PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

4.3. CONSTRAINTS FROM WALL CROSSING 43
act linearly on the variables ˇρ, ˇσ and ˇv. In order to show that the function ̂Φ has non-trivial
modular properties we need to find symmetries of ̂Φ which have non-vanishing C. This will
also determine the weight of ̂Φ under the modular transformation. To get a hint about any
possible additional symmetries of ̂Φ we need to make use of the wall crossing formula for the
dyon spectrum of N = 4 supersymmetric string theories. This will be the subject of discussion
in §4.3.
4.3 Constraints from wall crossing
As has already been discussed, the index associated with the quarter BPS dyon spectrum
in N = 4 supersymmetric string theories can undergo discontinuous jumps across walls of
marginal stability. A priori the formula for the dyon spectrum in different domains labelled by
the vector ⃗c could be completely different. However the study of dyon spectrum in a variety
of N = 4 supersymmetric string theories shows that in different domains the index continues
to be given by an expression similar to (4.1.2), the only difference being that the choice of the
3 real dimensional subspace (contour) over which we carry out the integration in the complex
(ˇρ, ˇσ, ˇv) plane is different in different domains. As a result the difference between the indices in
two different domains is given by the sum of residues of the integrand at the poles we encounter
while deforming the contour associated with one domain to the contour associated with another
domain. As a special example of this we can consider the decay (Q, P ) → (Q, 0) + (0, P ). In
all known examples change in the index across this wall of marginal stability is accounted for
by the residue of a double pole of the integrand at ˇv = 0, ı.e. as we cross this particular wall
of marginal stability in the moduli space, the integration contour crosses the pole at ˇv = 0.
Since the change in the index as we cross a given wall can be found using the wall crossing
formula [44,49–54], this provides information on the residue of the integrand at the ˇv = 0 pole.
There are many other possible decays of a quarter BPS state into a pair of half BPS
states. All such decays may be parametrized 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 ) , a 0 d 0 −b 0 c 0 = 1 .
(4.3.1)
a 0 , b 0 , c 0 , d 0 are not necessarily all integers, but must be such that the charges carried by
the decay products belong to the charge lattice. One can try to use the wall crossing formulæ
associated with these decays to further constrain the form of ̂Φ. For unit torsion states in
heterotic string theory on T 6 , a 0 , b 0 , c 0 and d 0 are integers and the decay given in (4.3.1) is
(
a0 b
related to the decay (Q, P ) → (Q, 0)+(0, P ) via an S-duality transformation
0
c 0 d 0
)
. Thus
the change in the index across the wall is controlled by the residue of the partition function at
a new pole that is related to the ˇv = 0 pole by the S-duality transformation (4.2.5). This gives
the location of the pole to be at
ˇρc 0 d 0 + ˇσa 0 b 0 + ˇv(a 0 d 0 + b 0 c 0 ) = 0 . (4.3.2)