supply the two Abelian fields, and oppositely oriented interbrane strings supply the off-diagonal modes. Joining strings multiplies these matrices, producing the non-Abelian commutators. Transverse vector polarizations become adjoint scalar fields. This is Coincident-D-brane gauge enhancement.

The massless gauge/scalar dynamics reduce to dimensionally reduced Yang-Mills theory in a flat background with constant dilaton and vanishing pulled-back $B$, at energies and field variations well below the string scale. More precisely, derivative corrections satisfy ${\alpha }^{\prime }{E}^2\ll 1$, and one expands in small dimensionless field strengths and scalar gradients rather than retaining the full nonlinear determinant. Weak string coupling suppresses loop corrections. After subtracting the constant brane vacuum energy and using the scalar normalization fixed below, the coefficient isA bosonic brane additionally has an open-string tachyon; the displayed action governs its massless sector and is not a stable complete truncation. For supersymmetric branes the corresponding leading action also has the massless fermions.

There is a sign qualification in the printed scalar potential. With the stated $D_a\varphi ={\partial }_a\varphi +i[A_a,\varphi ]$, take $A_a,{\varphi }^I$ Hermitian and $[{\varphi }^I,{\varphi }^J{]}^2$ to mean the literal matrix product. Then dimensional reduction hasConsequently $F_{IJ}{F}^{IJ}=-[{\varphi }^I,{\varphi }^J{]}^2$. The consistent low-energy action isThis is the Hermitian commutator potential in D-brane Yang-Mills theory. Its potential energy is $-\sum \mathrm{Tr}[{\varphi }^I,{\varphi }^J{]}^2/(4g_{\mathrm{YM}}^2)\ge 0$, because a Hermitian-matrix commutator is anti-Hermitian. For a concrete check, take ${\varphi }^1=v{\sigma }_1$, ${\varphi }^2=v{\sigma }_2$ with Pauli matrices. Then $[{\varphi }^1,{\varphi }^2{]}^2=-4v^{4}I$, so the literal printed plus sign would make the static energy negative and unbounded below. The printed plus sign is consistent only if its square denotes the positive norm $[{\varphi }^I,{\varphi }^J{]}^{†}[{\varphi }^I,{\varphi }^J]$, rather than a literal square. The gauge-boson mass calculation below uses the unambiguous kinetic terms.

In an ordinary separated-brane vacuum the scalar expectation values commute and can be simultaneously diagonalized:Their eigenvalue vectors specify the transverse positions, up to the normalization to be determined. The trace encodes the center of mass; their difference encodes separation. When the two vectors coincide, every $U(2)$ transformation commutes with the vacuum. When they are distinct, only independent diagonal phases do, so the unbroken group is $U(1)\times U(1)$. The off-diagonal vectors acquire mass by the Higgs mechanism.

Write the off-diagonal entries of $A_a$ as $W_a$ and $W_a^{\ast }$ and put $\Delta v^I=v_1^I-v_2^I$. At a constant diagonal vacuum, $D_a{\varphi }^I=i[A_a,⟨{\varphi }^I⟩]$, soThe quadratic off-diagonal gauge kinetic term is $-f_{ab}^{\ast }{f}^{ab}/(2g_{\mathrm{YM}}^2)$ with $f_{ab}={\partial }_a{W}_b-{\partial }_b{W}_a$, and the scalar kinetic term supplies $-{\sum }_I(\Delta v^I{)}^2{W}_a^{\ast }{W}^a/g_{\mathrm{YM}}^2$. Comparing these with the complex massive-vector action givesHere the term W-boson means this broken-generator vector, not specifically an electroweak particle. The normalization places the gauge coupling outside the action, so no extra $g_{\mathrm{YM}}$ multiplies these $v_r^I$; rescaling to canonically normalized scalar fields moves that coupling into the usual Higgs mass formula.

The corresponding open-string vector oscillator stretched over distance $L=|X_1-X_2|$ has classical stretching mass $TL=L/(2\pi {\alpha }^{\prime })$. Its unstretched mass contribution vanishes because it is the vector level, rather than the bosonic tachyon level. Equating it with the gauge-theory mass givesThe component relation follows by rotational covariance and a common choice of transverse origin; a uniform translation adds the same constant to both brane coordinates. This is the D-brane scalar-position normalization. Together with the string tension determined in Question 1, it fixes the exact normalization of scalar eigenvalues as brane positions.

Use the free-boson operator product expansionWrite $E_p=:e^{ip\cdot X}:$ and $ϵ=z-w$. The needed Wick contractions are $\partial X^{\mu }(z)\partial X^{\nu }(w)\sim -{\alpha }^{\prime }{\eta }^{\mu \nu }/(2{ϵ}^2)$ and $\partial X^{\mu }(z)E_p(w)\sim -i{\alpha }^{\prime }{p}^{\mu }{E}_p(w)/(2ϵ)$. For $V_{\zeta }={\zeta }_{\mu }:\partial X^{\mu }{E}_p:$, the cross double contraction in the worldsheet stress tensor has a cubic pole: either of its two derivatives can contract with the derivative in $V_{\zeta }$, while the other contracts with $E_p$. Including the factor $-1/{\alpha }^{\prime }$ givesA primary operator has no cubic pole. Hence the first vertex is primary precisely whenThe antiholomorphic conformal weight is $\bar{h}={\alpha }^{\prime }{p}^2/4$. Primarity alone puts no mass-shell condition on $p$.

For $W_{\zeta }={\zeta }_{\mu \nu }:\partial X^{\mu }\bar{\partial }{X}^{\nu }{E}_p:$, the holomorphic cubic pole is $-i{\alpha }^{\prime }{p}^{\mu }{\zeta }_{\mu \nu }:\bar{\partial }{X}^{\nu }{E}_p:/(2{ϵ}^3)$. The antiholomorphic operator product expansion gives the analogous pole involving $p^{\nu }{\zeta }_{\mu \nu }$. Thus the transversality of derivative-exponential primary vertices requiresThere is no trace-free requirement for primarity. A physical integrated string vertex operator must additionally have conformal weights $(1,1)$; for this vertex that gives $p^2=0$.

Now keep the fermionic signs in the real chiral Majorana fermion calculation. Contracting the external $\psi (w)$ through $:\psi (z)\partial \psi (z):$ givesMultiplying by $-1/2$ and Taylor expanding $\psi (z)$ at $w$ producesThis proves that $\psi$ is a primary operator with weight $h_{\psi }=1/2$. To find the central charge, in $T(z)T(w)$ the two double-contraction pairings contribute, before the overall factor $1/4$,The single Wick contractions give $2T(w)/{ϵ}^2+\partial T(w)/ϵ$, soComparing the leading term with $c/(2{ϵ}^4)$ proves $c_{\psi }=1/2$, as in the free chiral Majorana fermion conformal field theory.