PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

58 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
This leads to the following behaviour of ̂Φ near ˇv s = 0:
⎡ { ( ( )) −24 ( (
̂Φ(ˇρ, ˇσ, ˇv) ∝ ⎣ˇv s 2 η(ˇρ) 24 ˇσs
ˇσs
η + η
4
4 + 1 )) } ⎤
−24
−1
+ O(ˇv 4
2
s) ⎦ (4.5.2.22)
̂Φ given in (5.0.27) can be shown to satisfy this property.
φ m (ˇρ) given in (4.5.2.20) transforms as a modular form of weight 12 under
ˇρ → αˇρ + β ( )
α β
γ ˇρ + δ , ∈ SL(2, Z) . (4.5.2.23)
γ δ
On the other hand φ e (ˇσ) given in (4.5.2.21) can be shown to transform as a modular form of
weight 12 under
ˇσ s → pˇσ ( )
s + q p q
rˇσ s + s , ∈ Γ 0 (2) , (4.5.2.24)
r s
ı.e. SL(2, Z) matrices with q even. (4.5.2.23) and (5.0.25) can be represented as symplectic
transformations of (ˇρ, ˇσ s , ˇv s ) generated by the Sp(2, Z) matrices
⎛
⎞ ⎛
⎞
α 0 β 0
1 0 0 0
⎜ 0 1 0 0
⎟
⎝ γ 0 δ 0 ⎠ and ⎜ 0 p 0 q
⎟
⎝ 0 0 1 0 ⎠ , q ∈ 2 Z, α, β, γ, δ, p, r, s ∈ Z . (4.5.2.25)
0 0 0 1
0 r 0 s
We now note that these transformations fall in the class given in (4.5.2.16). Thus in this case
the modular symmetries of the half-BPS partition function associated with pole at ˇv = 0 are
lifted to symmetries of the full partition function.
In this case there is one additional wall which is not related to the wall considered above by
the Γ 0 (2) S-duality transformation (4.5.2.17) acting on the original variables. This corresponds
to the decay (Q, P ) → (Q − P, 0) + (P, P ). Comparing this with (4.3.1) we see that here
( )
a0 b 0
=
c 0 d 0
(
1 1
0 1
)
. (4.5.2.26)
Following (4.3.14), (4.3.15) and the relationship (4.5.2.4) between the original variables and
the rescaled variables we have
ˇρ ′ = ˇρ + 1 4 ˇσ s + ˇv s , ˇσ ′ = 1 4 ˇσ s, ˇv ′ = 1 2 ˇv s + 1 4 ˇσ s , (4.5.2.27)
Q ′ = Q − P, P ′ = P . (4.5.2.28)
Thus the pole of the partition function is at ˇv s + 1 2 ˇσ s = 0. Furthermore since from the relations
(4.5.2.2) we see that the allowed values of (Q − P ) 2 /2 = J + K − 2m and P 2 /2 = K are