PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

4.3. CONSTRAINTS FROM WALL CROSSING 45
As shown in [44], the right hand side of (4.3.5) vanishes on the wall of marginal stability
associated with the decay given in (4.3.1). Thus it follows from (4.3.5) that as we cross this
wall of marginal stability, the contour (4.3.4) crosses the pole at (4.3.2) in accordance with our
postulate.
This postulate allows us to identify the possible poles of the partition function besides
those related to the ˇv = 0 pole by the S-duality transformation (4.2.5), – they occur at (4.3.2)
for those values of a 0 , b 0 , c 0 and d 0 for which the decay (4.3.1) is consistent with the charge
quantization laws. One can also get information about the residues at these poles since they
are given by the jumps in the index. This jump can be expressed using the wall crossing
formula [44, 49–54] that tells us that as we cross a wall of marginal stability associated with
the decay (Q, P ) → (Q 1 , P 1 ) + (Q 2 , P 2 ) the index jumps by an amount
(−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 ) (4.3.7)
up to a sign, where d h (Q, P ) denotes the index measuring the number of bosonic minus the
number of fermionic half BPS supermultiplets carrying charges (Q, P ). Thus this relates the
residues at the poles of the integrand to the indices of half BPS states.
We shall now study the consequence of (4.3.7) on the residue at the pole (4.3.2). First
let us consider the special case associated with the decay (Q, P ) → (Q, 0) + (0, P ). In this case
the jump in the index is given by
(−1) Q·P +1 Q · P d e (Q) d m (P ) , (4.3.8)
where d e (Q) = d h (Q, 0) is the index of purely electrically charged states and d m (P ) = d h (0, P )
is the index of purely magnetically charged state. This jump is to be accounted for by the
residue of a pole of the integrand at ˇv = 0. The result (4.3.8) is reproduced if near ˇv = 0, ̂Φ
behaves as
̂Φ(ˇρ, ˇσ, ˇv) −1 ∝ {φ m (ˇρ) −1 φ e (ˇσ) −1ˇv −2 + O(ˇv 0 )} , (4.3.9)
where 1/φ m (ˇρ) and 1/φ e (ˇσ) denote respectively the partition functions of purely magnetic and
purely electric states:
d m (P ) = 1 ∫ iM1 +T 1 /2
dˇρ e −iπP 2 ˇρ 1
T 1 iM 1 −T 1 /2 φ m (ˇρ) , d e(Q) = 1 ∫ iM2 +T 2 /2
dˇσ e −iπQ2ˇσ 1
T 2 iM 2 −T 2 /2
φ e (ˇσ) .
(4.3.10)
Substituting (4.3.10) into the integrand in (4.1.2) and picking up the residue from the pole at
ˇv = 0 we get the change in the index to be
(−1) Q·P +1 Q · P d e (Q) d m (P ) , (4.3.11)
in agreement with (4.3.8), provided we choose the constant of proportionality in (4.3.9) appropriately.
Note that the Q · P factor comes from the ˇv derivative of the exponential factor in
(4.1.2) arising due to the double pole of ̂Φ −1 at ˇv = 0.