PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

34 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
3. Wall crossing formula: Given that the indices in different domains in the moduli space
are given by different choices of the 3-dimensional integration contour in the (ˇρ, ˇσ, ˇv)
space, the jump in the index as we cross a wall of marginal stability must be given by
the residue of the integrand at the pole(s) encountered while deforming one contour to
another. The walls across which the index jumps are the ones associated with decays
into a pair of half-BPS states. 3 We can label the decay products as [44]
(Q, P ) → (a 0 d 0 Q − a 0 b 0 P, c 0 d 0 Q − c 0 b 0 P ) + (−b 0 c 0 Q + a 0 b 0 P, −c 0 d 0 Q + a 0 d 0 P ) , (4.0.9)
where a 0 , b 0 , c 0 and d 0 are normalized so that a 0 d 0 − b 0 c 0 = 1. In a generic situation a 0 ,
b 0 , c 0 and d 0 are not necessarily integers but are constrained by the fact that the final
charges satisfy the charge quantization laws. In all known examples there is a specific
correlation between a wall corresponding to a given decay and the location of the pole
of the integrand that the contour crosses as we cross the wall in the moduli space. The
location of the pole associated with the decay (4.0.9) is given by:
ˇρc 0 d 0 + ˇσa 0 b 0 + ˇv(a 0 d 0 + b 0 c 0 ) = 0 . (4.0.10)
We shall assume that this formula continues to hold in all cases. This then relates the
jump in the index across a given wall of marginal stability to the residue of the partition
function at a specific pole. An explicit choice of moduli dependent contour that satisfies
this requirement can be found by generalizing the result of Cheng and Verlinde [46] to
generic quarter BPS dyons in generic N = 4 supersymmetric string theories:
(
|τ| 2
I(ˇρ) = Λ +
τ 2
(
1
I(ˇσ) = Λ +
τ 2
(
τ 1
I(ˇv) = −Λ +
τ 2
)
Q 2 R
√
Q
2
R
PR 2 − (Q ,
R · P R ) 2
)
PR
2 √
Q
2
R
PR 2 − (Q ,
R · P R ) 2
)
Q R · P
√ R
Q
2
R
PR 2 − (Q , (4.0.11)
R · P R ) 2
3 For a certain class of dyons kinematics allows decay into a pair of quarter BPS states or a half BPS and a
quarter BPS states on a codimension 1 subspace of the moduli space. These correspond to decays of the form
(Q, P ) → (αQ+βP, γQ+δP )+((1−α)Q−βP, −γQ+(1−δ)P ) with some of the α, β, γ, δ fractional so that we
can have 0 < (αδ − βγ) < 1 and 0 ≤ ((1 − α)(1 − δ) − βγ < 1 [42,43]. However a naive counting of the number
of fermion zero modes on a half BPS - quarter BPS and quarter BPS - quarter BPS combination suggests that
there are additional fermion zero modes besides the ones associated with the broken supersymmetry generators.
This makes the index associated with such a configuration vanish. Although a rigorous analysis of this system
is lacking at present, we shall proceed with the assumption that the result is valid so that such decays do not
change the index. Otherwise the dyon partition function will have additional poles associated with the jump
in the index across these additional walls of marginal stability. We wish to thank F. Denef for a discussion on
this point.