PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

62 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
Next we turn to the constraints from the wall crossing formula. We begin with the wall
associated with the decay (Q, P ) → (Q, 0)+(0, P ), – this controls the behaviour of ̂Φ at ˇv = 0.
The analysis is straightforward. We note that both electric and magnetic partition functions
involve summing over all possible even Q 2 /2 and P 2 /2 values. An analysis similar to the one
leading to (5.0.19) give
and
Thus we have
⎡
̂Φ(ˇρ, ˇσ, ˇv) ∼ ⎣ˇv s
2
{
η
φ e (ˇσ) −1 = 1 2
φ m (ˇρ) −1 = 1 2
( ) −24 ˇσs
+ η
4
{
{
η
η
( ) −24 ˇσs
+ η
4
( ) −24 ˇρs
+ η
4
( ) } −24 −1 {
ˇσs + 2
η
4
( ) } −24 ˇσs + 2
, (4.5.3.10)
4
( ) } −24 ˇρs + 2
. (4.5.3.11)
4
( ) −24 ˇρs
+ η
4
( ) } ⎤
−24
−1
ˇρs + 2
+ O(ˇv 4
4
s) ⎦ ,
(4.5.3.12)
near ˇv = 0. One can easily verify that the functions φ e (ˇσ) and φ m (ˇρ) transform as modular
forms of weight 12 ( under ) the transformation ( ) ˇσ s → (pˇσ s + q)/(rˇσ s + s) and ˇρ s → (αˇρ s +
p q
α β
β)/(γ ˇρ s +δ) with ∈ Γ
r s
0 (2) and ∈ Γ
γ δ
0 (2). These can be regarded as symplectic
transformations of the form
⎛ ⎞
⎞
α 0 β 0
1 0 0 0
⎜ ⎜⎝ 0 1 0 0 ⎟ ⎜ 0 p 0 q ⎟
acting on (ˇρ s , ˇσ s , ˇv s ).
γ 0 δ 0
0 0 0 1
⎟
⎠ and ⎜
⎝
⎛
0 0 1 0
0 r 0 s
αδ − βγ = 1, ps − qr = 1, p, r, s, α, γ, δ ∈ Z, q, β ∈ 2 Z , (4.5.3.13)
Next we consider the wall associated with the decay (Q, P ) → ((Q − P )/2, (P − Q)/2) +
((Q + P )/2, (Q + P )/2). From (4.3.1) we see that the associated matrix can be taken to be
( ) (
a0 b √2 1 1 ) √2
0
=
. (4.5.3.14)
c 0 d 0 √2
− 1 √
2
1
⎟
⎠ ,
According to (4.3.2) this controls the behaviour of ̂Φ(ˇρ, ˇσ, ˇv) near its zero at
ˇρ − ˇσ = 0 . (4.5.3.15)
Following the procedure outlined in eqs.(4.3.14)-(4.3.22) we can find the coefficient of (ˇρ − ˇσ) 2
in the expression for ̂Φ. One can see from (4.5.3.1) that in this case 1 2 ((Q + P )/2)2 and