PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

4.3. CONSTRAINTS FROM WALL CROSSING 47
here
(Q 1 , P 1 ) = (a 0 d 0 Q−a 0 b 0 P, c 0 d 0 Q−c 0 b 0 P ), (Q 2 , P 2 ) = (−b 0 c 0 Q+a 0 b 0 P, −c 0 d 0 Q+a 0 d 0 P ) ,
(4.3.12)
we have
Q 1 · P 2 − Q 2 · P 1 = −Q 2 c 0 d 0 − P 2 a 0 b 0 + Q · P (a 0 d 0 + b 0 c 0 ) . (4.3.13)
Let us now make a change of variables
ˇρ ′ = d 2 0 ˇρ + b 2 0ˇσ + 2b 0 d 0ˇv, ˇσ ′ = c 2 0 ˇρ + a 2 0ˇσ + 2a 0 c 0ˇv, ˇv ′ = c 0 d 0 ˇρ + a 0 b 0ˇσ + (a 0 d 0 + b 0 c 0 )ˇv ,
(4.3.14)
and define
Q ′ = d 0 Q − b 0 P, P ′ = −c 0 Q + a 0 P . (4.3.15)
Under this change of variables
dˇρ ∧ dˇσ ∧ dˇv = dˇρ ′ ∧ dˇσ ′ ∧ dˇv ′ , (4.3.16)
Q 1 · P 2 − Q 2 · P 1 = Q ′ · P ′ , (4.3.17)
(Q 1 , P 1 ) = (a 0 Q ′ , c 0 Q ′ ), (Q 2 , P 2 ) = (b 0 P ′ , d 0 P ′ ) , (4.3.18)
and
1
2 ˇρP 2 + 1 2 ˇσQ2 + ˇvQ · P = 1 2 ˇρ′ P ′2 + 1 2 ˇσ′ Q ′2 + ˇv ′ Q ′ · P ′ . (4.3.19)
Thus the jump in the index given in (4.3.7) can be expressed as
(−1) Q′·P ′ +1 Q ′ · P ′ d h (a 0 Q ′ , c 0 Q ′ )d h (b 0 P ′ , d 0 P ′ ) . (4.3.20)
Furthermore in these variables the pole at (4.3.2) is at ˇv ′ = 0. Thus we can identify (4.3.20)
with the residue of the integrand from ˇv ′ = 0. Using (4.3.16), (4.3.19) the latter may be
expressed as
∫
(−1) Q·P +1 dˇρ ′ dˇσ ′ dˇv ′ e iπ(ˇρ′ P ′2 +ˇσ ′ Q ′2 +2ˇv ′ Q ′·P ′ ) 1
̂Φ(ˇρ, ˇσ, ˇv) , (4.3.21)
where the integration contour is around ˇv ′ = 0. We now note that this result can be reproduced
if we assume that near the pole (4.3.2) the partition function behaves as 8
̂Φ(ˇρ, ˇσ, ˇv) −1 ∝ {φ e (ˇσ ′ ; a 0 , c 0 ) −1 φ m (ˇρ ′ ; b 0 , d 0 ) −1ˇv ′−2 + O(ˇv ′0 )} , (4.3.22)
where 1/φ e,m (τ; k, l) denote the partition functions of half BPS dyons in the set B such that
d h (a 0 Q ′ , c 0 Q ′ ) = 1 T
∫ iM+T/2
iM−T/2
d h (b 0 P ′ , d 0 P ′ ) = 1 T ′ ∫ iM+T ′ /2
iM−T ′ /2
dτ e −iπQ′2 τ 1
φ e (τ; a 0 , c 0 ) ,
dτ e −iπP ′2 τ 1
φ m (τ; b 0 , d 0 ) . (4.3.23)
8 This formula suffers from the same type of subtleties described below eq.(4.3.11) with (Q, P ) replaced by
(Q ′ , P ′ ) and (ˇρ, ˇσ, ˇv) replaced by (ˇρ ′ , ˇσ ′ , ˇv ′ ).