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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.

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 $F=e^{2\Phi }$.

Retain only fields independent of the circle coordinate $y$, with coordinate period $2\pi r_0$. Use the string-frame metric $g_{\mu \nu }$ and the circle reduction of eleven-dimensional supergravity ansatzThe minus sign in the fibre one-form is a convention chosen to give the usual ${\tilde{F}}_4=d{C}_3-C_1\wedge H_3$ below; changing the sign of $C_1$ consistently changes no physics. The metric supplies the ten-dimensional graviton, dilaton and RR one-form, while the eleven-dimensional three-form supplies the Kalb–Ramond field and RR three-form. DefineThen $F_4^{(11)}={\tilde{F}}_4+H_3\wedge \eta$. In particular $d{\tilde{F}}_4=-F_2\wedge H_3$, the reduced Bianchi identity. The field strength, not bare $d{C}_3$ alone, is the gauge-covariant horizontal component.

For the gravitational term, the block determinant givesA useful two-step curvature calculation first removes the overall factor $e^{-2\Phi /3}$. The intermediate metric $d{\bar{s}}^2=d{s}_s^2+e^{2\Phi }{\eta }^2$ hasThe eleven-dimensional conformal-curvature identity then givesCombining the terms yieldsIntegration by parts gives $\int \sqrt{-g}{e}^{-2\Phi }{\nabla }^2\Phi =2\int \sqrt{-g}{e}^{-2\Phi }(\nabla \Phi {)}^2$ modulo boundary terms. Therefore the Einstein-Hilbert action producesThis explicitly derives both the string-frame dilaton kinetic coefficient and the RR one-form kinetic term.

Use the normalized form norm $|F_k{|}^2=F_{{\mu }_1\cdots {\mu }_k}{F}^{{\mu }_1\cdots {\mu }_k}/k!$, interpreting the printed four-form square in this convention. The horizontal and fibre pieces are orthogonal in the $\eta$ basis. Four horizontal inverse metrics contribute $e^{8\Phi /3}$ to $|{\tilde{F}}_4{|}_{11}^2$. Three horizontal inverse metrics and one fibre inverse metric contribute $e^{2\Phi /3}$ to $|H_3\wedge \eta {|}_{11}^2$. ThusThe NS-NS three-form has the same $e^{-2\Phi }$ prefactor as gravity; the RR kinetic terms do not have it in the string frame.

Finally reduce the topological term using the coordinate decomposition, which makes its signs transparent. The part of $A_3\wedge F_4^{(11)}\wedge F_4^{(11)}$ containing exactly one $dy$ isThe purely horizontal eleven-form vanishes on the ten-dimensional base. Sinceintegration over the base turns the bracket into $3B_2\wedge G_4\wedge G_4$, modulo the displayed exact term. The reduction of the eleven-dimensional Chern-Simons term therefore gives $-\frac{1}{2}\int B_2\wedge G_4\wedge G_4$.

Absorbing the coordinate circumference into ${\kappa }_{10}^2={\kappa }_{11}^2/(2\pi r_0)$, the complete bosonic massless type IIA supergravity action in these conventions isDropping boundary terms assumes appropriate boundary conditions; nontrivial flux bundles may require patchwise potentials. Higher circle modes and higher-derivative corrections are excluded by this low-energy zero-mode reduction. No Romans-mass term arises from the ordinary ansatz.

The M-theory origin of type IIA D-branes follows by momentum, wrapping and magnetic geometry:

An M2-brane wrapped around the circle gives the fundamental string, and an unwrapped M5-brane gives the NS5-brane. These are NS objects rather than D-branes. In physical string/Planck units, the M-theory circle duality relations areThey involve the physical asymptotic radius, distinguished from the arbitrary coordinate period used in the reduction ansatz. For example, $m_{D0}=1/(g_s{\ell }_s)=1/R_{11}$ is one momentum quantum. With $T_{M2}=1/[(2\pi {)}^2{\ell }_p^3]$ and $T_{M5}=1/[(2\pi {)}^5{\ell }_p^6]$,The D6-brane is purely gravitational in eleven dimensions, rather than an elementary six-dimensional membrane; the circle-fibration magnetic charge gives its ten-dimensional interpretation.

A D8-brane is the important qualification. It sources the zero-form flux of massive type IIA supergravity and is not obtained as an ordinary wrapped M2/M5 or Kaluza-Klein object of the undeformed circle reduction above. Its extended/generalized lifts are outside this simple dictionary. The massless-circle interpretation of the other branes is developed in Four Lectures on M-theory.

Take an oriented closed string in flat, critical bosonic string theory, with ${\alpha }^{\prime }=1/(2\pi T)$ and a mostly-plus target Minkowski metric. After continuation of the worldsheet to Euclidean signature, the Polyakov path integral sums over embeddings $X$ and worldsheet metrics, divided by worldsheet diffeomorphisms and Weyl transformations. In a flat target its kinetic action isTarget-time continuation or analytic continuation of external momenta defines the Lorentzian scattering amplitude; a naive real Euclidean Gaussian for timelike $X^0$ would not be convergent.

An external tachyon is represented by the tachyon vertex operator $V_p(z,\bar{z})=:e^{ip\cdot X}:$, with conformal weights $h=\bar{h}={\alpha }^{\prime }{p}^2/4$. The physical integrated vertex has $(h,\bar{h})=(1,1)$, so $p^2=4/{\alpha }^{\prime }$. Schematically its tree amplitude isThe Faddeev-Popov determinant from conformal gauge is represented by the worldsheet ghost fields. At tree level the worldsheet is a Riemann sphere, whose unpunctured complex structure has no moduli. Its residual conformal automorphisms are the Möbius transformations, $PSL(2,\mathbb{C})$. Fix three insertion points, accompanying their unintegrated vertices by the required $c\tilde{c}$ ghost factors. The remaining $N-3$ complex insertion positions are integrated over the sphere; equivalently one integrates all positions and divides by the residual conformal group.

The embedding fields are free, withTheir zero-mode integral gives momentum conservation, and their nonzero-mode Gaussian integral gives the Koba-Nielsen factorThis explains the sphere tachyon position integral without needing its evaluation. For four tachyons the resulting Virasoro–Shapiro amplitude displays the exchanged string spectrum directly.

Use all-incoming external momenta and introduce $u=-(p_1+p_4{)}^2/(8\pi T)$ alongside the two printed dimensionless Mandelstam variables. Since $p_a^2=8\pi T$ and ${\sum }_a{p}_a=0$, one obtains $s+t+u=-4$. The physical center-of-mass energy squared in the $s$ channel is $s_{\mathrm{phys}}=8\pi Ts=4s/{\alpha }^{\prime }$. The $t$ channel measures the analogous crossed momentum transfer, with sign determined by the mostly-plus convention. Rewriting the Gamma factors in a symmetric form givesThe Gamma function poles imply, at generic fixed values of the other invariant,The denominator Gamma factors can remove residues at special intersecting channel kinematics; the statement concerns a generic single-channel limit. An $s$-channel pole occurs when the intermediate momentum $p_1+p_2$ satisfies the mass-shell condition for a closed-string state:The pole at $s=-1$ exchanges the ground-state tachyon, the pole at $s=0$ exchanges massless states, and the positive integer poles exchange the infinite massive tower. The $t$-channel interpretation is the crossed version. Factorization means that each residue is a sum of products of couplings to intermediate physical states that couple to the chosen external particles. It need not expose every representation at that mass.

For example, the Gamma function recurrence and Gamma function residue at a nonpositive integer give the dimensionless Virasoro–Shapiro amplitude pole residueThe residue polynomial has degree $2n$, consistent with maximum spin $2n$ in the exchanged level. The amplitude also has the corresponding $u$-channel poles by crossing symmetry.

The massless fields can be treated as target backgrounds rather than separate asymptotic insertions. Write the target metric as $G_{mn}={\eta }_{mn}+h_{mn}$, introduce a Kalb–Ramond field $B_{mn}$, and a dilaton $\Phi$. In conventional Euclidean signs their string nonlinear sigma model action isHere ${ϵ}^{ab}$ is the antisymmetric tensor density. The three backgrounds correspond to the graviton, antisymmetric tensor and scalar states at closed-string level one. Expanding the vacuum functional in $h,B,\Phi$, then Fourier expanding the backgrounds, produces exactly their integrated string vertex operators. Its functional derivatives therefore generate the amplitudes with massless external strings. The connected vacuum functional organizes connected amplitudes; the spacetime effective action organizes the corresponding vertices after treating massless propagation consistently.

At momenta small compared with $1/\sqrt{{\alpha }^{\prime }}$, massive string propagators can be expanded in powers of momenta over their masses. The analytic part of the amplitudes consequently determines local higher-derivative interactions, ordered by powers of ${\alpha }^{\prime }$. Massless exchange poles are retained through propagation of the massless fields, rather than expanded into local contact terms. Up to field redefinitions, the leading massless-sector action in the string-frame metric isThe terms denoted $O({\alpha }^{\prime })$ contain additional derivatives, including curvature-squared terms in the bosonic theory. The expansion concerns the massless sector around the perturbative bosonic background; the tachyon instability remains and is not cured by omitting its field from this displayed action. Thus this is a formal perturbative effective description, not a claim of a stable bosonic vacuum.

Finally split the dilaton into a constant and its variation, $\Phi ={\Phi }_0+\varphi$. By the Gauss-Bonnet theorem, its dilaton Euler-characteristic weighting on a connected closed oriented surface follows fromDefine the string coupling by $g_s=e^{{\Phi }_0}$. A genus-$g$ path integral is weighted byThe sphere carries $g_s^{-2}$, the torus carries $g_s^0$, and each extra handle adds $g_s^2$. At higher genus one integrates over complex-structure moduli as well as insertion points, with the associated antighost insertions supplying the correct moduli measure. With canonically normalized external vertices an $N$-point genus-$g$ amplitude scales as $g_s^{2g-2+N}$.

Summing connected worldsheets of every genus of a surface yields the string-loop effective action expansionEach coefficient has its own low-energy ${\alpha }^{\prime }$ expansion. The two parameters have different roles: ${\alpha }^{\prime }$ resolves finite string size through higher derivatives, while $g_s^2$ counts additional string loops.

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.