Type II T-Duality and Ramond--Ramond Fields

The previous pages made D-branes unavoidable: T-duality turns Neumann boundary conditions into Dirichlet boundary conditions, and the endpoints of open strings become constrained to lie on dynamical hypersurfaces. We now add an essential new fact. In superstring theory, the same T-duality also acts nontrivially on spacetime spinors and on Ramond—Ramond fields. This is why D-branes are not merely geometric defects; they are charged BPS objects.

The headline result is beautifully compact:

$$\boxed{\text{T-duality along one circle exchanges type IIA and type IIB.}}$$

The reason is that a single T-duality multiplies one chiral half of the spacetime spin fields by one gamma matrix. One gamma matrix flips ten-dimensional chirality. Since type IIA has opposite left- and right-moving chiralities, while type IIB has equal chiralities, a single T-duality maps one theory into the other.

This page develops the precise dictionary:

• how T-duality acts on worldsheet fermions and spin fields,
• how Ramond—Ramond potentials transform as differential forms,
• why D$p$-branes become D$(p-1)$- or D$(p+1)$-branes,
• and why a D$p$-brane preserves exactly half of the type II supersymmetry.

T-duality on worldsheet fields

Let $y=X^9$ be a compact coordinate of radius $R$. On a closed string we decompose

$$Y(z,\bar z)=Y_L(z)+Y_R(\bar z).$$

In the convention used here, T-duality along $y$ acts as

$$Y_L \longrightarrow Y_L,\qquad Y_R \longrightarrow -Y_R.$$

Worldsheet supersymmetry requires the same sign flip for the right-moving fermion:

$$\psi^y(z)\longrightarrow \psi^y(z),\qquad \tilde\psi^y(\bar z)\longrightarrow -\tilde\psi^y(\bar z).$$

For the noncompact directions $\mu=0,\ldots,8$,

$$X^\mu_L\to X^\mu_L,\qquad X^\mu_R\to X^\mu_R,\qquad \psi^\mu\to\psi^\mu,\qquad \tilde\psi^\mu\to \tilde\psi^\mu.$$

Thus T-duality is not just a relabeling of the compact coordinate. It is a spacetime reflection acting only on one chiral half of the worldsheet theory. In the bosonic sector, this reflection exchanges momentum and winding. In the fermionic Ramond sector, it acts on spin fields.

Spin fields and the chirality flip

The Ramond-sector vertex operators for spacetime fermions contain spin fields. In the $(-1/2)$ picture, schematically,

$$V_A^{(-1/2)}(z)=u_A\, e^{-\phi/2} S^A(z) e^{ik\cdot X(z)},$$

for the left-moving side, and

$$\tilde V_B^{(-1/2)}(\bar z)=\tilde u_B\, e^{-\tilde\phi/2} \tilde S^B(\bar z) e^{ik\cdot \tilde X(\bar z)}$$

for the right-moving side. The ten-dimensional chirality matrix is

$$\Gamma_{11}=\Gamma^0\Gamma^1\cdots\Gamma^9,\qquad \Gamma_{11}^2=1.$$

Because $\Gamma^y$ anticommutes with $\Gamma_{11}$,

$$\Gamma_{11}\Gamma^y=-\Gamma^y\Gamma_{11}.$$

Therefore multiplication by $\Gamma^y$ reverses chirality. Under the T-duality above, the right-moving Ramond spin field transforms as

$$\tilde S \longrightarrow \Gamma^y \tilde S.$$

Consequently,

$$\Gamma_{11}\tilde S=\pm \tilde S \qquad\Longrightarrow\qquad \Gamma_{11}(\Gamma^y\tilde S)=\mp(\Gamma^y\tilde S).$$

This proves the exchange of the two type II theories:

$$\boxed{ \begin{array}{ccl} \text{type IIA} & \xleftrightarrow{\;T_y\;} & \text{type IIB},\\ \text{opposite chiralities} && \text{same chiralities}. \end{array} }$$

More generally, an odd number of T-dualities exchanges IIA and IIB, while an even number maps each theory to itself.

A single T-duality along $y$ exchanges the two type II theories by flipping one Ramond chirality. In the Ramond—Ramond sector, components with a $y$ index lose it, while components without a $y$ index gain one.

Ramond—Ramond vertex operators as bispinors

Ramond—Ramond states are obtained by taking a Ramond ground state on both the left and right. In the $(-1/2,-1/2)$ picture,

$$V_{\text{RR}}^{(-1/2,-1/2)} = \mathcal F_{AB}\, e^{-\phi/2}S^A(z)\, e^{-\tilde\phi/2}\tilde S^B(\bar z)\, e^{ik\cdot X}.$$

The polarization $\mathcal F_{AB}$ is a spacetime bispinor. Gamma matrices identify such bispinors with antisymmetric differential forms:

$$\mathcal F_{AB} = \sum_n {1\over n!} F_{\mu_1\cdots\mu_n} \left(C\Gamma^{\mu_1\cdots\mu_n}\right)_{AB},$$

where $C$ is the charge-conjugation matrix and

$$\Gamma^{\mu_1\cdots\mu_n} = \Gamma^{[\mu_1}\Gamma^{\mu_2}\cdots\Gamma^{\mu_n]}.$$

After imposing the type II GSO projection, the physical Ramond—Ramond field strengths are

$$\begin{array}{c|c|c} \text{theory} & \text{R--R potentials} & \text{R--R field strengths}\\ \hline \text{IIA} & C_1,\ C_3,\ C_5,\ C_7,\ C_9 & F_0,\ F_2,\ F_4,\ F_6,\ F_8,\ F_{10}\\ \text{IIB} & C_0,\ C_2,\ C_4,\ C_6,\ C_8 & F_1,\ F_3,\ F_5,\ F_7,\ F_9 \end{array}$$

This is the democratic notation. In the minimal two-derivative supergravity description, one keeps only the independent fields and imposes duality relations,

$$F_n = (-1)^{n(n-1)/2} * F_{10-n},$$

with the special self-duality condition in type IIB,

$$F_5=*F_5.$$

In particular, the familiar independent potentials are

$$\text{IIA}:\quad C_1,\ C_3, \qquad \text{IIB}:\quad C_0,\ C_2,\ C_4^+,$$

where the superscript reminds us that the corresponding five-form field strength is self-dual.

The R—R T-duality rule

Let $y$ be the T-duality direction. Split an $n$-form potential into components with and without a $dy$ leg:

$$C_n = C_n^{\perp} + dy\wedge C_{n-1}^{\parallel},$$

where $C_n^\perp$ has no $dy$ index. Then, ignoring $B$-field refinements and signs depending on form-ordering conventions, T-duality acts as

$$\boxed{ C_n^{\perp}\longrightarrow dy'\wedge C_n^{\perp}, \qquad dy\wedge C_{n-1}^{\parallel}\longrightarrow C_{n-1}^{\parallel}. }$$

Equivalently, in components with $\mu_i\neq y$,

$$\boxed{ C'_{\mu_1\cdots\mu_n\, y} = C_{\mu_1\cdots\mu_n}, \qquad C'_{\mu_1\cdots\mu_n} = C_{\mu_1\cdots\mu_n y}. }$$

A component without a $y$ index gains one. A component with a $y$ index loses one. This simple rule turns odd-degree potentials into even-degree potentials and vice versa.

For example,

$$\begin{aligned} \text{IIA } C_\mu &\longrightarrow \begin{cases} \text{IIB } C_{\mu y}, & \mu\neq y,\\ \text{IIB } C_0, & \mu=y, \end{cases} \\ \text{IIA } C_{\mu\nu\rho} &\longrightarrow \begin{cases} \text{IIB } C_{\mu\nu\rho y}, & \mu,\nu,\rho\neq y,\\ \text{IIB } C_{\mu\nu}, & \text{one index is } y. \end{cases} \end{aligned}$$

This form-degree rule is the spacetime counterpart of the spin-field rule $\tilde S\to \Gamma^y\tilde S$.

A compact way to write the same statement is to combine the R—R potentials into a polyform

$$C=\sum_n C_n.$$

For backgrounds with no $B$-field, the T-dual polyform is schematically

$$C'=\iota_y C + dy'\wedge C,$$

where terms with two $dy$ legs vanish automatically. With a nonzero NS—NS two-form, the clean invariant object is $e^B C$, and the formula is correspondingly twisted by $B$. For most elementary D-brane applications, the component rule above is the safest way to use the dictionary.

D-branes as R—R electric sources

A D$p$-brane couples electrically to a Ramond—Ramond $(p+1)$-form potential:

$$S_{\text{WZ}}=\mu_p\int_{\mathcal W_{p+1}} C_{p+1}.$$

Here $\mathcal W_{p+1}$ is the brane worldvolume and $\mu_p$ is the R—R charge. This single formula explains the allowed D-branes in the two type II theories:

$$\boxed{ \begin{array}{c|c} \text{theory} & \text{BPS D-branes}\\ \hline \text{IIA} & p=0,2,4,6,8\\ \text{IIB} & p=-1,1,3,5,7,9 \end{array} }$$

The D$(-1)$-brane is a D-instanton. The D9-brane fills all spacetime.

Now apply T-duality along $y$. There are two cases.

T-duality along a brane direction

If the D$p$-brane wraps the circle $y$, then $dy$ appears in the pulled-back volume form. The Wess—Zumino coupling contains

$$\int_{\mathcal W_{p+1}} C_{\mu_0\cdots\mu_{p-1}y}\, dx^{\mu_0}\wedge\cdots\wedge dx^{\mu_{p-1}}\wedge dy.$$

After T-duality, $C_{\mu_0\cdots\mu_{p-1}y}$ becomes a $p$-form potential $C'_{\mu_0\cdots\mu_{p-1}}$. The brane loses the wrapped direction:

$$\boxed{ T_y\text{ along the brane:}\qquad D p \longrightarrow D(p-1). }$$

This is the R—R version of the open-string statement that a Neumann direction becomes Dirichlet.

T-duality transverse to the brane

If $y$ is transverse to the D$p$-brane, then the coupling uses a component $C_{\mu_0\cdots\mu_p}$ with no $y$ index. Under T-duality this becomes $C'_{\mu_0\cdots\mu_p y}$, so the brane gains the dual direction:

$$\boxed{ T_y\text{ transverse to the brane:}\qquad D p \longrightarrow D(p+1). }$$

This is the R—R version of the open-string statement that a Dirichlet direction becomes Neumann.

Allowed BPS D-branes alternate between type IIA and type IIB. T-duality along a worldvolume circle lowers $p$ by one, while T-duality along a transverse circle raises $p$ by one.

Supercharges in type II theories

Flat type II string theory has two ten-dimensional Majorana—Weyl supercharges,

$$Q_L,\qquad Q_R,$$

each with 16 real components. In worldsheet language they arise from integrated Ramond spin-field vertices,

$$Q_L^A=\oint dz\, e^{-\phi/2} S^A(z), \qquad Q_R^B=\oint d\bar z\, e^{-\tilde\phi/2}\tilde S^B(\bar z).$$

Their chiralities distinguish the two theories:

$$\begin{array}{c|c} \text{theory} & \text{chirality relation}\\ \hline \text{IIA} & Q_L \text{ and } Q_R \text{ have opposite chirality}\\ \text{IIB} & Q_L \text{ and } Q_R \text{ have the same chirality} \end{array}$$

The full flat-space theory has 32 real supercharges. A D-brane preserves only a diagonal half.

Supersymmetry preserved by a D$p$-brane

A D$p$-brane extended along $x^0,x^1,\ldots,x^p$ imposes boundary conditions that identify left- and right-moving worldsheet degrees of freedom at the boundary. The corresponding relation on spacetime supersymmetry parameters is

$$\boxed{ \epsilon_L = \eta\, \Gamma^{0\cdots p}\epsilon_R, }$$

where

$$\Gamma^{0\cdots p}=\Gamma^0\Gamma^1\cdots\Gamma^p,$$

and $\eta=+1$ for a brane, $\eta=-1$ for an antibrane, up to a convention-dependent overall sign.

Equivalently, the preserved supercharge is a linear combination

$$\boxed{ Q_{\text{pres}} = Q_L + \eta\, \Gamma^{0\cdots p} Q_R. }$$

This imposes 16 independent conditions on the original 32 supercharges, so a single flat D$p$-brane is half-BPS.

The chirality of $\Gamma^{0\cdots p}$ is exactly right for the allowed branes:

• if $p$ is even, then $p+1$ is odd, so $\Gamma^{0\cdots p}$ flips chirality; this matches type IIA, where $Q_L$ and $Q_R$ have opposite chirality;
• if $p$ is odd, then $p+1$ is even, so $\Gamma^{0\cdots p}$ preserves chirality; this matches type IIB, where $Q_L$ and $Q_R$ have the same chirality.

Thus the supersymmetry projection knows the same parity rule as the R—R potentials.

A flat D$p$-brane identifies the two type II supersymmetry parameters by $\epsilon_L=\eta\Gamma^{0\cdots p}\epsilon_R$. This leaves 16 of the original 32 real supercharges unbroken.

The D9-brane and type I strings

A useful special case is a D9-brane in type IIB. It fills all spacetime, so the preserved supersymmetry condition is

$$\epsilon_L=\eta\,\Gamma^{0\cdots 9}\epsilon_R =\eta\,\Gamma_{11}\epsilon_R.$$

For type IIB, $\epsilon_R$ has a definite chirality equal to that of $\epsilon_L$. Choosing the conventional sign gives a diagonal $N=1$ supersymmetry in ten dimensions. This is the same structure that appears in the type I orientifold: the worldsheet parity projection identifies the two type IIB supercharges and leaves a single ten-dimensional Majorana—Weyl supercharge.

T-dualizing a D9-brane along one spatial direction gives a D8-brane in type IIA. The D8-brane is special because it couples to a nine-form potential $C_9$, whose ten-form field strength $F_{10}$ is dual to the Romans mass $F_0$. In other words, the D8-brane naturally belongs to massive type IIA supergravity. This is the first sign that the full D-brane spectrum includes objects that are invisible in the smallest field-content list but are required by duality.

Worldvolume fields from dimensional reduction

The low-energy fields on a single D$p$-brane form the dimensional reduction of ten-dimensional $N=1$ super-Yang—Mills theory to $p+1$ dimensions.

The ten-dimensional gauge field decomposes as

$$A_M = (A_a,A_i), \qquad a=0,\ldots,p,\qquad i=p+1,\ldots,9.$$

On the brane,

$$A_a \quad\text{is a }(p+1)\text{-dimensional gauge field},$$

while

$$\Phi^i = {1\over 2\pi\alpha'} X^i$$

are scalar fields describing transverse fluctuations. Thus a D$p$-brane carries

$$9-p$$

real scalar fields. For $N$ coincident branes these fields become $N\times N$ matrices, and the diagonal entries encode brane positions.

The fermions reduce in the same way. The result is a maximally supersymmetric vector multiplet in $p+1$ dimensions, with 16 supercharges. This is precisely what one expects from a half-BPS brane inside a 32-supercharge bulk.

Summary

A single T-duality is more than the map $R\leftrightarrow \alpha'/R$. In type II string theory it also acts on the Ramond sector:

$$\tilde S\longrightarrow \Gamma^y\tilde S,$$

so it flips one spacetime chirality and exchanges type IIA with type IIB. Ramond—Ramond potentials transform by adding or removing an index along the dualized circle,

$$C'_{\mu_1\cdots\mu_n\,y}=C_{\mu_1\cdots\mu_n}, \qquad C'_{\mu_1\cdots\mu_n}=C_{\mu_1\cdots\mu_n y}.$$

D-branes are the electric sources for these potentials:

$$S_{\text{WZ}}=\mu_p\int C_{p+1}.$$

Therefore type IIA contains even-dimensional BPS D-branes and type IIB contains odd-dimensional BPS D-branes. T-duality along a worldvolume direction sends $D p$ to $D(p-1)$; T-duality along a transverse direction sends $D p$ to $D(p+1)$.

Finally, a flat D$p$-brane preserves the diagonal half of the type II supersymmetry,

$$Q_{\text{pres}}=Q_L+\eta\,\Gamma^{0\cdots p}Q_R,$$

leaving 16 real supercharges. The compatibility of this projection with ten-dimensional chirality is the supersymmetry version of the same IIA/IIB parity rule.

Exercises

Exercise 1

Show explicitly that multiplying a ten-dimensional Weyl spinor by $\Gamma^y$ reverses its chirality.

Solution

Let $\lambda$ be a Weyl spinor satisfying

$$\Gamma_{11}\lambda=s\lambda,\qquad s=\pm1.$$

Since

$$\Gamma_{11}\Gamma^y=-\Gamma^y\Gamma_{11},$$

we find

$$\Gamma_{11}(\Gamma^y\lambda) = -\Gamma^y\Gamma_{11}\lambda = -s\,\Gamma^y\lambda.$$

Thus $\Gamma^y\lambda$ has chirality $-s$. This is why a single T-duality, which multiplies one Ramond spin field by $\Gamma^y$, exchanges type IIA and type IIB.

Exercise 2

Use the R—R T-duality component rule to determine what type IIB fields arise from the type IIA three-form potential $C_3$ after T-duality along $y$.

Solution

Write the components of $C_3$ as

$$C_{\mu\nu\rho},\qquad C_{\mu\nu y},$$

where $\mu,\nu,\rho\neq y$. The T-duality rule gives

$$C_{\mu\nu\rho}\longrightarrow C'_{\mu\nu\rho y},$$

which is a component of the type IIB four-form $C_4$, and

$$C_{\mu\nu y}\longrightarrow C'_{\mu\nu},$$

which is a component of the type IIB two-form $C_2$. Thus

$$C_3^{\text{IIA}} \quad\xrightarrow{T_y}\quad C_4^{\text{IIB}}\oplus C_2^{\text{IIB}},$$

with the split determined by whether the original component carried a $y$ index.

Exercise 3

A D4-brane in type IIA wraps a circle $y$. What brane is obtained after T-duality along $y$? What happens if $y$ is instead transverse to the D4-brane?

Solution

If $y$ lies along the D4-brane worldvolume, T-duality turns a Neumann direction into a Dirichlet direction. The brane loses one spatial dimension:

$$D4 \longrightarrow D3.$$

The resulting D3-brane belongs to type IIB, as expected because one T-duality exchanges IIA and IIB.

If $y$ is transverse to the D4-brane, T-duality turns a Dirichlet direction into a Neumann direction. The brane gains one spatial dimension:

$$D4 \longrightarrow D5.$$

Again the result belongs to type IIB.

Exercise 4

Explain why the supersymmetry projection

$$\epsilon_L=\eta\,\Gamma^{0\cdots p}\epsilon_R$$

is compatible with type IIA only for even $p$ and with type IIB only for odd $p$.

Solution

The product $\Gamma^{0\cdots p}$ contains $p+1$ gamma matrices. Since each gamma matrix anticommutes with $\Gamma_{11}$, the product satisfies

$$\Gamma_{11}\Gamma^{0\cdots p} = (-1)^{p+1}\Gamma^{0\cdots p}\Gamma_{11}.$$

If $p$ is even, then $p+1$ is odd, so $\Gamma^{0\cdots p}$ flips chirality. This can relate $\epsilon_R$ to $\epsilon_L$ only if the two spinors have opposite chirality, which is type IIA.

If $p$ is odd, then $p+1$ is even, so $\Gamma^{0\cdots p}$ preserves chirality. This can relate $\epsilon_R$ to $\epsilon_L$ only if the two spinors have the same chirality, which is type IIB.

Therefore BPS D-branes have even $p$ in IIA and odd $p$ in IIB.

Exercise 5

A single D$p$-brane has a maximally supersymmetric vector multiplet on its worldvolume. Derive the number of scalar fields in this multiplet from dimensional reduction of ten-dimensional super-Yang—Mills theory.

Solution

Ten-dimensional super-Yang—Mills has a gauge field $A_M$ with

$$M=0,\ldots,9.$$

On a D$p$-brane, split the index as

$$M=(a,i), \qquad a=0,\ldots,p, \qquad i=p+1,\ldots,9.$$

The components $A_a$ remain a gauge field on the $(p+1)$-dimensional worldvolume. The transverse components $A_i$ become scalar fields from the lower-dimensional viewpoint:

$$A_i\longrightarrow \Phi^i.$$

The number of transverse directions is

$$10-(p+1)=9-p.$$

Hence the D$p$-brane worldvolume theory contains $9-p$ real scalar fields, describing the transverse position of the brane.

Exercise 6

Why is the D8-brane naturally associated with massive type IIA supergravity?

Solution

A D8-brane couples electrically to a nine-form Ramond—Ramond potential:

$$S_{\text{WZ}}=\mu_8\int_{\mathcal W_9} C_9.$$

The corresponding field strength is a ten-form,

$$F_{10}=dC_9.$$

In ten dimensions, a ten-form field strength is Hodge dual to a zero-form:

$$F_{10}=*F_0.$$

The zero-form $F_0$ is the Romans mass parameter of massive type IIA supergravity. Therefore a D8-brane is a domain wall across which the Romans mass can jump. This is why D8-branes are naturally described in massive type IIA rather than in the smallest massless type IIA supergravity field list.