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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 TWO79<br />

(4.6.8). One finds that the only term that has a zero at the desired location is the first term<br />

inside the first square bracket in (4.6.1). Furthermore this term is proportional to the dyon<br />

partition function 1/Φ 10 (ˇρ, ˇσ, ˇv) of the unit torsion states discussed in §4.5.1. Thus this term<br />

clearly will have the desired factorization property given in (4.5.3.25).<br />

Our proposal for the dyon partition function can be easily generalized to the torsion<br />

2, primitive Q, P and odd Q 2 /2, P 2 /2 dyons discussed at the end of §4.5.3. This requires<br />

changing the signs of appropriate terms in (4.6.1) so that the partition function is odd under<br />

ˇρ → ˇρ + 1 and also under ˇσ → ˇσ + 1 . The result is<br />

2 2<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) . (4.6.32)<br />

2 2 2<br />

This together with (4.6.1) exhausts all the dyons of torsion two with Q, P primitive since there<br />

are no dyons of this type with Q 2 /2 even, P 2 /2 odd or vice versa. To see this we note that<br />

since (Q ± P ) are 2× primitive vectors, (Q ± P ) 2 /2 must be multiples of four. Taking the sum<br />

and difference we find that (Q 2 + P 2 )/2 and Q · P must be even.<br />

Since (4.6.1) and (4.6.32) contains information about all the torsion two dyons with<br />

primitive (Q, P ), the full partition function for such dyons in obtained by taking the sum of<br />

these two functions. This gives the result quoted in (4.0.23).<br />

Given this result on torsion two dyons in string theory we can go to appropriate gauge<br />

theory limit to extract information about torsion two dyons in gauge theories as in [63,68]. For<br />

simplicity we shall consider SU(3) gauge theories. If we denote by α 1 and α 2 the two simple<br />

roots of SU(3), then, since the metric L reduces to the negative of the Cartan metric of the

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