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PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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

Next we turn to the constraints from the wall crossing formula. We begin with the wall<br />

associated with the decay (Q, P ) → (Q, 0)+(0, P ), – this controls the behaviour of ̂Φ at ˇv = 0.<br />

The analysis is straightforward. We note that both electric and magnetic partition functions<br />

involve summing over all possible even Q 2 /2 and P 2 /2 values. An analysis similar to the one<br />

leading to (5.0.19) give<br />

and<br />

Thus we have<br />

⎡<br />

̂Φ(ˇρ, ˇσ, ˇv) ∼ ⎣ˇv s<br />

2<br />

{<br />

η<br />

φ e (ˇσ) −1 = 1 2<br />

φ m (ˇρ) −1 = 1 2<br />

( ) −24 ˇσs<br />

+ η<br />

4<br />

{<br />

{<br />

η<br />

η<br />

( ) −24 ˇσs<br />

+ η<br />

4<br />

( ) −24 ˇρs<br />

+ η<br />

4<br />

( ) } −24 −1 {<br />

ˇσs + 2<br />

η<br />

4<br />

( ) } −24 ˇσs + 2<br />

, (4.5.3.10)<br />

4<br />

( ) } −24 ˇρs + 2<br />

. (4.5.3.11)<br />

4<br />

( ) −24 ˇρs<br />

+ η<br />

4<br />

( ) } ⎤<br />

−24<br />

−1<br />

ˇρs + 2<br />

+ O(ˇv 4<br />

4<br />

s) ⎦ ,<br />

(4.5.3.12)<br />

near ˇv = 0. One can easily verify that the functions φ e (ˇσ) and φ m (ˇρ) transform as modular<br />

forms of weight 12 ( under ) the transformation ( ) ˇσ s → (pˇσ s + q)/(rˇσ s + s) and ˇρ s → (αˇρ s +<br />

p q<br />

α β<br />

β)/(γ ˇρ s +δ) with ∈ Γ<br />

r s<br />

0 (2) and ∈ Γ<br />

γ δ<br />

0 (2). These can be regarded as symplectic<br />

transformations of the form<br />

⎛ ⎞<br />

⎞<br />

α 0 β 0<br />

1 0 0 0<br />

⎜ ⎜⎝ 0 1 0 0 ⎟ ⎜ 0 p 0 q ⎟<br />

acting on (ˇρ s , ˇσ s , ˇv s ).<br />

γ 0 δ 0<br />

0 0 0 1<br />

⎟<br />

⎠ and ⎜<br />

⎝<br />

⎛<br />

0 0 1 0<br />

0 r 0 s<br />

αδ − βγ = 1, ps − qr = 1, p, r, s, α, γ, δ ∈ Z, q, β ∈ 2 Z , (4.5.3.13)<br />

Next we consider the wall associated with the decay (Q, P ) → ((Q − P )/2, (P − Q)/2) +<br />

((Q + P )/2, (Q + P )/2). From (4.3.1) we see that the associated matrix can be taken to be<br />

( ) (<br />

a0 b √2 1 1 ) √2<br />

0<br />

=<br />

. (4.5.3.14)<br />

c 0 d 0 √2<br />

− 1 √<br />

2<br />

1<br />

⎟<br />

⎠ ,<br />

According to (4.3.2) this controls the behaviour of ̂Φ(ˇρ, ˇσ, ˇv) near its zero at<br />

ˇρ − ˇσ = 0 . (4.5.3.15)<br />

Following the procedure outlined in eqs.(4.3.14)-(4.3.22) we can find the coefficient of (ˇρ − ˇσ) 2<br />

in the expression for ̂Φ. One can see from (4.5.3.1) that in this case 1 2 ((Q + P )/2)2 and

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