PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

4.6. A PROPOSAL FOR THE PARTITION FUNCTION OF DYONS OF TORSION TWO73
Thus (4.4.6) takes the fom
̂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)
where (ρ, σ, v) and (ˇρ, ˇσ, ˇv) are related via (4.4.7).
4.6 A proposal for the partition function of dyons of
torsion two
In this section we shall consider the set of dyons described in §4.5.3, carrying charge vectors
(Q, P ) with torsion 2, Q, P primitive and Q 2 /2, P 2 /2 even, and propose a form of the partition
function that satisfies all the constraints derived in §4.5.3. The proposed form of the partition
function is
1
̂Φ(ˇρ, ˇσ, ˇv)
[
= 1 1
16 Φ 10 (ˇρ, ˇσ, ˇv) + 1
Φ 10 (ˇρ, ˇσ + 1, ˇv) + 1
Φ
2 10 (ˇρ + 1 , ˇσ, ˇv)
2
1
+
Φ 10 (ˇρ + 1, ˇσ + 1, ˇv) + 1
Φ
2 2 10 (ˇρ + 1, ˇσ + 1, ˇv + 1)
4 4 4
1
+
Φ 10 (ˇρ + 1, ˇσ + 3, ˇv + 1) + 1
Φ
4 4 4 10 (ˇρ + 3, ˇσ + 1, ˇv + 1)
4 4 4
1
+
Φ 10 (ˇρ + 3, ˇσ + 3, ˇv + 1) + 1
Φ
4 4 4 10 (ˇρ + 1, ˇσ + 1, ˇv + 1)
2 2 2
1
+
Φ 10 (ˇρ + 1, ˇσ, ˇv + 1) + 1
Φ
2 2 10 (ˇρ, ˇσ + 1, ˇv + 1) + 1
Φ
2 2 10 (ˇρ, ˇσ, ˇv + 1)
2
1
+
Φ 10 (ˇρ + 3, ˇσ + 3, ˇv + 3) + 1
Φ
4 4 4 10 (ˇρ + 3, ˇσ + 1, ˇv + 3)
4 4 4
]
1
+
Φ 10 (ˇρ + 1, ˇσ + 3, ˇv + 3) + 1
Φ
4 4 4 10 (ˇρ + 1, ˇσ + 1, ˇv + 3) 4 4 4
[
1
+
Φ 10 (ˇρ + ˇσ + 2ˇv, ˇρ + ˇσ − 2ˇv, ˇσ − ˇρ)
]
1
+
Φ 10 (ˇρ + ˇσ + 2ˇv + 1, ˇρ + ˇσ − 2ˇv + 1, ˇσ − ˇρ + 1) . (4.6.1)
2 2 2
The index d(Q, P ) is computed from this partition function using the formula
d(Q, P ) = 1 ∫
(−1)Q·P +1 dˇρ s dˇσ s dˇv s e −i π 4 (ˇσsQ2 +ˇρ sP 2 +2ˇv sQ·P ) 1
4 ̂Φ(ˇρ, ˇσ, ˇv) ,
C
(ˇρ s , ˇσ s , ˇv s ) ≡ (4ˇρ, 4ˇσ, 4ˇv) , (4.6.2)