PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

78 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
The behaviour of ̂Φ near ˇv s ′ = 0 can be found by the usual procedure and result is
{
̂Φ(ˇρ, ˇσ, ˇv) −1 = − 1 ( ) ˇρ
′ −24 ( ) }
η s ˇρ
′ −24
+ η s + 2
4π2ˇv ′2
s 4
4
{ ( ) ˇσ
′ −24 ( )
× η s ˇσ
′ −24 ( )
+ η s + 1 ˇσ
′ −24 ( ) }
+ η s + 2 ˇσ
′ −24
+ η s + 3
4
4
4
4
{
− 1 ( ) ˇρ
′ −24 ( ) }
η s ˇρ
′ −24
+ η s + 2
)
η(ˇσ
π2ˇv ′ ′2
s 4
4
s) −24 ′0
+ O
(ˇv s (4.6.28)
Note that the first set of terms represent correctly the factorization behaviour given in (4.5.3.21),
but the second set of terms are extra. Thus the wall crossing formula gets modified for the
decay into non-primitive states. Using (4.6.28) we can compute the jump in the index across
the wall
∆d(Q, P ) = 1 ∫ {
iM ′ (−1)Q·P +1 Q ′ · P ′ 1 +1
( ) ˇρ
dˇρ ′ ′ −24 ( ) }
16 iM 1 ′ −1
s η s ˇρ
′ −24
+ η s + 2
e −iπ ˇρ′ sP ′2 /4
4
4
∫ {
iM ′
2 +1/2
( ) ˇσ
dˇσ ′ ′ −24 ( )
iM 2 ′ −1/2
s η s ˇσ
′ −24
+ η s + 1
4
4
( ) ˇσ
′ −24 ( ) }
+η s + 2 ˇσ
′ −24
+ η s + 3
e −iπˇσ′ sQ ′2 /4
Defining
4
+ 1 ∫ {
iM ′ (−1)Q·P +1 Q ′ · P ′ 1 +1
dˇρ ′ s η
4
×
∫ iM ′
2 +1/2
we can express (4.6.29) as
iM ′ 1 −1
4
( ) ˇρ
′ −24
s
+ η
4
( ) }
ˇρ
′ −24
s + 2
4
e −iπ ˇρ′ sP ′2 /4
iM ′ 2 −1/2 dˇσ ′ s η(ˇσ ′ s) −24 e −iπˇσ′ sQ ′2 /4 . (4.6.29)
∆d(Q, P ) = (−1) Q 1·P 2 −Q 2·P 1 +1 (Q 1 · P 2 − Q 2 · P 1 )
(Q 1 , P 1 ) = (Q − P, 0), (Q 2 , P 2 ) = (P, P ) , (4.6.30)
{
( 1
d h (Q 1 , P 1 ) + d h
2 Q 1, 1 )}
2 P 1 d h (Q 2 , P 2 ) .
(4.6.31)
The second term is extra compared to (4.3.7); it represents the effect of non-primitivity of the
final state dyons.
Finally let us turn to the analysis of the black hole entropy. For this we need to identify
the zeroes of ̂Φ at ˇρˇσ − ˇv 2 + ˇv = 0 and show that ̂Φ has the behaviour given in (4.5.3.25)
near this pole. This is easily done using (4.6.1) and the locations of the zeroes of Φ 10 given in