The D-brane charge and action

There is no force between static BPS objects of like charge. The multi-object state is still supersymmetric and so its energy is determined only by its charge and is independent of the separations. For parallel Dp-branes, the unbroken supersymmetry (13.2.5) is the same as for a single Dp-brane.

The vanishing of the force comes about from a cancellation between attraction due to the graviton and dilaton and repulsion due to the R–R tensor. We can calculate these forces explicitly from the usual cylinder vacuum amplitude. The exchange of light NS–NS closed strings was isolated in eq. (10.8.4). Modify this expression by removing the factors for the momentum integrations in the Dirichlet directions and introducing a term for the tension of a string stretched over a separation y^µ:

A^NS–NS ≈ iVp+14× 16

8π(8π²α^)⁵

_∞

0

πdt

t² (8π²α^t)^(9−p)/2exp



− ty² 2πα^



= iVp+12π(4π²α^)^3−pG_9−p(y) (13.3.1) with Gd(y) = 2⁻²π^−d/2Γ(¹₂d− 1)y^2−d the scalar Green’s function. The Chan–Paton weight is 2 here, from the two orientations of the open string, and there is no factor of ¹₂ from the orientation projection because the physics is locally oriented. Due to supersymmetric cancellation in the trace, the R–R exchange amplitude is

AR–R=−ANS–NS (13.3.2)

and so the total force vanishes as expected.

The field theory calculation (8.7.25) for the dilaton–graviton potential changes only by the substitution 6 = (D− 2)/4 → 2, and so is

2iκ²τ²_pG9−p(y) . (13.3.3) Thus

τ²_p= π

κ²(4π²α^)^3−p . (13.3.4) This satisfies the same T -duality relation as in the bosonic string. For the R–R exchange, the low-energy action is

− 1

4κ²₁₀

d¹⁰x (−G)^1/2|Fp+2|²+ µp

Cp+1 . (13.3.5)

13.3 The D-brane charge and action 147 The kinetic term is canonically normalized, so the propagator for any given component (such as the one parallel to the D-brane) is 2κ²₁₀i/k², and the field theory amplitude is

− 2κ²10iµ²_pG9−p(y) . (13.3.6) Hence

µ²_p = π

κ²₁₀(4π²α^)^3−p = e^2Φ0τ²_p = T_p² . (13.3.7) The reader can carry out a similar calculation of the force between a D-brane and an orientifold plane and show that it has an additional−(2^5−k).

We deduced from the cancellation of divergences that the charge of the orientifold plane should have a factor of −(2^4−k); the extra factor of 2 in the force arises because the orientifold geometry squeezes the flux lines into half the solid angle.

The calculation of the interaction confirms our earlier deduction that D-branes carry the R–R charges. It is interesting to see how this is consistent with our earlier discussion of string vertex operators. The R–R vertex operator (12.1.14) is in the (−¹₂,−¹₂) picture, which can be used in almost all processes. On the disk, however, the total left- plus right-moving ghost number must be −2. With two or more R–R vertex operators, all can be in the (−¹₂,−¹₂) picture (with PCOs included as well), but a single vertex operator must be in either the (−³₂,−¹₂) or the (−¹₂,−³₂) picture.

The (−¹₂,−¹₂) vertex operator is essentially e^−φG₀ times the (−³₂,−¹₂) operator, so besides the shift in the ghost number the latter has one less power of momentum and one less Γ-matrix. The missing factor of momentum turns F into C, and the missing Γ-matrix gives the correct Lorentz representations for the potential rather than the field strength.

Dirac quantization condition

There is an important consistency check on the value of the R–R charge, which generalizes the Dirac quantization condition for magnetic monopole charge. Let us review the Dirac condition, shown in figure 13.3. Consider a magnetic charge µm at the origin. The integrated flux is

S2

F2 = µm . (13.3.8)

Because the integral over a closed surface is nonzero, we cannot write F2 = dA1 for any vector potential. However, we can write F2 = dA1

except along a Dirac string ending on the monopole. Now consider an electric charge µe moving in this field. Its coupling to the field produces a

P D

Fig. 13.3. Sphere surrounding monopole, with a Dirac string running upward.

The particle path P is bounded by the lower cap D.

phase when the particle moves on a closed path P . The surface D spans P and does not intersect the Dirac string. Now consider the limit as the path is contracted to a small circle around the Dirac string. The phase becomes

exp The Dirac string must be invisible, so this phase must be 1. Equivalently, this is the condition that the phase (13.3.9) is unchanged if we instead take the surface D^ = S2− D spanning P in the upper hemisphere. The result is the Dirac quantization condition,

µ_eµ_m = 2πn (13.3.11)

for some integer n.

A p-brane and (6− p)-brane are sources for Fp+2 and F8−p respectively.

These two field strengths are Poincar´e dual to one another, so again there is a Dirac quantization condition that must be satisfied by the product of their charges. Let us think about Fp+2 as the field strength, so that the p-brane is an electric source and the (6−p)-brane a magnetic source. In nine dimensions a (6− p)-dimensional object is surrounded by a (p + 2)-sphere, so by analogy to the magnetic flux (13.3.8),

Sp+2

Fp+2 =µ_6−p 2κ²₁₀. (13.3.12) One can then repeat the same argument. For example, let the p-brane be extended in the directions 4 ≤ µ ≤ p + 3 and the (6 − p)-brane in the

/

13.3 The D-brane charge and action 149 directions p + 4 ≤ µ ≤ 9. The system essentially reduces to the three-dimensional situation of figure 13.3 in the directions µ = 1, 2, 3, and the charges must satisfy

µ_pµ_6−p =πn κ²₁₀ . (13.3.13) Remarkably, the charges (13.3.7), arrived at in an entirely different way, satisfy this relation with the minimum quantum n = 1.

D-brane actions

The coupling of a D-brane to NS–NS closed string fields is the same Dirac–Born–Infeld action as in the bosonic string,

SDp=−µp

d^p+1ξ Tr e^−Φ[− det(Gab+ Bab+ 2πα^F_ab)]^1/2

!

, (13.3.14) where Gab and Bab are the components of the spacetime NS–NS fields parallel to the brane and Fab is the gauge field living on the brane. The argument leading to this form is exactly as in the bosonic case, section 8.7.

Recall that for n D-branes at small separation, where the strings stretched between them are light enough to be included in the low energy action, the collective coordinates X^µ(ξ), gauge fields Aa(ξ), and their fermionic partners λ(ξ) all become n× n matrices. The trace in the action is in this n× n space. In addition there is a term

O([X^m, Xⁿ]²) (13.3.15) in the potential. As discussed in chapter 8, the effect of this potential is that in the flat directions the collective coordinates become diagonal. They can then be interpreted as n ordinary collective coordinates for n objects.

At small separation the full matrix dynamics is crucial, as we will see.

The coupling to the R–R background also includes corrections involving the gauge field on the brane. Like the Born–Infeld action, these can be deduced via T -duality. Consider, as an example, a 1-brane in the (1,2) plane. The action is

C2 =

dx⁰(dx¹C₀₁+ dx²C₀₂) =

dx⁰dx^1C₀₁+ ∂1X²C₀₂^. (13.3.16) Under a T -duality in the 2-direction this becomes

dx⁰dx^{1 }C₀₁₂+ 2πα^F₁₂C₀ . (13.3.17) We have used the T -transformation of the C fields, eq. (13.1.5). A D-brane at an angle is T -dual to one with a magnetic field, as in figure 13.2.

We are not keeping track of the normalization but one could, with the result µp = µp−1/2πα^1/2 in agreement with the explicit calculation. The

/

dx²

generalization of (13.3.17) to an arbitrary configuration, and to multiple D-branes, gives the Chern–Simons-like result

iµ_p

p+1

Tr



exp(2πα^F2 + B2)∧^

q

Cq



. (13.3.18) The expansion of the integrand (13.3.18) involves forms of various rank;

the integral picks out precisely the terms that are proportional to the volume form of the p-brane world-volume. There are similar couplings with the spacetime curvature in addition to the field strength; these can be obtained from a string calculation.

Thus far we have given only the action for the bosonic fields on the brane. For the leading fluctuations around a flat D-brane in flat spacetime the fermionic action is of the usual Dirac form

− i
d^p+1ξ Tr(λΓ^aD_aλ) . (13.3.19) The full nonlinear supersymmetric form is left to the references.

Coupling constants

The ratio of the F-string tension to the D-string tension is τ_F1

τD1

= 1

2πα^ κ

4π^5/2α^ = κ

8π^7/2α^2 . (13.3.20) Up to now there has been no natural convention for defining the additive normalization of the dilaton field or the multiplicative normalization of the closed string coupling g = e^Φ. The dimensionless ratio (13.3.20) is proportional to the closed string coupling, and it turns out to be very convenient to take it as the definition of the coupling,

g = τ_F1 τD1

. (13.3.21)

Then the gravitational coupling is

κ² = ¹₂(2π)⁷g²α^4 (13.3.22) and the D-brane tension is

τ_p= 1

g(2π)^pα^(p+1)/2 = (2κ²)^−1/2(2π)^{(7−2 )/2}α^(3−p)/2 . (13.3.23) Also, the constant appearing in the low energy actions of section 12.1 is

κ²₁₀= ¹₂(2π)⁷α^4 ; (13.3.24) this differs from the physically measured κ because the latter depends on the dilaton background.

p

13.3 The D-brane charge and action 151 Expanding the action (13.3.14) gives the coupling of the Yang–Mills theory on the Dp-brane,

g_Dp² = 1 (2πα^)²τp

= (2π)^p−2gα^(p−3)/2 . (13.3.25) Notice that for p = 3 this coupling is dimensionless, as expected in a (3 + 1)-dimensional gauge theory. For p < 3 the coupling has units of length to a negative power, and for p > 3 length to a positive power.

We now wish to obtain the relation among κ, gYM, and α^ in the type I theory. We cannot quite identify gDp for p = 9 with gYM, because the former has been obtained in a locally oriented theory and there are some additional factors of 2 in the type I case. Rather than repeat the string calculation we will make a more roundabout but possibly instructive argument using T -duality.

First, we should note that the coupling (13.3.25) is for the U(n) gauge theory of coincident branes in the oriented theory: it appears in the form

1

4g²_DpTr_f , (13.3.26)

where the trace is in the n× n fundamental representation. Now let us consider moving the branes to an orientifold plane so that the gauge symmetry is enlarged to S O(2n). An S U(n) generator t is embedded in S O(2n) as

˜t =

 t 0 0 −t^T



, (13.3.27)

because the orientation projection reverses the order of the Chan–Paton factors and the sign of the gauge field. Comparing the low energy actions gives

1

4g²_DpTr_f(t²) = 1

4g²_{Dp,S O(2n)}Trv(˜t²) (13.3.28) and so g_{Dp,S O(2n)}² = 2g_Dp² .

Now consider the type I theory compactified on a k-torus with all radii equal to R. The couplings in the lower-dimensional S O(32) theory are related to those in the type I theory by

κ²_10−k = (2πR)^−kκ² (type I) , g²_10−k,YM= (2πR)^−kg_YM² (type I) . (13.3.29) In the T -dual picture, the bulk theory is of type II and the gauge fields live on a D( − k)-brane, and

κ²_10−k = 2(2πR^)^−kκ² , g₁₀²_−k,YM= g²_{D(9−k),SO(32)} . (13.3.30) The dimensional reduction for κ²_10−k has an extra factor of 2 because the compact space is an orientifold, its volume halved. The gauge coupling is

9

independent of the volume because the fields are localized on the D-brane.

Combining these results with the relations (13.3.22) and (13.3.25) gives, independent of k, the type I relation

g_YM²

κ = 2(2π)^7/2α^ (type I) . (13.3.31) As one final remark, the Born–Infeld form for the gauge action applies by T -duality to the type I theory:

S = − 1

(2πα^)²g²_YM

d¹⁰x Tr [− det(ηµν+ 2πα^F_µν)]^1/2! , (13.3.32) whose normalization is fixed by the quadratic term in F. In the previous chapter we obtained the tree-level string correction (12.4.28) to the type I effective action. If the gauge field lies in an Abelian subgroup, the tensor structure simplifies to

This is indeed the quartic term in the expansion of the Born–Infeld action, as one finds by using tr(x^2k+1) = 0 for antisymmetric x. Note that only when the gauge field can be diagonalized can we give a geometric interpretation to the T -dual configuration and so derive the Born–Infeld form.

In document Polchinski J. String Theory. Vol. 2. Superstring Theory and Beyond (Page 168-174)