String Theory and M-Theory

616 Gauge theory/string theory dualities
where now dx · dx is the six-dimensional Lorentz metric, and dy · dy =
dr2 + r2dΩ2 4 is the five-dimensional Euclidean metric. As before, the powers
of H are chosen such that a supersymmetric solution is obtained if H solves
Laplace’s equation (this time in five dimensions), so that
where
The four-form flux in this case is magnetic
H = 1 + r3 5
, (12.9)
r3 r 3 5 = πN5ℓ 3 p. (12.10)
F4 = ⋆ dx 0 ∧ dx 1 ∧ . . . ∧ dx 5 ∧ dH −1 , (12.11)
as expected for the black M5-brane solution.
Near-horizon limits
The extremal M2-brane solution has a horizon at r = 0. Let us write the
perpendicular part of the metric in spherical coordinates
dy · dy = dr 2 + r 2 dΩ 2 7. (12.12)
Then as r → 0, the coefficient of dΩ2 7 has a finite limit
r 2 H 1/3 → r 2 2. (12.13)
Therefore, r2 is the radius of horizon, which in this case has topology S 7 ס 2
times a null line. The 11-dimensional near-horizon geometry is
ds 2 ∼ (r/r2) 4 dx · dx + (r2/r) 2 dr 2 + r 2 2dΩ 2 7. (12.14)
The first two terms describe four-dimensional anti-de Sitter space, so altogether
the near-horizon geometry of this extremal black M2-brane is AdS4 ×
S 7 . 5
Anti-de Sitter space in (d + 1) dimensions
To understand the near-horizon geometry, let us describe (d+1)-dimensional
anti-de Sitter space (AdSd+1) of radius R by the metric
5 You are asked to construct this solution in Problem 8.2.
ds 2 2 dx · dx + dz2
= R
z2 , (12.15)