In this section we consider D-brane solutions in supergravity. In section 2.2.5 we have seen that R-R fields are sourced by D-branes. From table 2.3 we know that type IIB supergravity contains R-R tensor fields of rank 0, 2 and 4 and their magnetic duals, a field of rank 8, 6 and 4. Accordingly, the possible D-branes are D(−1)-, D1-,
D3-, D5- and D7-branes. In addition, the NS-NS fieldB2 can also source a 1-brane which is just the fundamental stringF1. Its magnetic dual is called NS5-brane. We
summarize the possiblep-branes in type IIB supergravity in table 2.5.
Since the D-branes have tension and, thus, are massive, they contribute to the energy-momentum tensor and appear in Einstein’s equations which determine the
2.2. Superstring Theory 35
spacetime metric. So in addition to a non-zero flux of the field strength, we expect the spacetime to be curved if we include these branes.
Let us now calculate the metric and the field strength for a Dp-brane. First
we describe the geometry of Dp-brane solutions. It has a (p+ 1)-dimensional flat
hypersurface which is Poincar´e invariant. The transverse space is (9−p)-dimensional
and rotational invariant. Therefore, Dp-branes are solutions with a symmetry group Rp+1×SO(1, p)×SO(9−p). It is convenient to denote the coordinates as follows
coordinates parallel to the brane: xα α= 0, . . . , p
coordinates orthogonal to the brane: yA =xp+A A= 1, . . . ,9−p . (2.76)
Due to the Poincar´e invariance in (p+ 1) dimensions, the metric in the directions
parallel to the brane has to be a rescaling of the flat Minkowski metric, while rota- tional invariance in the transverse space forces the metric in those directions to be a rescaling of the flat Euclidean metric. Furthermore, the metric rescaling factors must be independent of the coordinates parallel to the brane, xα forα = 0, . . . , p.
Substituting an ansatz with the above restrictions into the field equations, we find that the solution can be expressed in terms of a single function H,
ds2=H(y)−12dxαdxα+H(y) 1
2dy2 and eΦ =H(y) 3−p
4 , (2.77)
where the function H must be harmonic with respect to y. Using the rotational symmetry SO(9−p) in the transverse space and the fact that the metric should
become flat as y=|y| → ∞, the most general solution becomes
H(y) = 1 +R
7−p
y7−p . (2.78)
The factorRhas dimension length and must therefore be proportional to the square
root of the inverse string tension√α0 sinceα0 is the only dimensionful parameter of
the theory. ForNc coincident Dp-branes the exact relation is given by
R7−p=Ncgs(4π) 5−p 2 Γ 7 −p 2 α07−2p. (2.79)
D3-branes are worth to consider in more detail since they have a (3 + 1)-dimen-
sional worldvolume as the observed universe surrounding us. Let us now look at the result for D3-branes. We obtain
ds2=H(y)−12dxµdxµ+H(y)12 dy2+y2dΩ2 5 , (2.80) with H(y) = 1 +R 4 y4 , R 4= 4πN cgsα02. (2.81)
36 Chapter 2. Introduction to String Theory
Furthermore, we find that the D3-brane solution has constant axion and dilaton fields, the two-formsB2 and C2 are zero and the field strength of the four-formC4 is given by
F5 = (1 +∗)dx0∧dx1∧dx2∧dx3∧dH−1. (2.82) Since we consider here the supergravity approximation of string theory we have to determine the parameter region in which this approximation is valid. For this purpose, we compare the string lengthls∝
√
α0 with the radius R of the D3-brane
solution. Their ratio is given by
R4
α02 = 4πgsNc. (2.83)
ForgsNc1, the radiusR is much smaller than the string lengthls and, thus, full string theory has to be considered in this region, while forgsNc1, the radiusRis much bigger than the string lengthls and the supergravity approximation is valid. In the following we only consider the regiongsNc1.
To study this geometry further, we take two different limits into account. For
y R, we recover flat space R1,9 and F
5 = 0. Next we consider the near-horizon regiony R. Its geometry is often denoted as the throat. In this limit the solution
becomes ds2 =R2 d xµdxµ+ dz2 z2 + dΩ 2 5 , F5 =−4 R4 z5(1 +∗)dx 0∧dx1∧dx2∧dx3∧dz. (2.84) where we use the coordinate z = R2/y. The first term is the metric of a five- dimensionalAnti-de Sitter space AdS5 while the second term is the metric of a five sphereS5. In conclusion, the geometry close to a stack of D3-branes, i. e. y R,
may be summarized asAdS5×S5 where both components have the identical radius
R.