String Theory and M-Theory

11.4 Statistical derivation of the entropy 583
manner, but that is left as a homework problem. The derivation was first
given by Strominger and Vafa in the context of type IIB compactifications
on K3×S 1 . The discussion that follows analyzes the somewhat simpler case
of the toroidal compactification described in Section 11.3. The analysis can
be carried out either for the D1-D5-P system or for the S-dual F1-NS5-P
system. For definiteness, the discussion that follows refers to the former
set-up.
The fact that there are Q1 units of charge associated with D1-branes
means that there are Q1 windings of D1-branes around the circle. However,
the way this is achieved has not been specified. The two extreme possibilities
are (1) there are Q1 D1-branes each of which wraps around the circle
once and (2) there is a single D1-brane that wraps around the circle Q1
times. Altogether, the distinct possibilities correspond to the partitions of
Q1. When there is more than one D1-brane, it is important that they form
a bound state in order to give a single black hole. The Q5 units of D5-brane
charge also can be realized in various ways. In all cases, one wants the D1-
D5-P system to form a bound state, so that one ends up with a localized
object in the noncompact dimensions.
The low-energy physics of these bound states is described by an orbifold
conformal field theory that is defined on the circle of radius R. The fields
in the conformal field theory correspond to the zero modes of open strings
that connect the D1-branes to the D5-branes. There are Q1Q5 distinct such
strings, since each strand of D1-branes can connect to each strand of D5branes.
That is the picture locally. However, imagine displacing this (small)
connecting string repeatedly around the circle. If there is a single multiply
wound D1-brane and a single multiply wound D5-brane (along the circle),
and if Q1 and Q5 have no common factors, then one must go around the
circle Q1Q5 times to get back to where one started. Thus, the excitations
of this system are the same as what one gets from having a single string
wound around the circle Q1Q5 times. Since this string is localized in the
noncompact dimensions, the only bosonic zero modes in its world-volume
theory correspond to its position in the four transverse compact dimensions.
Since the system is supersymmetric, there must therefore be four boson and
four fermion zero modes on the string world volume.
The system described above can be represented as an orbifold conformal
field theory that is obtained by taking the tensor product of Q1Q5 theories
describing singly wound strings and then modding out by all of their
(Q1Q5)! permutations. This orbifold theory has many twisted sectors, 15 and
15 They are given by the conjugacy classes of the permutation group SQ1Q5 .