String Theory and M-Theory

11.5 The attractor mechanism 587
clear: it can be interpreted as brane-antibrane annihilation. Specifically, for
the set-up in the preceding subsection, where there are both left-moving and
right-moving Kaluza–Klein excitations, they can collide to give a massless
closed-string state, which is then emitted from the black hole. The calculation
has been carried out for ¯n ≪ n with the conclusion that the decay rate
as a function of frequency is
where the temperature is
A
dΓ(ω) =
eω/T d
− 1
4k , (11.108)
(2π) 4
T = 2√¯n . (11.109)
πR
If one considers D1-brane anti-D1-brane or D5-brane/anti-D5-brane annihilations
instead, then a different viewpoint is convenient. When a brane
and an antibrane coincide, their common world volume contains a tachyonic
mode that arises as the lowest mode of the open string that connects the
brane to the antibrane. This tachyon signals an instability of the worldvolume
theory, which results in the emission of closed-string radiation as
in the previous discussion. In fact, one can test this reasoning by using
Witten’s string field theory to describe the open string. Sen has argued persuasively
that this theory gives a potential for the tachyon field, and that the
decay corresponds to sliding down this potential from a local maximum to a
local minimum, that is, tachyon condensation. Furthermore, the value of the
potential at the minimum should be lower than its value at the maximum
by exactly twice the brane tension. Thus the world-volume tachyon rolling
to the minimum of its potential precisely corresponds to brane–antibrane
annihilation. This results in the emission of closed-string quanta. In the
black-hole setting considered here, these quanta comprise the Hawking radiation.
This prediction for the gap between the maximum and minimum
of the potential has been tested numerically in Witten’s bosonic string field
theory, and it has been verified to high precision. Moreover, it has recently
been derived analytically.
11.5 The attractor mechanism
Moduli fields
As has been discussed in previous sections, black holes can appear when
a superstring theory or M-theory is compactified to lower dimensions and
when branes are wrapped on nontrivial cycles of the compact manifold. The