PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

48 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
The integration over τ run parallel to the real axis over unit period with the imaginary part
fixed at some large positive value M. Substituting (4.3.22) into (4.3.21) and picking up the
residue from the pole at ˇv ′ = 0 we get the change in the index to be
in agreement with (4.3.20).
(−1) Q′·P ′ +1 Q ′ · P ′ d h (a 0 Q ′ , c 0 Q ′ )d h (b 0 P ′ , d 0 P ′ ) , (4.3.24)
To summarize, (4.3.2) gives us the locations of the zeroes of ̂Φ, whereas eq.(4.3.9) and
more generally (4.3.22) give us information about the behaviour of ̂Φ near this zero. We
shall now show that these results suggest additional symmetries of ̂Φ of the type described
in (4.2.15). Typically in any theory the partition functions of half BPS states have modular
properties. Let us for definiteness consider the decay (Q, P ) → (Q, 0) + (0, P ). In this case the
functions φ m (ˇρ) and φ e (ˇσ) transform as modular forms of a subgroup of SL(2, Z) since they
arise from quantization of a fundamental string or a dual magnetic string. These relations take
the form
φ m ((αˇρ + β)(γ ˇρ + δ) −1 ) = (γ ˇρ + δ) k+2 φ m (ˇρ), φ e ((pˇσ + q)(rˇσ + s) −1 ) = (rˇσ + s) k+2 φ e (ˇσ) ,
( ) ( ) (4.3.25)
α β p q
where k is an integer specific to the theory under study, and and belong to
γ δ r s
appropriate subgroups of SL(2, Z). Given that φ m and φ e have these symmetries, we conclude
from (4.3.9) that near ˇv = 0, ̂Φ also has some additional symmetries. Even though there is no
guarantee that these will be symmetries of the full quarter BPS partition function, one could
hope that some part of these do lift to symmetries of the partition function and hence of ̂Φ.
Those which do can be represented by symplectic transformations of the type (4.2.15) with
⎛
⎞
⎛
⎞
( )
α 0 β 0
( )
1 0 0 0
A B
= ⎜ 0 1 0 0
⎟ A B
C D ⎝ γ 0 δ 0 ⎠ and = ⎜ 0 p 0 q
⎟
C D ⎝ 0 0 1 0 ⎠ . (4.3.26)
0 0 0 1
0 r 0 s
The first transformation generates
while the second transformation generates
ˇρ → αˇρ + β
γˇv2
ˇv
, ˇσ → ˇσ − , ˇv →
γ ˇρ + δ γ ˇρ + δ γ ˇρ + δ , (4.3.27)
ˇρ → ˇρ −
rˇv2
rˇσ + s ,
pˇσ + q ˇv
ˇσ → , ˇv →
rˇσ + s rˇσ + s . (4.3.28)
Both transformations leave the ˇv = 0 surface invariant. Furthermore applying these transformations
on (4.2.15) and using (4.3.9) near ˇv = 0 we generate the transformation laws (4.3.25).