PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

4.5. EXAMPLES 57
We shall now set aside this result for a while and study the implications of S-duality
symmetry and the wall crossing formula on the partition function. The eventual goal is to
test the conclusions drawn from the general arguments along the lines of §4.2 and §4.3 against
the known results for ̂Φ given above. It follows ( from ) (4.2.2) and (4.5.2.2) that in order that
a b
an S-duality transformation generated by takes an arbitrary element of the set B
c d
to another element of the set B we must have b even. Thus S-duality transformations which
preserve the set B take the form:
Q → Q ′′ = aQ+bP, P → P ′′ = cQ+dP, a, c, d ∈ Z, b ∈ 2 Z, ad−bc = 1 . (4.5.2.17)
On the original variables (ˇρ, ˇσ, ˇv) the associated transformation can be represented by the
symplectic matrix (4.2.14). After conjugation by the matrix (4.5.2.14) we get the symplectic
matrix acting on the rescaled variables (ˇρ, ˇσ s , ˇv s ):
⎛
d ˜b
⎞
0 0
⎜ ˜c a 0 0
⎟
⎝ 0 0 a −˜c ⎠ , a,˜b ≡ b/2, d ∈ Z, ˜c ≡ 2c ∈ 2 Z . (4.5.2.18)
0 0 −˜b d
This clearly has the form given in (4.5.2.16). Also the periodicities along the (ˇρ, ˇσ s , ˇv s ) directions,
as given in (4.5.2.5), are represented by the symplectic transformation
⎛
⎞
1 0 ã 1 ã 3
⎜ 0 1 ã 3 ã 2
⎟
⎝ 0 0 1 0 ⎠ , ã 1 ∈ Z, ã 2 , ã 3 ∈ 2 Z . (4.5.2.19)
0 0 0 1
These also are of the form given in (4.5.2.16).
Next we turn to the information obtained from the wall crossing relations. Consider first
the wall associated with decay (Q, P ) → (Q, 0) + (0, P ), – this controls the behaviour of ̂Φ
near ˇv = 0 via eq.(4.3.9). Since Q 2 = −4m and P 2 = 2K can vary independently inside
the set A, and since any two charge vectors of the same norm can be related by a T-duality
transformation [41], there is no subtlety of the type described below (4.3.11). The inverse of
the magnetic partition function φ m entering (4.3.9) is the same as the one that appeared in
(4.5.1.7):
φ m (ˇρ) = (η(ˇρ)) 24 . (4.5.2.20)
The electric partition function gets modified from the corresponding expression given in (4.5.1.7)
due to the fact that ( we are only including even Q 2 /2 states. As a result the partition function
now becomes 1 η(ˇσ) −24 + η (ˇσ ) )
+ 1 −24
. Replacing ˇσ by ˇσ
2
2
s /4 we get
φ e (ˇσ) −1 = 1 ( ( )) −24 ˇσs
η
+ 1 ( ( ˇσs
η
2 4 2 4 + 1 )) −24
. (4.5.2.21)
2