PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

4.7. REVERSE APPLICATIONS 83
For the N = 6 model, taking
m 1 = −6, n 1 = 1, m 2 = n 2 = 0, j = 5 , (4.7.6)
we see that ̂Φ indeed has a zero at (4.7.4). Thus by examining the known expression for ̂Φ
near this zero we can determine the half-BPS partition functions of interest. This can be done
in a straightforward manner following the general procedure described in [15, 24].
We note in passing that in an arbitrary Z N model with the set A chosen as
⎛ ⎞ ⎛ ⎞
0
K
Q = ⎜ m/N
⎟
⎝ 0 ⎠ , P = ⎜ J
⎟
⎝ 1 ⎠ , m, K, J ∈ Z , (4.7.7)
−1
0
the walls of marginal stability are controlled by matrices
[44]
( )
a0 b 0
subject to the conditions
c 0 d 0
a 0 d 0 − b 0 c 0 = 1, a 0 , b 0 , c 0 , d 0 ∈ Z, c 0 d 0 ∈ N Z . (4.7.8)
According to our hypothesis this decay will be controlled by a double zero of ̂Φ at
This corresponds to the choice
ˇρc 0 d 0 + ˇσa 0 b 0 + ˇv(a 0 d 0 + b 0 c 0 ) = 0 . (4.7.9)
m 1 = −c 0 d 0 , n 1 = a 0 b 0 , m 2 = n 2 = 0, j = a 0 d 0 + b 0 c 0 , (4.7.10)
in eq.(4.7.5). We now see that the m i ’s, n i ’s and j given in (4.7.10) satisfies all the constraints
mentioned in (4.7.5) ( as a consequence ) of (4.7.8). Thus our proposal that the decay associated
a0 b
with the matrix
0
is always controlled by the zero at (4.7.9) is at least consistent
c 0 d 0
with the locations of the zeroes of ̂Φ for Z N orbifold models.