PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

1.4. DEGENERACY OF QUARTER-BPS DYONIC BLACK HOLES 7
degeneracy does not depend on the moduli and it jumps by a known amount once the moduli
cross the boundary of the domain (these boundaries are usually called the walls of marginal
stability).Therefore the dependence on such mixed invariants can be determined in terms of
jumps across the walls of marginal stability. This makes the job somewhat easier. One can
calculate the degeneracy for a specific charge vector and then can extend the result to other
charge vectors by duality symmetry. This turns out to be crucial for what we are going to do
in the next few chapters.
We shall now state the result for the degeneracy of quarter-BPS dyonic black holes
carrying charges (Q, P ) [7]. Let d(Q, P ) stand for the degenaracy in a specific domain of the
moduli space. Then,
d(Q, P ) = (−1) Q·P +1 ∫
C
dρdσdve −iπ(σQ2 +ρP 2 +2vQ·P ) 1
Φ 10 (ρ, σ, v)
(1.4.6)
where C is the contour of integration in three complex dimensional space parametrized by (ρ, σ,
v) and
0 ≤ ρ 1 ≤ 1, 0 ≤ σ 1 ≤ 1, 0 ≤ v 1 ≤ 1, ρ 2 = M 1 , σ 2 = M 2 , v 2 = M 3 , (1.4.7)
M 1 , M 2 and M 3 are three real constants. Φ 10 is the unique weight ten Igusa cusp form of
Sp(2, Z) and it is invariant under both T and S duality transformations. The effect of the
asymptotic moduli is contained in the choice of the contour of integration. We have already
seen that the degeneracy in a given domain does not depend on the asymptotic moduli, but
it can jump if we cross a wall of marginal stability. There exists a specific moduli dependent
choice of contour [46] which beautifully captures both the moduli independence in a given
domain and the wall crossing phenomenon. It is given by,
where,
{
ρ 2
|τ| 2
= Λ +
τ 2
{
σ 2
1
= Λ +
τ 2
{
v 2 = −Λ
τ 1
+
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 , (1.4.8)
R · P R ) 2
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 . (1.4.9)
For Λ sufficiently large it gives us the correct contour of integration associated with a
specific value of the moduli M and τ. As long as we are varying the moduli inside a given