PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

equiring that we reproduce the black hole entropy π √ Q 2 P 2 − (Q · P ) 2 that arises in the
supergravity approximation one finds that ̂Φ is required to have a zero at [7,8,12,15,24]
37
ˇρˇσ − ˇv 2 + ˇv = 0 . (4.0.18)
In order to find the behaviour of ̂Φ near this zero one needs to calculate the first nonleading
correction to the black hole entropy and compare this with the first non-leading
correction to the formula for the index. In general the former requires the knowledge
of the complete set of four derivative terms in the effective action, but in all known
examples one can reproduce the answer for the index just by taking into account the
effect of the Gauss-Bonnet term in the action. If we assume that this continues to hold
in general then by matching the first non-leading corrections on both sides one can relate
the behaviour of ̂Φ near (4.0.18) to the coefficient of the Gauss-Bonnet term. The result
is
̂Φ(ˇρ, ˇσ, ˇv) ∝ (2v − ρ − σ) k {v 2 g(ρ) g(σ) + O(v 4 )} , (4.0.19)
where
ˇρˇσ − ˇv2 ˇρˇσ − (ˇv − 1)2
ρ = , σ = , v = ˇρˇσ − ˇv2 + ˇv
, (4.0.20)
ˇσ
ˇσ
ˇσ
and g(τ) is a modular form of weight k + 2 of the S-duality group, related to the Gauss-
Bonnet term
∫
d 4 x √ − det g φ(a, S) { R µνρσ R µνρσ − 4R µν R µν + R 2} , (4.0.21)
via the relation
φ(a, S) = − 1 ((k + 2) ln S + ln g(a + iS) + ln g(−a + iS)) + constant . (4.0.22)
64π2 Here τ = a + iS is the axion-dilaton modulus.
In §4.5 we apply the considerations described above to several examples. These include
known examples involving unit torsion dyons in heteroric string theory on T 6 and CHL orbifolds
and also some unknown cases like dyons of torsion 2 in heterotic string theory on T 6 (ı.e.
dyons for which gcd(Q ∧ P )=2 [20]) and dyons carrying untwisted sector charges in Z 2 CHL
orbifold [55, 56]. In the latter cases we determine the constraints imposed by the S-duality
invariance and wall crossing formulæ and also try to use the known modular properties of
half-BPS states to guess the symmetry group of the quarter BPS dyon partition function.
In §4.6 we propose a formula for the dyon partitions function of torsion two dyons in
heterotic string theory on T 6 . The formula for the partition function when Q and P are both
primitive but (Q ± P ) are twice primitive vectors is
1
̂Φ(ˇρ, ˇσ, ˇv)
= 1 8
[
1
Φ 10 (ˇρ, ˇσ, ˇv) + 1
Φ 10 (ˇρ + 1 2 , ˇσ + 1 2 , ˇv) + 1
Φ 10 (ˇρ + 1 4 , ˇσ + 1 4 , ˇv + 1 4 )