Low-Energy Limits: DBI, Type I, and Type II Supergravity

String theory contains infinitely many massive states, but its long-distance physics is governed by a small set of massless fields. The organizing limit is

$$E\sqrt{\alpha'}\ll 1,$$

or equivalently distances much larger than the string length $\ell_s=\sqrt{\alpha'}$. Massive string modes then have masses of order $1/\sqrt{\alpha'}$ and decouple from external states. Their virtual effects do not disappear completely; they become higher-derivative corrections such as $\alpha'^2 F^4$, $\alpha'^3R^4$, and derivative terms. The two-derivative theory that remains is the low-energy effective theory of the massless string modes.

The essential dictionary is simple:

$$\boxed{ \begin{array}{ccl} \text{massless open strings} &\longrightarrow& \text{gauge fields and gauginos},\\ \text{massless NS--NS closed strings} &\longrightarrow& G_{\mu\nu},\ B_{\mu\nu},\ \Phi,\\ \text{massless R--R closed strings} &\longrightarrow& p\text{-form gauge potentials } C_p. \end{array}}$$

The first line gives ten-dimensional super-Yang—Mills theory and, for slowly varying fields on D-branes, the Dirac—Born—Infeld action. The second and third lines give the massless bosonic fields of type IIA and type IIB supergravity. The result is not merely a plausible field-theory guess: it is fixed by string scattering amplitudes, gauge invariance, BRST invariance, spacetime supersymmetry, and worldsheet topology.

The low-energy limit keeps the massless open- and closed-string sectors. Disk amplitudes produce open-string gauge dynamics, while sphere amplitudes produce closed-string supergravity.

Throughout this page we use mostly-plus spacetime signature and write differential-form norms as

$$|F_p|^2={1\over p!}F_{\mu_1\cdots\mu_p}F^{\mu_1\cdots\mu_p}.$$

The fundamental string tension is

$$T_F={1\over 2\pi\alpha'}.$$

The string coupling is the expectation value of the dilaton,

$$g_s=e^{\Phi_0}.$$

We will distinguish the string frame, in which the worldsheet sigma model couples directly to the spacetime metric $G^{(S)}_{\mu\nu}$, from the Einstein frame, in which the gravitational action has the canonical Einstein—Hilbert form.

Why scattering amplitudes determine the action

A local field theory is determined, up to field redefinitions, by its on-shell scattering amplitudes. In string theory the amplitudes are computed first; the effective action is the local action whose tree-level Feynman diagrams reproduce the low-energy expansion of those amplitudes.

For an amplitude involving massless external states, the schematic expansion has the form

$$\mathcal A_{\rm string}(k_i,\epsilon_i) = \mathcal A_{\rm EFT}^{(2\partial)}(k_i,\epsilon_i) +\alpha'\mathcal A_{\rm EFT}^{(4\partial)} +\alpha'^2\mathcal A_{\rm EFT}^{(6\partial)}+\cdots.$$

The first term is the two-derivative action. The later terms are higher-derivative corrections. For superstrings, spacetime supersymmetry eliminates many possible lower-order corrections; for example, the first purely gravitational correction in type II theories starts at order $\alpha'^3R^4$, not at order $\alpha'R^2$.

There is a second expansion, controlled by worldsheet topology. For a closed oriented worldsheet of genus $g$ with $b$ boundaries, the Euler characteristic is

$$\chi=2-2g-b.$$

A constant dilaton contributes a factor

$$e^{-\Phi_0\chi}=g_s^{-\chi}$$

to the path integral. Therefore a sphere has factor $g_s^{-2}$, a disk has factor $g_s^{-1}$, and each closed-string handle brings $g_s^2$. This explains the characteristic dilaton factors in the effective action:

$$\text{closed-string tree level: } e^{-2\Phi}, \qquad \text{open-string disk level: } e^{-\Phi}.$$

This is one of the cleanest ways to remember why the NS—NS supergravity action begins with $e^{-2\Phi}$, while the D-brane DBI action begins with $e^{-\Phi}$.

Open strings: from disk amplitudes to ten-dimensional Yang—Mills

The massless NS open-string vertex operator describes a gauge boson. In the zero picture it is schematically

$$V_A^{(0)}(x) =\lambda^a\,\epsilon_\mu\left(\partial_tX^\mu+i\alpha' k\cdot\psi\,\psi^\mu\right)e^{ik\cdot X}(x),$$

with Chan—Paton matrix $\lambda^a$. BRST invariance gives the familiar massless conditions

$$k^2=0, \qquad k\cdot\epsilon=0, \qquad \epsilon_\mu\sim \epsilon_\mu+k_\mu.$$

Thus the string vertex already knows the gauge redundancy of a Yang—Mills field.

The color-ordered four-gluon disk amplitude has the universal form

$$\mathcal A_4^{\rm open} = \mathcal A_4^{\rm YM}\,\mathcal F_{\rm open}(s,t,u),$$

where $\mathcal A_4^{\rm YM}$ is the corresponding color-ordered Yang—Mills amplitude and $\mathcal F_{\rm open}$ is a string form factor built from gamma functions. With a common convention,

$$\mathcal F_{\rm open}(s,t) ={\Gamma(1-\alpha's)\Gamma(1-\alpha't)\over \Gamma(1-\alpha's-\alpha't)}.$$

For small $\alpha's$ and $\alpha't$,

$$\mathcal F_{\rm open}(s,t) =1-\zeta(2)\alpha'^2st+O(\alpha'^3k^6).$$

The leading term is exactly Yang—Mills theory. The subleading terms are higher-derivative string corrections. The absence of an order-$\alpha'$ correction in the maximally supersymmetric open superstring is a useful diagnostic: supersymmetry is already doing substantial work.

The two-derivative open-string effective action is ten-dimensional $N=1$ super-Yang—Mills,

$$S_{\rm SYM} ={1\over g_{\rm YM,10}^2} \int d^{10}x\ {\rm Tr}\left( -\frac14F_{\mu\nu}F^{\mu\nu} -\frac{i}{2}\bar\chi\Gamma^\mu D_\mu\chi \right),$$

where

$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu-i[A_\mu,A_\nu], \qquad D_\mu\chi=\partial_\mu\chi-i[A_\mu,\chi].$$

The fermion $\chi$ is a ten-dimensional Majorana—Weyl spinor in the adjoint representation. The on-shell bosonic degrees of freedom are the $D-2=8$ transverse polarizations of a massless vector in ten dimensions. A Majorana—Weyl spinor in ten dimensions has $16$ real components, and the massless Dirac equation removes half of them, leaving $8$ physical fermionic degrees of freedom. This is the $10$d $N=1$ vector multiplet.

Dimensional reduction of this action is extremely important. Reducing from ten dimensions to $p+1$ dimensions gives a gauge field $A_a$ along the brane and scalars $\Phi^i$ from the components $A_i$ transverse to the brane:

$$A_\mu\quad\longrightarrow\quad A_a,\quad \Phi^i, \qquad a=0,\ldots,p, \quad i=p+1,\ldots,9.$$

On a stack of D$p$-branes, the scalar matrices $\Phi^i$ describe transverse brane positions. Their commutators encode the fact that coincident branes have nonabelian geometry.

The DBI action: the open-string square root

Yang—Mills theory is the leading term in a more nonlinear open-string action. For a single D$p$-brane and slowly varying worldvolume fields, the disk-level action is the Dirac—Born—Infeld action

$$\boxed{ S_{\rm DBI} =-\mu_p\int d^{p+1}\xi\ e^{-\Phi} \sqrt{-\det\left(P[G+B]_{ab}+2\pi\alpha' F_{ab}\right)}. }$$

Here $\xi^a$ are worldvolume coordinates, $P[G+B]_{ab}$ is the pullback of the bulk tensor $G+B$ to the brane, and $F_{ab}$ is the worldvolume gauge-field strength. The constant

$$\mu_p={1\over (2\pi)^p\alpha'^{(p+1)/2}}$$

is the D$p$-brane R—R charge in a common convention. The physical tension in a background with constant dilaton is

$$\tau_p={\mu_p\over g_s}.$$

For $B=0$, constant dilaton, and small $F$, expand the determinant:

$$\sqrt{-\det(g+2\pi\alpha'F)} = \sqrt{-\det g}\left[ 1+{(2\pi\alpha')^2\over4}F_{ab}F^{ab}+O(F^4) \right].$$

The DBI action therefore contains

$$S_{\rm DBI} =-\tau_p\int d^{p+1}\xi\sqrt{-g} -{1\over4g_{\rm YM,p}^2}\int d^{p+1}\xi\sqrt{-g}\,F_{ab}F^{ab} +O(\alpha'^2F^4),$$

with

$${1\over g_{\rm YM,p}^2}=\mu_p e^{-\Phi_0}(2\pi\alpha')^2,$$

or

$$\boxed{ g_{\rm YM,p}^2=(2\pi)^{p-2}g_s\,\alpha'^{(p-3)/2} }$$

up to the normalization chosen for ${\rm Tr}(T^aT^b)$. For $p=3$ this gives a dimensionless gauge coupling, as expected for four-dimensional Yang—Mills theory.

The DBI action resums powers of the abelian field strength at disk level. The Yang—Mills term is just the first nontrivial term in the expansion.

There are two important caveats.

First, the DBI square root is exact at disk level only for a single brane with slowly varying fields, or more precisely for the class of backgrounds for which derivative corrections may be neglected. Terms involving $D_aF_{bc}$ are not captured by simply expanding the determinant.

Second, for a stack of coincident branes the fields become matrices. The nonabelian DBI action is not obtained by naively putting a trace around the abelian square root. The symmetrized-trace prescription captures part of the low-energy expansion, but additional commutator and derivative terms are required. The clean principle is this: the full nonabelian action is defined by matching open-string disk amplitudes order by order in $\alpha'$.

Closed strings: the NS—NS action in string frame

The massless NS—NS closed-string states arise from the tensor product of left- and right-moving NS oscillators. Their polarization tensor decomposes into

$$\epsilon_{\mu\nu} =\epsilon_{(\mu\nu)} +\epsilon_{[\mu\nu]} +{1\over D}\eta_{\mu\nu}\epsilon^\rho{}_{\rho}.$$

These pieces become, respectively,

$$G_{\mu\nu}, \qquad B_{\mu\nu}, \qquad \Phi.$$

The closed-string three-point amplitude of NS—NS states fixes the two-derivative string-frame action

$$\boxed{ S_{\rm NSNS}^{(S)} ={1\over 2\kappa_{10}^2} \int d^{10}x\sqrt{-G^{(S)}}\,e^{-2\Phi} \left( R^{(S)}+4\partial_\mu\Phi\partial^\mu\Phi -\frac12|H_3|^2 \right), }$$

where

$$H_3=dB_2.$$

This action also follows from demanding Weyl invariance of the interacting worldsheet sigma model. The sigma-model couplings are

$$S_\sigma={1\over4\pi\alpha'}\int d^2\sigma\sqrt{h} \left( h^{ab}G_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu +i\epsilon^{ab}B_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu \right) +{1\over4\pi}\int d^2\sigma\sqrt{h}\,\Phi(X)R^{(2)}.$$

The beta functions of this two-dimensional quantum field theory reproduce the spacetime field equations derived from $S_{\rm NSNS}^{(S)}$, up to field redefinitions. At leading order in $\alpha'$, they are schematically

$$\beta^G_{\mu\nu} =\alpha'\left( R_{\mu\nu}+2\nabla_\mu\nabla_\nu\Phi -\frac14H_{\mu\rho\sigma}H_\nu{}^{\rho\sigma} \right)+O(\alpha'^2),$$ $$\beta^B_{\mu\nu} =\alpha'\left( -\frac12\nabla^\rho H_{\rho\mu\nu} +\nabla^\rho\Phi\,H_{\rho\mu\nu} \right)+O(\alpha'^2),$$

and $\beta^\Phi=0$ gives the dilaton equation. Thus the condition that the worldsheet theory remain conformal is the spacetime equation of motion.

Einstein frame

The string-frame action is natural from the worldsheet, but the Einstein-frame action is often better for discussing gravity. In $D$ spacetime dimensions, define

$$G_{\mu\nu}^{(E)}=e^{-4\Phi/(D-2)}G_{\mu\nu}^{(S)}.$$

In ten dimensions this is

$$\boxed{ G_{\mu\nu}^{(E)}=e^{-\Phi/2}G_{\mu\nu}^{(S)}. }$$

The NS—NS action becomes

$$S_{\rm NSNS}^{(E)} ={1\over2\kappa_{10}^2} \int d^{10}x\sqrt{-G^{(E)}} \left( R^{(E)} -\frac12(\partial\Phi)^2 -\frac12e^{-\Phi}|H_3|^2_E \right).$$

The Einstein frame makes the dilaton dependence of different form fields transparent. A $p$-form field strength appearing without an $e^{-2\Phi}$ prefactor in string frame typically obtains a different exponential of $\Phi$ in Einstein frame from the metric rescaling alone. This is why R—R kinetic terms in type II supergravity have dilaton factors distinct from the NS—NS $H_3$ term.

Type I: $N=1$ supergravity plus super-Yang—Mills

Type I string theory is obtained by an unoriented projection of type IIB together with an open-string sector. Its low-energy field content is ten-dimensional $N=1$ supergravity coupled to $N=1$ super-Yang—Mills. Consistency at the quantum level selects the gauge group $SO(32)$.

The bosonic fields include

$$G_{\mu\nu}, \qquad \Phi, \qquad C_2, \qquad A_\mu^a.$$

The NS—NS two-form $B_2$ is projected out by worldsheet parity, while an R—R two-form remains. The schematic string-frame action is

$$S_{\rm I}^{(S)} ={1\over2\kappa_{10}^2}\int d^{10}x\sqrt{-G} \left[e^{-2\Phi}\left(R+4(\partial\Phi)^2\right) -\frac12|\widetilde F_3|^2 \right] -{1\over2g_{10}^2}\int d^{10}x\sqrt{-G}\,e^{-\Phi}\,{\rm Tr}|F_2|^2+ \cdots.$$

Here

$$F_2=dA-iA\wedge A,$$

and $\widetilde F_3$ is a gauge-invariant R—R three-form field strength. In the full theory it is modified by Yang—Mills and Lorentz Chern—Simons terms, schematically

$$\widetilde F_3=dC_2+\omega_3^{\rm YM}-\omega_3^{\rm L}.$$

This modification is tied to the Green—Schwarz anomaly-cancellation mechanism. From the viewpoint of this course, the important lesson is that the same consistency conditions that determined the open-string gauge group also determine the low-energy supergravity couplings.

In Einstein frame, the Yang—Mills term carries a characteristic $e^{-\Phi/2}$ factor:

$$S_{\rm YM}^{(E)} =-{1\over2g_{10}^2}\int d^{10}x\sqrt{-G^{(E)}}\,e^{-\Phi/2}\,{\rm Tr}|F_2|^2.$$

This factor is another way of seeing that open-string tree amplitudes come from disks rather than spheres.

Type II supergravity: the common NS—NS sector and the R—R sector

Type IIA and type IIB strings are oriented closed superstrings. They share the same NS—NS fields

$$G_{\mu\nu}, \qquad B_2, \qquad \Phi,$$

but differ in the chirality of their R-sector ground states. This chirality difference determines the R—R potentials.

Type IIA is non-chiral and contains odd R—R potentials in the minimal formulation. Type IIB is chiral and contains even R—R potentials, including the self-dual four-form potential.

The rule is

$$\boxed{ \begin{array}{rcl} \text{type IIA} &:& C_1,\ C_3,\\ \text{type IIB} &:& C_0,\ C_2,\ C_4, \end{array}}$$

where $C_4$ in type IIB has a self-dual five-form field strength. In a democratic formulation one also introduces magnetic dual potentials, but then imposes duality constraints. For example, IIA may be described using $C_1,C_3,C_5,C_7,C_9$ and IIB using $C_0,C_2,C_4,C_6,C_8$, with duality relations removing the doubled degrees of freedom.

The natural electric coupling of a D$p$-brane to R—R fields is

$$\mu_p\int_{\mathcal W_{p+1}} C_{p+1},$$

more generally extended to the Wess—Zumino coupling

$$S_{\rm WZ} =\mu_p\int_{\mathcal W_{p+1}} C\wedge e^{B_2+2\pi\alpha'F},$$

where $C=\sum_q C_q$ is the formal sum of R—R potentials. This compact formula anticipates several later results: lower-dimensional brane charge can be dissolved as worldvolume flux, and T-duality acts naturally on the entire R—R polyform.

Type IIB supergravity

In Einstein frame, the bosonic type IIB pseudo-action is

$$\boxed{ \begin{aligned} S_{\rm IIB}^{(E)}={1\over2\kappa_{10}^2} \int d^{10}x\sqrt{-G}\bigg[& R -\frac12(\partial\Phi)^2 -\frac12e^{-\Phi}|H_3|^2 -\frac12e^{2\Phi}|F_1|^2 \\ &-\frac12e^{\Phi}|\widetilde F_3|^2 -\frac14|\widetilde F_5|^2 \bigg] -{1\over4\kappa_{10}^2}\int C_4\wedge H_3\wedge F_3. \end{aligned}}$$

The field strengths are

$$H_3=dB_2, \qquad F_1=dC_0, \qquad F_3=dC_2,$$

and the gauge-invariant combinations are conventionally taken to be

$$\widetilde F_3=F_3-C_0H_3,$$ $$\widetilde F_5=dC_4-\frac12C_2\wedge H_3+\frac12B_2\wedge F_3.$$

The five-form satisfies the self-duality condition

$$\boxed{\widetilde F_5=*\widetilde F_5.}$$

This condition cannot be derived from an ordinary covariant action with only the field $C_4$ and no extra auxiliary structure. The expression above is therefore called a pseudo-action: vary it first, and impose self-duality afterward. The coefficient $-\frac14|\widetilde F_5|^2$ rather than $-\frac12|\widetilde F_5|^2$ is part of this convention and prevents double-counting once the self-duality constraint is imposed.

Type IIB has a remarkable nonperturbative $SL(2,\mathbb Z)$ duality. The axion and dilaton combine into

$$\tau=C_0+ie^{-\Phi},$$

and the NS—NS and R—R two-forms transform as a doublet. This duality will later exchange fundamental strings and D1-branes, and it makes the D3-brane special: the D3-brane couples to $C_4$, whose field strength is self-dual, and the D3-brane maps to itself under S-duality.

Type IIA supergravity and its eleven-dimensional origin

The bosonic type IIA action is often written in string frame as

$$\boxed{ \begin{aligned} S_{\rm IIA}^{(S)}={1\over2\kappa_{10}^2} \int d^{10}x\sqrt{-G}\bigg[& e^{-2\Phi}\left(R+4(\partial\Phi)^2-\frac12|H_3|^2\right)\\ &-\frac12|F_2|^2 -\frac12|\widetilde F_4|^2 \bigg] -{1\over4\kappa_{10}^2}\int B_2\wedge F_4\wedge F_4. \end{aligned}}$$

The field strengths are

$$H_3=dB_2, \qquad F_2=dC_1, \qquad F_4=dC_3,$$

with gauge-invariant combination

$$\widetilde F_4=F_4-C_1\wedge H_3.$$

The Chern—Simons term is inherited from eleven-dimensional supergravity. Indeed, type IIA has a geometric interpretation as M-theory compactified on a circle. Let $y$ be the eleventh coordinate. The eleven-dimensional metric and three-form decompose as

$$\boxed{ ds_{11}^2=e^{-2\Phi/3}ds_{10,S}^2+e^{4\Phi/3}(dy+C_1)^2, }$$ $$\boxed{ A_3^{(11)}=C_3+B_2\wedge dy. }$$

Thus the ten-dimensional fields arise as

$$\begin{array}{ccl} G_{MN}^{(11)} &\longrightarrow& G_{\mu\nu}^{(10)},\ C_{1\mu},\ \Phi,\\ A_{MNP}^{(11)} &\longrightarrow& C_{3\mu\nu\rho},\ B_{2\mu\nu}. \end{array}$$

The relation between the M-theory circle radius and the type IIA coupling is

$$R_{11}=g_s\sqrt{\alpha'},$$

and the eleven-dimensional Planck length obeys

$$\ell_p^3=g_s\alpha'^{3/2}.$$

At weak coupling the M-theory circle is small and the ten-dimensional string description is appropriate. At strong coupling the circle grows, and type IIA reveals an eleventh dimension. This is the first sign of a unifying structure that will become important when we discuss branes and dualities.

Degree-of-freedom checks

A useful way to build confidence in these field lists is to count physical polarizations.

For a massless particle in $D$ dimensions, the little group is $SO(D-2)$. In ten dimensions the little group is $SO(8)$. The NS—NS sector contains

$$8_v\otimes 8_v=35\oplus 28\oplus 1,$$

which correspond to the graviton, two-form, and dilaton:

$$35: G_{\mu\nu}, \qquad 28: B_{\mu\nu}, \qquad 1: \Phi.$$

For type II, the R—R sector is built from spinor tensor products. The chirality choice determines whether the resulting forms have odd or even degree. Opposite chiralities give type IIA; equal chiralities give type IIB. This is the representation-theoretic origin of the IIA/IIB table above.

For eleven-dimensional supergravity, the little group is $SO(9)$. The graviton has

$${9\cdot10\over2}-1=44$$

physical components, and the three-form has

$$\binom{9}{3}=84.$$

Together they give

$$44+84=128$$

bosonic degrees of freedom, matching the $128$ fermionic degrees of freedom of the eleven-dimensional gravitino. This matching is the eleven-dimensional ancestor of type IIA spacetime supersymmetry.

What to remember

The low-energy actions are not optional embellishments of string theory; they are the first terms in its exact S-matrix.

The most important points are:

• Disk amplitudes of massless open strings reproduce ten-dimensional $N=1$ super-Yang—Mills and its $\alpha'$ corrections.
• For a single slowly varying D-brane gauge field, those open-string corrections are packaged by the DBI square root.
• Sphere amplitudes of massless closed strings reproduce the NS—NS supergravity action with its characteristic $e^{-2\Phi}$ factor.
• Type I combines $N=1$ supergravity with an $SO(32)$ open-string gauge sector.
• Type IIA and type IIB share the NS—NS fields but differ in R—R potentials because their Ramond ground-state chiralities differ.
• Type IIA is the circle reduction of eleven-dimensional supergravity; the IIA string coupling measures the size of the eleventh dimension.

These actions will be used constantly in the brane part of the course. D-brane tensions and R—R charges are read from DBI and Wess—Zumino terms; brane solutions are classical solutions of the corresponding supergravity actions; and AdS/CFT begins by comparing the open-string gauge theory on D3-branes with the closed-string geometry sourced by those same branes.

Exercises

Exercise 1: Expanding the DBI determinant

Let $F_{ab}$ be antisymmetric and let $g_{ab}$ be a fixed worldvolume metric. Show that

$$\sqrt{-\det(g+2\pi\alpha'F)} =\sqrt{-\det g}\left[1+{(2\pi\alpha')^2\over4}F_{ab}F^{ab}+O(F^4)\right].$$

Use this to identify $g_{\rm YM,p}$ in terms of $\mu_p$, $g_s$, and $\alpha'$.

Solution

Write

$$\det(g+2\pi\alpha'F)=\det g\,\det(1+M), \qquad M^a{}_b=2\pi\alpha' g^{ac}F_{cb}.$$

Since $F_{ab}$ is antisymmetric,

$${\rm tr}\,M=2\pi\alpha' g^{ab}F_{ba}=0.$$

Using

$$\det(1+M)=\exp\left({\rm tr}\log(1+M)\right)$$

and

$${\rm tr}\log(1+M)={\rm tr}\left(M-{1\over2}M^2+\cdots\right),$$

we get

$$\det(1+M)=1-{1\over2}{\rm tr}M^2+O(M^4).$$

For an antisymmetric field strength in Lorentzian signature,

$${\rm tr}M^2=-(2\pi\alpha')^2F_{ab}F^{ab}.$$

Therefore

$$\sqrt{\det(1+M)} =1+{(2\pi\alpha')^2\over4}F_{ab}F^{ab}+O(F^4),$$

with the overall $\sqrt{-\det g}$ restored. The quadratic DBI term is

$$-\mu_p e^{-\Phi_0}{(2\pi\alpha')^2\over4} \int d^{p+1}\xi\sqrt{-g}\,F_{ab}F^{ab}.$$

Comparing with

$$-{1\over4g_{\rm YM,p}^2}\int d^{p+1}\xi\sqrt{-g}\,F_{ab}F^{ab},$$

gives

$${1\over g_{\rm YM,p}^2}=\mu_pe^{-\Phi_0}(2\pi\alpha')^2.$$

Using $e^{\Phi_0}=g_s$ and $\mu_p=(2\pi)^{-p}\alpha'^{-(p+1)/2}$,

$$g_{\rm YM,p}^2=(2\pi)^{p-2}g_s\alpha'^{(p-3)/2}.$$

Exercise 2: The Einstein-frame metric

In $D$ dimensions, the string-frame NS—NS action contains

$$\int d^Dx\sqrt{-G^{(S)}}e^{-2\Phi}R^{(S)}.$$

Show that the Weyl rescaling

$$G_{\mu\nu}^{(E)}=e^{-4\Phi/(D-2)}G_{\mu\nu}^{(S)}$$

removes the dilaton prefactor in front of the Ricci scalar. Specialize to $D=10$.

Solution

Let

$$G_{\mu\nu}^{(S)}=e^{2\omega}G_{\mu\nu}^{(E)}.$$

Then

$$\sqrt{-G^{(S)}}=e^{D\omega}\sqrt{-G^{(E)}},$$

and the Ricci scalar transforms as

$$R^{(S)}=e^{-2\omega}\left(R^{(E)}+\text{terms involving derivatives of }\omega\right).$$

The coefficient multiplying $R^{(E)}$ is therefore

$$\sqrt{-G^{(S)}}e^{-2\Phi}R^{(S)} =\sqrt{-G^{(E)}}e^{(D-2)\omega-2\Phi}R^{(E)}+\cdots.$$

To remove the prefactor, require

$$(D-2)\omega-2\Phi=0, \qquad \omega={2\Phi\over D-2}.$$

Thus

$$G_{\mu\nu}^{(S)}=e^{4\Phi/(D-2)}G_{\mu\nu}^{(E)},$$

or

$$G_{\mu\nu}^{(E)}=e^{-4\Phi/(D-2)}G_{\mu\nu}^{(S)}.$$

For $D=10$,

$$G_{\mu\nu}^{(E)}=e^{-\Phi/2}G_{\mu\nu}^{(S)}.$$

Exercise 3: Eleven-dimensional degree counting

Show that the bosonic fields of eleven-dimensional supergravity have $128$ physical degrees of freedom.

Solution

For massless fields in eleven dimensions, the little group is $SO(9)$.

A massless graviton transforms as a symmetric traceless tensor of $SO(9)$. A symmetric tensor in $9$ dimensions has

$${9\cdot10\over2}=45$$

components, and removing the trace leaves

$$45-1=44.$$

A three-form gauge field has physical polarizations counted by choosing three indices in the transverse $9$-dimensional little-group space:

$$\binom{9}{3}=84.$$

Therefore the bosonic fields have

$$44+84=128$$

physical degrees of freedom. This matches the fermionic degrees of freedom of the eleven-dimensional gravitino, as required by supersymmetry.

Exercise 4: Reducing eleven-dimensional fields to type IIA

Starting from the eleven-dimensional metric $G_{MN}^{(11)}$ and three-form $A_{MNP}^{(11)}$, identify the ten-dimensional type IIA fields obtained by compactifying one direction $y$ on a circle.

Solution

The metric decomposes into components with zero, one, or two indices along the circle:

$$G_{MN}^{(11)} \longrightarrow G_{\mu\nu}^{(10)},\quad G_{\mu y}^{(11)},\quad G_{yy}^{(11)}.$$

The off-diagonal metric component is a Kaluza—Klein vector. In type IIA this is the R—R one-form $C_1$. The circle radius is encoded in the dilaton $\Phi$. In string frame the decomposition is

$$ds_{11}^2=e^{-2\Phi/3}ds_{10,S}^2+e^{4\Phi/3}(dy+C_1)^2.$$

The eleven-dimensional three-form decomposes as

$$A_3^{(11)}=C_3+B_2\wedge dy.$$

Thus the components with no $y$ index give the R—R three-form $C_3$, while components with one $y$ index give the NS—NS two-form $B_2$.

Therefore the IIA bosonic fields are

$$G_{\mu\nu},\quad B_2,\quad \Phi,\quad C_1,\quad C_3.$$

Exercise 5: Why type IIB uses a pseudo-action

Explain why the type IIB five-form field strength cannot be treated like an ordinary unconstrained form field in a standard covariant action.

Solution

The type IIB five-form field strength obeys

$$\widetilde F_5=*\widetilde F_5.$$

In ten-dimensional Lorentzian signature, this self-duality condition halves the number of independent degrees of freedom. If one writes an ordinary kinetic term

$$-\frac12\int |\widetilde F_5|^2$$

and varies an unconstrained four-form potential $C_4$, the resulting equation of motion is

$$d*\widetilde F_5=0,$$

which is distinct from the Bianchi identity

$$d\widetilde F_5=\cdots.$$

Self-duality identifies these two equations. A simple covariant action with only $C_4$ either double-counts the degrees of freedom or fails to impose self-duality directly.

The practical convention is to use a pseudo-action with coefficient

$$-\frac14|\widetilde F_5|^2,$$

vary it to obtain the equations, and then impose

$$\widetilde F_5=*\widetilde F_5$$

by hand. More sophisticated formulations introduce auxiliary fields, but the pseudo-action is the standard compact way to present type IIB supergravity.

Exercise 6: Dilaton powers from worldsheet topology

A constant dilaton couples to the worldsheet by

$$S_\Phi={\Phi_0\over4\pi}\int_\Sigma d^2\sigma\sqrt{h}\,R^{(2)}.$$

Use the Gauss—Bonnet theorem to show that a worldsheet with Euler characteristic $\chi$ contributes a factor $g_s^{-\chi}$ to the path integral. What are the factors for the sphere and the disk?

Solution

For a closed worldsheet, Gauss—Bonnet gives

$${1\over4\pi}\int_\Sigma d^2\sigma\sqrt{h}\,R^{(2)}=\chi.$$

For worldsheets with boundary, the boundary curvature term must also be included, but the final topological answer is again $\chi$.

The Euclidean path integral contains

$$e^{-S_\Phi}=e^{-\Phi_0\chi}.$$

Since

$$g_s=e^{\Phi_0},$$

this is

$$e^{-\Phi_0\chi}=g_s^{-\chi}.$$

For a sphere,

$$\chi=2, \qquad \text{factor}=g_s^{-2}=e^{-2\Phi_0}.$$

For a disk,

$$\chi=1, \qquad \text{factor}=g_s^{-1}=e^{-\Phi_0}.$$

This is why the closed-string tree-level NS—NS action has an $e^{-2\Phi}$ prefactor, while D-brane disk actions have an $e^{-\Phi}$ prefactor.