PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

44 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
As long as ̂Φ is manifestly S-duality invariant, ı.e. satisfies (4.2.14), (4.2.15), the residues at
these poles will automatically satisfy the wall crossing formula. Thus they do not provide any
new information. However in a generic situation new walls may appear, labelled by fractional
values of a 0 , b 0 , c 0 , d 0 . Also the S-duality group is smaller. As a result not all the walls can
be related to each other by S-duality transformation. It is tempting to speculate that the
jump across any wall of marginal stability associated with the decay (4.3.1) is described by the
residue of the partition function at the pole at (4.3.2). We shall proceed with this assumption
– this will be one of our key postulates. 7
Before we proceed we shall show that this postulate is internally consistent, ı.e. it is
possible to choose C at different points in the moduli space consistent with this postulate.
For this we generalize the contour prescription of [46], assuming that it holds for all N = 4
supersymmetric string theories. Let τ = τ 1 + iτ 2 be the axion-dilaton moduli, M be the usual
r × r symmetric matrix valued moduli satisfying MLM T = L, and
Q 2 R ≡ Q T (M + L)Q, P 2 R ≡ P T (M + L)P, Q R · P R ≡ Q T (M + L)P . (4.3.3)
Then at the point (τ, M) in the space of asymptotic moduli we choose C to be
(
)
|τ| 2
Q 2 R
I(ˇρ) = Λ + √
τ 2 Q
2
R
PR 2 − (Q ,
R · P R ) 2
(
1
I(ˇσ) = Λ +
τ 2
I(ˇv) = −Λ
where Λ is a large positive number. Then on C
I(c 0 d 0 ˇρ + a 0 b 0ˇσ + (a 0 d 0 + b 0 c 0 )ˇv)
{ (
Λ τ 2 + E ) 2
+
2c 0 d 0
= c 0d 0
τ 2
)
PR
2 √
Q
2
R
PR 2 − (Q ,
R · P R ) 2
(
)
τ 1 Q R · P
+ √ R
τ 2 Q
2
R
PR 2 − (Q , (4.3.4)
R · P R ) 2
(
τ 1 − a 0d 0 + b 0 c 0
2c 0 d 0
) 2 )
−
(1 } + E2
, (4.3.5)
4c 2 0d 2 0
where
E = c 0d 0 Q 2 R + a 0b 0 PR 2 − (a 0d 0 + b 0 c 0 )Q R · P
√ R
Q
2
R
PR 2 − (Q . (4.3.6)
R · P R ) 2
7 Of course the translation symmetries (4.2.16) allow us to shift a pole at (4.3.2) to other equivalent locations.
Our postulate asserts that the contribution comes from poles which can be brought to (4.3.2) using the
translation symmetries (4.2.16). In that case we can choose the unit cell over which we carry out the integration
in (4.1.2) in such a way that only the pole at (4.3.2) contributes to the jump in the index across the wall at
(4.3.1). A possible exception to this will be discussed in the paragraphs above eq.(4.3.12) where we address
some subtle issues.