String Theory and M-Theory

11.6 Small BPS black holes in four dimensions 601
Counting states
Let us now compute the corrections to Eq. (11.161). The degeneracy dN
denotes the number of ways that the 24 left-moving bosonic oscillators can
give NL = N + 1 units of excitation. This can be encoded in a partition
function
Z(β) = dNe −βN = 16
, (11.162)
∆(q)
where
q = e −β = e 2πiτ , (11.163)
and the factor of 16 is the degeneracy of right-moving ground states. The
factor ∆(q) is related to the Dedekind η function by
∆(q) = η(τ) 24 ∞
= q (1 − q n ) 24 . (11.164)
The large-N degeneracy depends on the value of this function as q → 1 or
β → 0. Under a modular transformation the Dedekind η function transforms
as
As a result,
∆(e −β ) =
n=1
η(−1/τ) = √ −iτη(τ). (11.165)
−12 β
∆(e
2π
−4π2 /β
), (11.166)
which, by using ∆(q) ≈ q for small q, gives the estimate
∆(e −β −12 β
) ≈ e
2π
−4π2 /β
. (11.167)
This result is extremely accurate, since all corrections are exponentially
suppressed.
Now one can compute dN, as in earlier chapters.
dN = 1
2πi
Z(β) dq 1
=
qN+1 2πi
16 dq
. (11.168)
∆(q) qN+1 Using Eq. (11.166), this can be approximated for large N by
where
Îν(z) = 1
2πi
dN ≈ 16 Î13(4π √ N), (11.169)
ε+i∞
(t/2π)
ε−i∞
−ν−1 e t+z2 /4t
dt (11.170)