PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

42 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
Comparing (4.2.10) and (4.2.7) we get
̂Φ(ˇρ, ˇσ, ˇv) = ̂Φ(ˇρ ′′ , ˇσ ′′ , ˇv ′′ ) . (4.2.11)
For future reference we shall rewrite the transformation laws (4.2.5) in a suggestive form.
We define
( )
ˇρ ˇv ˇΩ = . (4.2.12)
ˇv ˇσ
Then the transformations (4.2.5) may be written as
where A, B, C and D are 2 × 2 matrices, given by
⎛
( )
A B
= ⎜
C D ⎝
ˇΩ = (AˇΩ ′′ + B)(C ˇΩ ′′ + D) −1 , (4.2.13)
d b 0 0
c a 0 0
0 0 a −c
0 0 −b d
⎞
⎟
⎠ . (4.2.14)
Eq.(4.2.11) now gives (after replacing the dummy variable ˇΩ ′′ by ˇΩ on both sides),
̂Φ((AˇΩ + B)(C ˇΩ + D) −1 ) = det(C ˇΩ + D) k ̂Φ(ˇΩ) , (4.2.15)
for A, B, C, D given in (4.2.14). Here k is an arbitrary number. Since det(CΩ + D) = 1, we
cannot yet ascertain the value of k.
To this we can also append the translational symmetries of ̂Φ:
̂Φ(ˇρ, ˇσ, ˇv) = ̂Φ(ˇρ + a 1 , ˇσ + a 2 , ˇv + a 3 ) , (4.2.16)
where a i ’s are integer multiples of the T i ’s. It is convenient, although not necessary, to work
with appropriately rescaled Q and/or P so that the T i ’s and hence the a i ’s are integers. This
symmetry can also be rewritten as (4.2.15) with the choice
⎛
⎞
( )
1 0 a 1 a 3
A B
= ⎜ 0 1 a 3 a 2
⎟
C D ⎝ 0 0 1 0 ⎠ . (4.2.17)
0 0 0 1
Again since det(C ˇΩ + D) = 1 the choice of k is arbitrary.
The alert reader would have noticed that although we have expressed the consequences of
S-duality invariance and charge quantization conditions as symmetries of the function ̂Φ under
a symplectic transformation, the symplectic transformations arising this way are trivial, – for
all the transformations arising this way the matrix C vanishes and hence the transformations