D-branes are more than just hyperplanes on which open strings end. They are by themselves dynamical objects that
• gravitate by coupling to closed strings in the NS-NS sector; i.e. they have a mass. • are charged under RR p-form potentials.
By supersymmetry, they interact, of course, with the respective fermionic superpartners. In the sequel we restrict ourselves to describing the bosonic interactions.
There is the following evidence for the claimed dynamical nature of D-branes:
i) In the full quantum theory the worldvolume of a Dp-brane is not static, but it undergoes quantum fluctuations. These brane fluctuations in the normal directions are, in fact, de- scribed by the light open-string excitations.
1We say ’variant’ because we started from Type I, which includes quotienting by worldsheet parity Ω. Thus
The excitations ψn−1/2|0, k i normal to the Dp-brane describe a massless scalar field prop- agating along the Dp-brane. It is interpreted as a modulus field whose VEV determines the position of the brane. Its quantum fluctuations describe brane fluctuations. This is implied by the fact that there is non-zero momentum exchange between the DD string and the D-brane, as can be checked explicitly for the DD-solution.
The necessity to include such quantum fluctuations can be understood by the following analogy with the situation in the closed string sector:
• We start from a theory of closed strings in flat spacetime. This theory gives rise to gravitons in its massless spectrum, which are nothing but the quantum fluctuations of the dynamical metric.
• Similarly we start with an open string sector along an initially rigid hypersurface, which gives rise to scalar fields. Their fluctuations represent the fluctuations of the dynamical Dp-brane.
ii) A very direct argument was achieved in a seminal paper by Polchinski (1996), which com- puted the tree-level exchange of RR and NS-NS closed strings between two Dp-branes by considering instead a 1-loop open diagram.
∼ =
open 1-loop tree-level closed
As in the computation of the string-loop amplitude with NN boundary conditions in all dimensions, we can transform the 1-loop open string channel into a tree-level closed string channel amplitude. This allows one to compare the amplitude for exchange of NS-NS and RR states with the one ontained in an effective action of extended objects with mass and RR-charge.
More precisely, this dynamics is captured by a low-energy effective action for the world- volume of the Dp-brane of the form
Seff = SDBI | {z } coupling to NS-NS + SCS |{z} coupling to R-R . (7.37)
a) The Dirac-Born-Infeld action for the Dp-brane reads SDBI= −Tp Z dp+1ξe−Φ [− det (Gab+ 2πα0Fab+ Bab)] 1 2. (7.38) • Here, Gab= ∂Xµ ∂ξa ∂Xν ∂ξb Gµν(X(ξ)) (7.39)
is the pullback of the ambient space metric onto the brane worldvolume. Note that ξa
a = 0, 1, ..., p represent brane coordinates, while Xµ(ξ) describes the embedding of brane
world-volume in 10D. Thus,R dp+1ξ√− det G
abis the higher-dimensional generalisation
of the Nambu-Goto action and appears naturally.
• The factor of e−Φshows that closed strings couple at tree-level to the disk in the open-
closed CFT.
• The field strength of the U(1) gauge field propagating along a single Dp-brane, 2πα0F ab,
appears only in combination with the pullback of the Kalb-Ramond field, Bab=
∂Xµ ∂ξa
∂Xν
∂ξb Bµν(X(ξ)) . (7.40)
As we recall from Assignment 12, only the combination 2πα0Fµν = 2πα0Fµν+ Bµν is
invariant under the closed string U(1) symmetry
δBµν = ∂µξν− ∂νξµ, δAµ = −
1
2πα0ξµ (7.41)
due to the worldsheet coupling i 4πα0 Z Σ d2ξ√hab∂aXµ∂bXνBµν+ i Z ∂Σ dXµAµ. (7.42)
• The coupling strength is governed by the brane tension Tp= 2π `p+1s , `s= 2π √ α0. (7.43)
Note that expanding the square root in the DBI action leads to the kinetic term of Yang-Mills theory plus higher order curvature corrections. These match with an explicit computation of scattering results.
b) Chern-Simons action
So far the massless RR-sector of Type II superstrings contains the following p-forms: Type IIA: C(1), C(3) Type IIB: C(0), C(2), C(4)+.
By Hodge duality in 10 dimensions we can dualise the associated field strengths as
Note that in 10 dimensions the field strengths, not the potentials are dualised. Alternatively, we can dualise the potentials in the 8 transverse dimensions of light-cone quantisation. Recall that it was in this framework that we had found a self-dual 4-form.
In any case, the above argument shows that C(q) and ˜C(8−q) describe the same degrees of freedom. Thus, we can switch to a so-called ”democratic formulation” of Type II supergravity and consider the following field content in the massless RR sector,
Type IIA : C(1), C(3), C(5), C(7),
Type IIB : C(0), C(2), C(4), C(6), C(8). (7.45) Now, a (p + 1)-form couples naturally to the worldvolume of a Dp-brane via
Z
Dp
C(p+1)= Z
dξ0. . . dξpC01...(p+1). (7.46)
Indeed, to lowest order the Chern-Simons coupling is just SCS= −µp
Z
Dp
C(p+1). (7.47) Further curvature terms can be inferred, e.g., by T-duality. The charge of a Dp-brane under C(p+1) is therefore
µp=
2π `p+1s
. (7.48)
This explains the spectrum of D-branes observed at the end of the previous section:
IIB : D(2p + 1) ↔ C(2p+2) p = −1, 0, ..., 4 (7.49) IIA : D(2p) ↔ C(2p+1) p = 0, ..., 4. (7.50)
Only those Dp-branes exist as stable objects which have the matching RR-forms available. E.g. a D7-brane in IIB cannot decay because it carries C(8) charge; in IIA a D7-brane would decay (at least in R1,9). In fact the dynamics between Dp-branes is a rich and exciting topic by itself.
Remarks
• For a Dp-brane of the above type, the tension (mass) and charge coincide:
Tp= µp. (7.51)
Such objects are called BPS because they are extremely with respect to the Bogomolny’i- Prasad-Sommerfeld (BPS) bound
M ≥ Z (7.52)
with Z the charge.
• The description of D-branes with the help of open string+closed string CFT is adequate if gsis small so that a perturbative expansion makes sense. For large gsthe Dp-branes back-
react substantially on the geometry of the ambient spacetime due to their mass. They form so-called black brane solutions in supergravity, which are higher-dimensional generalisations of black hole solutions of 4-dimensional Einstein or Einstein-Maxwell theory. In fact, these solutions had been known entirely form a SUGRA persepctive before it was realised in 1996 by Polchinski that they describe the same objects as the hyperplanes associated with DD boundary conditions.