String Theory Demystified

252 String Theory Demystifi ed
Now the calculation of the entropy is actually quite straightforward. From the
metric in Eq. (14.42), there are three radii associated with the horizon. Using
Eq. (14.9) with D = 5 , we can write each of these as
r
mG g
RV m
2 8
16π
s s
= =
i
3Ω
2 5
i
3
(14.44)
where R is the radius of the circular dimension S 1 and V is the volume of the torus.
The individual masses can be calculated from string considerations. The fi rst two
masses are due to winding modes. First, the string winding around radius R gives
m QR 1 = 1 2
(14.45)
For the D5-brane, fi rst we have the winding mode which wraps around the circle
and torus:
m
2
g s s
QRV 5
= 6
g s s
(14.46)
Then we have a third mass, due to the Kaluza-Klein excitation of the D5-brane
along the circular dimension:
Now let’s calculate each of the radii:
r
r
1
2
r
m
3
n
= (14.47)
R
g
RV m
4
3
g Q
s s s s
= = 1
V
3
1
(14.48)
4
g s s
= m = g Q
(14.49)
2 s s 5
RV
g
RV m
g
R V n
4
4
s s s s
= = 3
(14.50)