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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.

The starting point is a physical massless closed-string stateIts symmetric traceless part is a spacetime spin-two field, the graviton; the antisymmetric part is the Kalb–Ramond field, and the scalar field part gives the dilaton. The physical-state conditions impose transverse polarizations, and longitudinal null states identifyfor the symmetric component. In position space this is the linearized diffeomorphism $\delta h_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$. Thus the massless spin-two state has the gauge redundancy required of a gravitational field, rather than an arbitrary massive tensor interaction.

Closed-string amplitudes factorize on this graviton pole. Its leading soft coupling is universal and couples to the energy-momentum tensor of the other string states. Decoupling longitudinal polarizations enforces the gravitational Ward identities. The two-derivative cubic graviton vertex extracted from the low-momentum three-point amplitude is the cubic vertex of the Einstein-Hilbert action; continuing these Ward identities requires the nonlinear diffeomorphism-invariant completion. A complementary way to derive the same dynamics is to demand quantum consistency of propagation in a curved background.

Replace the flat target metric in the worldsheet action by $G_{\mu \nu }(X)$, and allow the two-form and dilaton backgrounds. After Euclidean continuation, their couplings have the schematic normalizationThese backgrounds are couplings of a string nonlinear sigma model. Quantum Weyl invariance requires their Weyl anomaly coefficients to vanish. To first order in ${\alpha }^{\prime }$, the metric equation isThe dilaton and two-form equations accompany it. In a critical-dimensional background with $H=0$ and constant $\Phi$, the leading metric equation is $R_{\mu \nu }=0$, precisely the vacuum Einstein field equations. Matter and other massless string fields supply the corresponding stress-energy when retained. This argument also explains why arbitrary curved target metrics are not consistent string backgrounds.

The equations arise from the leading string-frame massless effective actionA constant-dilaton expansion of its curvature term directly gives the Einstein-Hilbert kinetic and self-interaction terms. For a varying dilaton, introduce the Einstein-frame metric $G_{\mu \nu }^E=e^{-4\Phi /(D-2)}{G}_{\mu \nu }$. The Weyl rescaling of the metric and the corresponding transformation of the Ricci scalar and an integration by parts giveThis displays ordinary Einstein gravity coupled to the dilaton and two-form. Gravity need not be the only low-energy field: those fields, compactification moduli and gauge fields have to be retained or stabilized as appropriate. Four-dimensional gravity additionally requires compactification from the critical target dimension, with its Newton constant depending on the compactification volume.

The approximation is controlled when external energies obey $E\sqrt{{\alpha }^{\prime }}\ll 1$ and curvatures obey ${\alpha }^{\prime }|R_{\mu \nu \rho \sigma }|\ll 1$. Massive string modes then produce suppressed higher-derivative interactions, such as higher-curvature corrections. Small string coupling $g_s=e^{{\Phi }_0}$ also suppresses higher-genus quantum corrections; dimensionally $G_D$ scales as $g_s^2{\alpha }^{\prime (D-2)/2}$. Thus Einstein gravity is the universal leading two-derivative dynamics of the massless closed-string spin-two field, with additional massless matter, string-length corrections and quantum corrections systematically controlled.

For the bosonic theory this is the massless-sector effective description in 26 dimensions; the tachyon means that it is not by itself a proof of a stable vacuum. Tachyon-free supersymmetric string theories have the same massless graviton mechanism in ten dimensions before compactification. Removing the tachyon and obtaining a realistic four-dimensional model require extra physical input beyond the emergence of the Einstein action.

The local gauge symmetries of the Polyakov action are worldsheet diffeomorphisms and Weyl transformations. Under a reparametrization ${\sigma }^{\alpha }\mapsto {\sigma }^{\prime \alpha }$, the embedding coordinates $X^{\mu }$ transform as scalars and the worldsheet metric as a covariant tensor; the contracted integrand is a scalar density. Infinitesimally, with the active convention, $\delta X=-{\xi }^{\alpha }{\partial }_{\alpha }X$ and $\delta \gamma =-{\mathcal{L}}_{\xi }\gamma$. A Weyl transformation sends ${\gamma }_{\alpha \beta }\mapsto e^{2\omega }{\gamma }_{\alpha \beta }$ and leaves $X$ fixed. In two dimensions the factors from $\sqrt{-\gamma }$ and ${\gamma }^{\alpha \beta }$ cancel. Target-space Poincaré transformations are additional global symmetries, rather than worldsheet gauge symmetries.

These gauge freedoms put the metric locally in conformal gauge, ${\gamma }_{\alpha \beta }={\eta }_{\alpha \beta }$. Its variation must still be imposed: it gives the worldsheet stress tensor constraints $T_{++}=T_{--}=0$. The field equation is the wave equation, and periodicity of a closed string gives independent left- and right-moving oscillator sets. In a normalization with ${\alpha }_0^{\mu }={\tilde{\alpha }}_0^{\mu }=\sqrt{{\alpha }^{\prime }/2}{k}^{\mu }$, their commutators and normal-ordered generators arewith the same formulas for the tilded sector and zero mixed commutators.

The closed-string physical-state Virasoro conditions in the critical flat-space theory areThey act on the oscillator Fock space, and physical spurious (null) states are quotiented out. Equivalently physical states are the suitable BRST cohomology classes. Positive modes constrain kets; their adjoints constrain bras, so one does not impose every positive and negative matter mode on the same ket. Subtracting the two zero-mode equations gives closed-string level matching, $N=\tilde{N}$, and adding their mass-shell content gives $M^2=4(N-1)/{\alpha }^{\prime }$. Quantum consistency fixes the intercept to one and, for just these flat coordinate fields, $d=26$: the matter central charge $d$ cancels the reparametrization ghost central charge $-26$. Other dimensions require additional worldsheet degrees of freedom or a noncritical completion.

For the residual symmetry, take ${\sigma }^{\pm }=\tau \pm \sigma$. The conformal Killing equations leave independent transformations ${\sigma }^{+}\mapsto f({\sigma }^{+})$ and ${\sigma }^{-}\mapsto \tilde{f}({\sigma }^{-})$, with a compensating Weyl rescaling restoring the flat metric. On the Euclidean plane the corresponding holomorphic vector fields are ${\ell }_n=-z^{n+1}{\partial }_z$ and their antiholomorphic partners. They satisfy the Witt algebra $[{\ell }_m,{\ell }_n]=(m-n){\ell }_{m+n}$. The conserved contour generators are the Virasoro modes $L_n=(2\pi i{)}^{-1}\oint z^{n+1}T(z)dz$. Their quantum Virasoro algebra iswith a second independent copy for the right movers. These are the generators of the residual conformal transformations, including their quantum central extension.

The two-dimensional quantum-gravity interpretation of strings follows from what is integrated, not from an analogy: at genus $h$, the Polyakov path integral integrates over both $X$ and the metric, divided by diffeomorphism and Weyl gauge volume. String perturbation theory sums these surface topologies, weighted by $g_s^{2h-2}$ before external-state coupling factors. Gauge fixing leaves worldsheet ghost determinants and integrals over worldsheet moduli. Thus it is two-dimensional quantum gravity coupled to the $d$ coordinate matter fields. A flat gauge is local; general higher-genus surfaces retain conformal-structure moduli and need not admit a globally flat representative.

Target-space gravity is a different conclusion. At $N=\tilde{N}=1$, the state ${ϵ}_{\mu \nu }{\alpha }_{-1}^{\mu }{\tilde{\alpha }}_{-1}^{\nu }|k⟩$ is massless, transverse in both indices, and has longitudinal gauge identifications. Its symmetric traceless transverse polarization is a graviton; the antisymmetric and scalar polarizations give the Kalb–Ramond field and dilaton. The graviton gauge identification is precisely the linearized metric diffeomorphism. Moreover the leading string-frame massless effective action contains $\int \sqrt{-G}{e}^{-2\Phi }R$, and worldsheet Weyl invariance with constant dilaton and zero two-form requires $R_{\mu \nu }=0$ at leading order in ${\alpha }^{\prime }$. Thus both the propagating massless spin-two state and its low-energy interactions reproduce gravity. Closed strings contain the graviton and yield Einstein gravity in their low-energy massless sector, together with other fields and higher-derivative corrections; the bosonic tachyon instability remains.

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.

Choose a Euclidean string worldsheet and the usual normalization of the Polyakov action. For a closed oriented worldsheet $\Sigma$, the string nonlinear sigma model isHere ${\epsilon }^{12}=1$ is the alternating density, not the tensor with components $1/\sqrt{g}$. The imaginary coefficient in the Kalb–Ramond field term is the Euclidean continuation; in Lorentzian signature this term is real. $G$ is the spacetime metric and $\Phi$ the dilaton.

All three terms have worldsheet diffeomorphism invariance, and are covariant under spacetime coordinate transformations when $X$, $G$, $B$ and $\Phi$ are transformed together. Under a Weyl transformation $g_{ab}\mapsto e^{2\omega }{g}_{ab}$, $\sqrt{g}{g}^{ab}$ is invariant in two dimensions, so the metric coupling has classical Weyl invariance. The antisymmetric coupling is independent of the worldsheet metric, so it too has classical Weyl invariance.

The curvature coupling requires care. An infinitesimal Weyl transformation gives ${\delta }_{\omega }(\sqrt{g}{R}^{(2)})=-2\sqrt{g}{\Delta }_g\omega$. Integration by parts on the closed worldsheet therefore gives the Weyl variation of a nonconstant dilaton coupling:A constant dilaton coupling is topological and Weyl invariant; a general varying dilaton coupling is not separately classically Weyl invariant. Its classical variation participates in canceling the quantum worldsheet Weyl anomaly, producing the improved background equations below.

To relate these background couplings to the flat-space bosonic string mass spectrum, expand $G_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }(X)$ and express $h$ and $B$ as superpositions of plane waves. The perturbation of the action is a sum of integrated string vertex operators of the formThis is the vertex of ${\alpha }_{-1}^{\mu }{\tilde{\alpha }}_{-1}^{\nu }|p⟩$ in the massless first closed-string level. Its conformal weights are $(1+{\alpha }^{\prime }{p}^2/4,1+{\alpha }^{\prime }{p}^2/4)$, so marginality gives $p^2=0$. The primary operator conditions give transversality, and the string-state gauge redundancy removes longitudinal components. The symmetric trace-free polarization tensor is the graviton and perturbs $G$; its antisymmetric part is the Kalb–Ramond field and perturbs $B$. The scalar physical polarization is the dilaton. At the level of background fields its vertex involves the metric trace and the curvature improvement, so an arbitrary off-shell metric trace should not simply be identified with the physical dilaton without this distinction.

The spacetime gauge invariance of the antisymmetric coupling isIndeed its action is $S_B=i(2\pi {\alpha }^{\prime }{)}^{-1}{\int }_{\Sigma }{X}^{\ast }B$. Its variation isfor a closed worldsheet, by Stokes theorem. In components this is the total derivative ${\partial }_a({\epsilon }^{ab}{\Lambda }_{\nu }(X){\partial }_b{X}^{\nu })$: the term containing ${\partial }_a{\partial }_b{X}^{\nu }$ vanishes by antisymmetry. Thus only the Kalb-Ramond field strength $H$ can enter the local spacetime equations. With boundaries, a compensating boundary gauge-field transformation is needed.

Separate the dilaton into a constant ${\Phi }_0$ and a varying part. The Gauss-Bonnet theorem givesfor a connected closed orientable genus-$g$ worldsheet. Its Euclidean path-integral weight containsConsequently the sum over worldsheet topologies is the string genus expansion ${\sum }_{g\ge 0}{g}_s^{2g-2}{\mathcal{A}}_g$, with each coefficient integrating over that genus's worldsheet moduli. Each extra handle costs $g_s^2$, which is the loop-counting factor of string perturbation theory. With the usual normalization of $n$ external closed-string vertices, the connected amplitude carries $g_s^{2g-2+n}$. This dilaton Euler-characteristic weighting is distinct from the expansion in ${\alpha }^{\prime }$ of a fixed-worldsheet string nonlinear sigma model.

Finally quantize that two-dimensional theory. The background fields are its couplings. The background field expansion of a string sigma model, $X=x+\xi$ in target-space normal coordinates, contains curvature vertices quadratic in the fluctuation $\xi$. Contracting their two fluctuation indices produces the Ricci tensor multiplying $\partial x^{\mu }\bar{\partial }{x}^{\nu }$. Logarithmic short-distance contractions renormalize $G$, $B$, and $\Phi$, giving functional sigma-model beta functions. Gauge independence requires vanishing of the worldsheet Weyl anomaly, including the classical improvement from the varying dilaton, not merely an arbitrary choice of renormalization scale.

For clarity, in conventional leading-order normalization the improved coefficients areHere $H^2=H_{\mu \nu \rho }{H}^{\mu \nu \rho }$. The constant term in the dilaton coefficient is the matter-plus-ghost central charge deficit. At $D=26$, setting these coefficients to zero gives the leading spacetime field equationsFor the last equation, take the trace of the first, $R-H^2/4+2{\nabla }^2\Phi =0$, and combine it with ${\overline{\beta }}^{\Phi }=0$; this verifies the coefficients and signs. These are equivalently the equations from the leading string-frame massless effective actionHigher worldsheet orders give higher-derivative ${\alpha }^{\prime }$ corrections, whereas additional handles give string-loop corrections controlled by $g_s$. Thus quantum consistency of a two-dimensional string nonlinear sigma model determines the dynamics of its spacetime background fields. These leading coefficient conventions agree with the primary string-theory lectures' background-field calculation.