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PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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4.6. A PROPOSAL FOR THE PARTITION FUNCTION OF DYONS OF TORSION TWO73<br />

Thus (4.4.6) takes the fom<br />

̂Φ(ˇρ, ˇσ, ˇv) ∝ (2v − ρ − σ) 6 {v 2 η(ρ) 8 η(2ρ) 8 η(σ) 8 η(2σ) 8 + O(v 4 )} , (4.5.5.19)<br />

where (ρ, σ, v) and (ˇρ, ˇσ, ˇv) are related via (4.4.7).<br />

4.6 A proposal for the partition function of dyons of<br />

torsion two<br />

In this section we shall consider the set of dyons described in §4.5.3, carrying charge vectors<br />

(Q, P ) with torsion 2, Q, P primitive and Q 2 /2, P 2 /2 even, and propose a form of the partition<br />

function that satisfies all the constraints derived in §4.5.3. The proposed form of the partition<br />

function is<br />

1<br />

̂Φ(ˇρ, ˇσ, ˇv)<br />

[<br />

= 1 1<br />

16 Φ 10 (ˇρ, ˇσ, ˇv) + 1<br />

Φ 10 (ˇρ, ˇσ + 1, ˇv) + 1<br />

Φ<br />

2 10 (ˇρ + 1 , ˇσ, ˇv)<br />

2<br />

1<br />

+<br />

Φ 10 (ˇρ + 1, ˇσ + 1, ˇv) + 1<br />

Φ<br />

2 2 10 (ˇρ + 1, ˇσ + 1, ˇv + 1)<br />

4 4 4<br />

1<br />

+<br />

Φ 10 (ˇρ + 1, ˇσ + 3, ˇv + 1) + 1<br />

Φ<br />

4 4 4 10 (ˇρ + 3, ˇσ + 1, ˇv + 1)<br />

4 4 4<br />

1<br />

+<br />

Φ 10 (ˇρ + 3, ˇσ + 3, ˇv + 1) + 1<br />

Φ<br />

4 4 4 10 (ˇρ + 1, ˇσ + 1, ˇv + 1)<br />

2 2 2<br />

1<br />

+<br />

Φ 10 (ˇρ + 1, ˇσ, ˇv + 1) + 1<br />

Φ<br />

2 2 10 (ˇρ, ˇσ + 1, ˇv + 1) + 1<br />

Φ<br />

2 2 10 (ˇρ, ˇσ, ˇv + 1)<br />

2<br />

1<br />

+<br />

Φ 10 (ˇρ + 3, ˇσ + 3, ˇv + 3) + 1<br />

Φ<br />

4 4 4 10 (ˇρ + 3, ˇσ + 1, ˇv + 3)<br />

4 4 4<br />

]<br />

1<br />

+<br />

Φ 10 (ˇρ + 1, ˇσ + 3, ˇv + 3) + 1<br />

Φ<br />

4 4 4 10 (ˇρ + 1, ˇσ + 1, ˇv + 3) 4 4 4<br />

[<br />

1<br />

+<br />

Φ 10 (ˇρ + ˇσ + 2ˇv, ˇρ + ˇσ − 2ˇv, ˇσ − ˇρ)<br />

]<br />

1<br />

+<br />

Φ 10 (ˇρ + ˇσ + 2ˇv + 1, ˇρ + ˇσ − 2ˇv + 1, ˇσ − ˇρ + 1) . (4.6.1)<br />

2 2 2<br />

The index d(Q, P ) is computed from this partition function using the formula<br />

d(Q, P ) = 1 ∫<br />

(−1)Q·P +1 dˇρ s dˇσ s dˇv s e −i π 4 (ˇσsQ2 +ˇρ sP 2 +2ˇv sQ·P ) 1<br />

4 ̂Φ(ˇρ, ˇσ, ˇv) ,<br />

C<br />

(ˇρ s , ˇσ s , ˇv s ) ≡ (4ˇρ, 4ˇσ, 4ˇv) , (4.6.2)

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