PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

38 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
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, ˇσ, ˇv + 1) + 1
Φ
2 10 (ˇρ + 3, ˇσ + 3, ˇv + 3) + 1
Φ
4 4 4 10 (ˇρ + 1, ˇσ + 1, ˇv + 3) 4 4 4
2
+
(4.0.23)
Φ 10 (ˇρ + ˇσ + 2ˇv, ˇρ + ˇσ − 2ˇv, ˇσ − ˇρ)
where Φ 10 is the weight 10 Igusa cusp form of Sp(2, Z) describing the inverse partition function
of torsion one dyons. The sum of the first eight terms on the right hand side of (4.0.23)
coincides with the partition function of unit torsion dyons subject to the constraints that
Q 2 + P 2 ± 2Q · P are multiples of 8; the last term is a new addition. We show that (4.0.23)
satisfies all the required consistency conditions. First of all it has the required S-duality
invariance. It also satisfies the wall crossing formulæ at all the walls of marginal stability
at which the original dyon decays into a pair of primitive dyons. It satisfies the constraint
(4.0.19) coming from the requirement that the statistical entropy and the black hole entropy
agrees up to the first non-leading order in inverse powers of charges. Furthermore by taking
an appropriate limit of this formula we can reproduce the known results for torsion two dyons
in gauge theories [57–59, 77, 78].
In the case of torsion two dyons with Q, P both primitive, the vectors Q ± P are not
primitive, but (Q ± P )/2 are primitive vectors [48]. As a result for the decay into
(Q 1 , P 1 ) = (Q − P, 0), (Q 2 , P 2 ) = (P, P ) , (4.0.24)
the charge vector (Q 1 , P 1 ) is not primitive. Computing the jump in the index from (4.0.23) we
find that in this case the change in the index across this wall of marginal stability is given by
{
(
∆d(Q, P ) = (−1) Q 1·P 2 −Q 2·P 1 1 +1 (Q 1 · P 2 − Q 2 · P 1 ) d h (Q 1 , P 1 ) + d h
2 Q 1, 1 )}
2 P 1 d h (Q 2 , P 2 ) .
(4.0.25)
This differs from the formula (4.0.13). A similar modification of the wall crossing formula for
decays into non-primitive states in N = 2 supersymmetric string theories has been suggested
in [54].
There are two more classes of dyons of torsion two, – one where Q is primitive and P is
twice a primitive vector and the other where P is primitive and Q is twice a primitive vector.
The partition functions for these dyons can be recovered from the one given above by S-duality
transformations (Q, P ) → (Q, P − Q) and (Q, P ) → (Q − P, P ) respectively [48]. This amount
to making replacements (ˇρ, ˇσ, ˇv) → (ˇρ, ˇσ + ˇρ + 2ˇv, ˇv + ˇρ) and (ˇρ, ˇσ, ˇv) → (ˇρ + ˇσ + 2ˇv, ˇσ, ˇv + ˇσ)
respectively in eq.(4.0.23).
Although we have presented most of our analysis as a way of extracting information about
the partition function of quarter BPS states from known spectrum of half-BPS states, we could
also use it in the reverse direction. In the final section §4.7 we provide some examples in the
context of Z N orbifold models where the knowledge of the quarter BPS partition function can
be used to compute the spectrum of a certain class of half-BPS states.