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PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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58 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION<br />

This leads to the following behaviour of ̂Φ near ˇv s = 0:<br />

⎡ { ( ( )) −24 ( (<br />

̂Φ(ˇρ, ˇσ, ˇv) ∝ ⎣ˇv s 2 η(ˇρ) 24 ˇσs<br />

ˇσs<br />

η + η<br />

4<br />

4 + 1 )) } ⎤<br />

−24<br />

−1<br />

+ O(ˇv 4<br />

2<br />

s) ⎦ (4.5.2.22)<br />

̂Φ given in (5.0.27) can be shown to satisfy this property.<br />

φ m (ˇρ) given in (4.5.2.20) transforms as a modular form of weight 12 under<br />

ˇρ → αˇρ + β ( )<br />

α β<br />

γ ˇρ + δ , ∈ SL(2, Z) . (4.5.2.23)<br />

γ δ<br />

On the other hand φ e (ˇσ) given in (4.5.2.21) can be shown to transform as a modular form of<br />

weight 12 under<br />

ˇσ s → pˇσ ( )<br />

s + q p q<br />

rˇσ s + s , ∈ Γ 0 (2) , (4.5.2.24)<br />

r s<br />

ı.e. SL(2, Z) matrices with q even. (4.5.2.23) and (5.0.25) can be represented as symplectic<br />

transformations of (ˇρ, ˇσ s , ˇv s ) generated by the Sp(2, Z) matrices<br />

⎛<br />

⎞ ⎛<br />

⎞<br />

α 0 β 0<br />

1 0 0 0<br />

⎜ 0 1 0 0<br />

⎟<br />

⎝ γ 0 δ 0 ⎠ and ⎜ 0 p 0 q<br />

⎟<br />

⎝ 0 0 1 0 ⎠ , q ∈ 2 Z, α, β, γ, δ, p, r, s ∈ Z . (4.5.2.25)<br />

0 0 0 1<br />

0 r 0 s<br />

We now note that these transformations fall in the class given in (4.5.2.16). Thus in this case<br />

the modular symmetries of the half-BPS partition function associated with pole at ˇv = 0 are<br />

lifted to symmetries of the full partition function.<br />

In this case there is one additional wall which is not related to the wall considered above by<br />

the Γ 0 (2) S-duality transformation (4.5.2.17) acting on the original variables. This corresponds<br />

to the decay (Q, P ) → (Q − P, 0) + (P, P ). Comparing this with (4.3.1) we see that here<br />

( )<br />

a0 b 0<br />

=<br />

c 0 d 0<br />

(<br />

1 1<br />

0 1<br />

)<br />

. (4.5.2.26)<br />

Following (4.3.14), (4.3.15) and the relationship (4.5.2.4) between the original variables and<br />

the rescaled variables we have<br />

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)<br />

Q ′ = Q − P, P ′ = P . (4.5.2.28)<br />

Thus the pole of the partition function is at ˇv s + 1 2 ˇσ s = 0. Furthermore since from the relations<br />

(4.5.2.2) we see that the allowed values of (Q − P ) 2 /2 = J + K − 2m and P 2 /2 = K are

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