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PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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4.5. EXAMPLES 67<br />

IIB description of the theory [24]. The results are<br />

Eq.(4.3.9) then gives, near ˇv = 0,<br />

φ e (ˇσ) = η(ˇσ) 8 η(ˇσ/2) 8 , φ m (ˇρ) = η(ˇρ) 8 η(2ˇρ) 8 . (4.5.4.4)<br />

̂Φ(ˇρ, ˇσ, ˇv) ∝ {ˇv 2 η(ˇσ) 8 η(ˇσ/2) 8 η(ˇρ) 8 η(2ˇρ) 8 + O(ˇv 4 ) } . (4.5.4.5)<br />

φ e (ˇσ) and φ m (ˇρ) transform as modular forms of weight 8 under<br />

and<br />

ˇσ → pˇσ + q , p, r, s ∈ Z, q ∈ 2 Z, ps − qr = 1 , (4.5.4.6)<br />

rˇσ + s<br />

ˇρ → αˇρ + β , α, β, δ ∈ Z, γ ∈ 2 Z, αδ − βγ = 1 . (4.5.4.7)<br />

γ ˇρ + δ<br />

The corresponding groups are Γ 0 (2) and Γ 0 (2) respectively. Thus from (??), (4.2.17), (4.3.26)<br />

we see that if (4.5.4.6) and (4.5.4.7) lift to symmetries of the full partition function then the<br />

partition function transforms as a modular form of weight 6 under the Sp(2, Z) transformations<br />

of the form<br />

⎛<br />

⎞ ⎛<br />

⎞ ⎛<br />

⎞ ⎛<br />

⎞<br />

d b 0 0 1 0 a 1 a 3 α 0 β 0<br />

1 0 0 0<br />

⎜ c a 0 0<br />

⎟<br />

⎝ 0 0 a −c ⎠ ,<br />

⎜ 0 1 a 3 a 2<br />

⎟<br />

⎝ 0 0 1 0 ⎠ ,<br />

⎜ 0 1 0 0<br />

⎟<br />

⎝ γ 0 δ 0 ⎠ and ⎜ 0 p 0 q<br />

⎟<br />

⎝ 0 0 1 0 ⎠ ,<br />

0 0 −b d 0 0 0 1 0 0 0 1<br />

0 r 0 s<br />

(4.5.4.8)<br />

with<br />

( ) ( ) ( )<br />

a b<br />

α β<br />

p q<br />

∈ Γ<br />

c d 0 (2),<br />

∈ Γ<br />

γ δ 0 (2),<br />

∈ Γ 0 (2),<br />

r s<br />

a 1 , a 3 ∈ Z, a 2 ∈ 2 Z .<br />

(4.5.4.9)<br />

All the Sp(2, Z) matrices in (4.5.4.8) subject to the constraints (4.5.4.9) have the form<br />

⎛<br />

⎞<br />

1 ∗ ∗ ∗<br />

⎜ 0 1 ∗ 0<br />

⎟<br />

⎝ 0 0 1 0 ⎠ mod 2 . (4.5.4.10)<br />

0 ∗ ∗ 1<br />

Furthermore the set of matrices (4.5.4.10) are closed under matrix multiplication. Thus the<br />

group generated by the set of Sp(2, Z) matrices (4.5.4.8) subject to the condition (4.5.4.9) is<br />

contained in the group Ǧ of Sp(2, Z) matrices (4.5.4.10).<br />

All the symmetries listed in (4.5.4.8) are indeed symmetries of the dyon partition function<br />

of this model proposed in [12] and proved in [15]. Furthermore near ˇv = 0 the partition function<br />

is known to have the factorization property given in (4.5.4.5) [44–46]. One question that one

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