PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

4.6. A PROPOSAL FOR THE PARTITION FUNCTION OF DYONS OF TORSION TWO77
Substituting (4.6.14) into (4.6.17) we get
∆d(Q, P ) = 1 4 (−1)Q·P +1 Q ′ · P ′ ∫ iM ′
1 +1
∫ iM ′
2 +1
iM ′ 1 −1 dˇρ ′ s η(ˇρ ′ s/2) −24 e −iπ ˇρ′ s P ′2 /4
iM ′ −1 dˇσ s ′ η(ˇσ s/2) ′ −24 e −iπˇσ′ s Q ′2/4 . (4.6.19)
On the other hand now the indices of the half-BPS decay products carrying charges
(Q 1 , P 1 ) = ((Q − P )/2, (P − Q)/2), (Q 2 , P 2 ) = ((Q + P )/2, (Q + P )/2) , (4.6.20)
are given by
d h (Q 1 , P 1 ) =
and
∫ iM+1/2
iM−1/2
dτ(η(τ)) −24 e −iπτ((Q−P )/2)2 = 1 2
∫ 2iM+1
2iM−1
dˇσ ′ s η(ˇσ ′ s/2) −24 e −iπˇσ′ s Q ′2 /4 ,
(4.6.21)
d h (Q 2 , P 2 ) =
∫ iM+1/2
iM−1/2
Using these results and the identities
dτ(η(τ)) −24 e −iπτ((Q+P )/2)2 = 1 2
∫ 2iM+1
2iM−1
dˇρ ′ s η(ˇρ ′ s/2) −24 e −iπ ˇρ′ s P ′2 /4 .
(4.6.22)
Q 1 · P 2 − Q 2 · P 1 = Q ′ · P ′ , (−1) Q 1·P 2 −Q 2·P 1
= (−1) (Q−P )2 /2−P 2 +Q·P = (−1) Q·P , (4.6.23)
we can express (4.6.19) as
∆d(Q, P ) = (−1) Q 1·P 2 −Q 2·P 1 +1 (Q 1 · P 2 − Q 2 · P 1 ) d h (Q 1 , P 1 ) d h (Q 2 , P 2 ) , (4.6.24)
in agreement with the wall crossing formula.
Next consider the decay (Q, P ) → (Q − P, 0) + (P, P ). This is controlled by the pole at
ˇσ + ˇv = 0. To analyze this contribution we define
Q ′ = Q − P, P ′ = P (4.6.25)
and
ˇρ ′ s = ˇρ s + ˇσ s + 2ˇv s , ˇσ ′ s = ˇσ s , ˇv ′ s = ˇv s + ˇσ s , (4.6.26)
so that ˇρ s P 2 + ˇσ s Q 2 + 2ˇv s Q · P = ˇρ ′ sP ′2 + ˇσ ′ sQ ′2 + 2ˇv ′ sQ ′ · P ′ . In terms of these variables the
periods are
(ˇρ ′ s, ˇσ ′ s, ˇv ′ s) → (ˇρ ′ s + 2, ˇσ ′ s, ˇv ′ s), (ˇρ ′ s, ˇσ ′ s + 1, ˇv ′ s), (ˇρ ′ s, ˇσ ′ s, ˇv ′ s + 2) . (4.6.27)