PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

70 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
Now suppose b in (4.5.5.6) is even. Then we can apply a T-duality transformation on the
charge vector given in (4.5.5.6) with the matrix
⎛
⎞ ⎛
⎞
1 l 0 d −2c
⎜ 1 −l 0
⎟ ⎜ −b/2 a
⎟
⎝ 0 1 ⎠ ⎝
a b/2 ⎠ , l 0 ≡ 1 2 bd(2K + 1) − c {(2m + 1)a + 2bJ} .
2
0 1
2c d
(4.5.5.8)
It is straightfoward to verify that this brings (4.5.5.6) back to the set A consisting of pairs of
charge vectors of the form given in (4.5.5.1). ( This)
shows that a sufficient condition for (4.5.5.6)
a b
to lie in the set B is to have b even, ı.e. ∈ Γ(2). Using (4.2.2) we can also see that
c d
this condition is necessary since acting on a pair (Q, P ) with Q 2 /2 odd, P 2 /2 odd and Q · P
even, an S-duality transformation produces a (Q ′ , P ′ ) with odd Q ′2 /2 only if b is even.
Thus we identify the subgroup Γ(2) of the S-duality group Γ 0 (2) as the symmetry of
the set B. The overall scaling of Q and P does not change the symmetry group. Thus the
quarter BPS dyon partition function associated with the set B must be invariant under the
Γ(2) S-duality symmetry. This in turn corresponds to symplectic transformations of the form
⎛
⎜
⎝
d b 0 0
c a 0 0
0 0 a −c
0 0 −b d
acting on (ˇρ, ˇσ, ˇv) and also on (ˇρ s , ˇσ s , ˇv s ).
⎞
⎟
⎠ , ad − bc = 1, a, d ∈ Z, b, c ∈ 2 Z , (4.5.5.9)
Next we turn to the analysis of the constraints from wall crossing. First consider the wall
corresponding to the decay (Q, P ) → (Q, 0) + (0, P ), and examine whether there are subtleties
of kind mentioned below eq.(4.3.11) in applying eqs.(4.3.9), (4.3.10). For this we note that
here Q 2 = −2(2m + 1) and P 2 = 2(2K + 1) are uncorrelated. For a given Q 2 = −2(2m + 1)
there is a unique charge vector in the list given in (4.5.5.1). On the other hand even though
for a given P 2 = 2(2K + 1) there is an infinite family of P labelled by J, they are all related
⎛
⎞
⎛
1 0 0 −J
by the T-duality transformation matrix ⎜ 0 1 J 0
⎟
⎝ 0 0 1 0 ⎠ to the vector ⎜
⎝
0 0 0 1
there are no subtleties of the kind mentioned below (4.3.11) and we have
2K + 1
0
1
0
⎞
⎟
⎠ . Thus
̂Φ(ˇρ, ˇσ, ˇv) ∝ {ˇv 2 φ m (ˇρ)φ e (ˇσ) + O(ˇv 4 ) } for ˇv ≃ 0 . (4.5.5.10)
The magnetic partition function is obtained from (4.5.4.4) by projection to odd values of P 2 /2
followed by ˇρ → ˇρ s /2 replacement. This gives
φ m (ˇρ) −1 = 1 2
{
η(ˇρs /2) −8 η(ˇρ s ) −8 − η((ˇρ s + 1)/2) −8 η(ˇρ s ) −8} . (4.5.5.11)