String Theory and M-Theory

604 Black holes in string theory
where p I denote magnetic charges and qI denote electric charges as before.
Moreover, in the conventions that are usually used, the graviphoton field
strength at the horizon takes the value
C 2 W 2 = 256. (11.176)
After taking the corrections into account, it can be shown that the black-hole
entropy is
S = πi
qICX
2
I − p I
CF I + π
2 Im C 3 ∂CF . (11.177)
The first term in this equation agrees with the attractor value S = π|Z⋆| 2
(for G4 = 1) derived in the previous section when one takes account the
rescaling mentioned in the footnote. The second term is a string theory
correction.
The first equation in (11.175) is solved by writing
In order to solve the second equation, we define
Using this definition,
where we have used
CX I = p I + i
π φI . (11.178)
F(φ, p) = −π ImF (p I + i
π φI , 256). (11.179)
qI = 1 ∂
CFI + CF I = − F(φ, p), (11.180)
2
∂φI ∂ i ∂ i ∂
= − . (11.181)
∂φI πC ∂X I I πC ∂X
The homogeneity relation for the prepotential then implies
C∂CF X I , 256
C2
I ∂
= X F − 2F. (11.182)
∂X I
As a result, the corrected entropy can be written in the form
I ∂
S(p, q) = F(φ, p) − φ F(φ, p). (11.183)
∂φI In other words, the entropy of the black hole is the Legendre transform of
F with respect to φ I . So it is more convenient to specify the φ I , which play
the role of chemical potentials, rather than the electric charges qI.