PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

4.5. EXAMPLES 51
Here k is the same integer that appeared in (4.2.15) and g(τ) transforms as a modular form
of weight k + 2 under the S-duality group. In a given theory g(τ) can be calculated in string
perturbation theory [61,62]. To the first non-leading order in the inverse power of charges, the
effect of this term is to change the black hole entropy to [24]
S BH = π √ ( √ )
Q 2 P 2 − (Q · P ) 2 + 64 π 2 Q · P
φ
P , Q2 P 2 − (Q · P ) 2
+ · · · (4.4.5)
2 P 2
The analysis of [7,8,12,15,24] shows that this behaviour can be reproduced if we assume that
near the zero at (4.4.2)
̂Φ(ˇρ, ˇσ, ˇv) ∝ (2v − ρ − σ) k {v 2 g(ρ) g(σ) + O(v 4 )} , (4.4.6)
where
ˇρˇσ − ˇv2 ˇρˇσ − (ˇv − 1)2
ρ = , σ = , v = ˇρˇσ − ˇv2 + ˇv
. (4.4.7)
ˇσ
ˇσ
ˇσ
If we assume that eq.(4.4.6) holds in general, then it gives us information about the behaviour
of ̂Φ(ˇρ, ˇσ, ˇv) near the zero at (4.4.2). On the other hand if we can determine ̂Φ from other
considerations then the validity of (4.4.6) would provide further evidence for the postulate that
in N = 4 supersymmetric string theories the Gauss-Bonnet term gives the complete correction
to black hole entropy to first non-leading order.
4.5 Examples
In this section we shall describe several applications of the general procedure described in
§4.1. Some of them will involve known cases and will provide a test for our procedure, while
others will be new examples where we shall derive a set of constraints on certain dyon partition
functions which have not yet been computed from first principles.
4.5.1 Dyons with unit torsion in heterotic string theory on T 6
We consider a dyon of charge (Q, P ) in the heterotic string theory on T 6 . Q and P take
values in the Narain lattice Λ [26, 27]. Let S 1 and ˜S 1 be two circles of T 6 , each labelled by a
coordinate with period 2π and let us denote by n ′ , ñ the momenta along S 1 and ˜S 1 , by −w ′ , − ˜w
the fundamental string winding numbers along S 1 and ˜S 1 , by N ′ , Ñ the Kaluza-Klein monopole
charges associated with S 1 and ˜S 1 , and by −W ′ , −˜W the H-monopole charges associated with
S 1 and ˜S 1 [24]. Then in the four dimensional subspace consisting of charge vectors
⎛ ⎞ ⎛ ⎞
ñ
˜W
Q = ⎜ n ′
⎟
⎝ ˜w ⎠ , P = ⎜ W ′
⎟
⎝ Ñ ⎠ , (4.5.1.1)
w ′ N ′