PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

68 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
can ask is: do the matrices given in (4.5.4.8) generate the full symmetry group of the partition
function (which is known in this case)? It turns out that the answer is no. This group does
not include the matrix
⎛
⎞
1 0 0 0
⎜ 0 1 0 0
⎟
⎝ 0 −1 1 0 ⎠ , (4.5.4.11)
−1 0 0 1
since this is not of the form given in (4.5.4.10). This generates the transformation
ˇρ →
ˇρ
(1 − ˇv) 2 − ˇσˇρ , ˇσ → ˇσ
ˇρˇσ + ˇv(1 − ˇv)
, ˇv →
(1 − ˇv) 2 − ˇσˇρ (1 − ˇv) 2 − ˇσˇρ , (4.5.4.12)
and is known to be a symmetry of the partition function. 12
Finally we turn to the constraints from black hole entropy. In this case the function g(τ)
is given by [62, 64]:
g(τ) = η(τ) 8 η(2τ) 8 . (4.5.4.13)
Thus (4.4.6) takes the fom
̂Φ(ˇρ, ˇσ, ˇv) ∝ (2v − ρ − σ) 6 {v 2 η(ρ) 8 η(2ρ) 8 η(σ) 8 η(2σ) 8 + O(v 4 )} , (4.5.4.14)
where (ρ, σ, v) and (ˇρ, ˇσ, ˇv) are related via (4.4.7). The dyon partition function of the Z 2 CHL
model is known to satisfy this property. In fact historically this is the property that was used
to guess the form of the partition function [12].
This analysis can be easily generalized to the dyons of Z N CHL orbifolds carrying twisted
sector electric charges.
4.5.5 Dyons in ZZ 2 CHL model with untwisted sector electric charge
We again consider the Z 2 CHL model introduced in §4.5.4, but now take the set A to consist
of dyons with charge vectors
⎛
⎞ ⎛ ⎞
0
2K + 1
Q = ⎜ (2m + 1)/2
⎟
⎝ 0 ⎠ , P = ⎜ J
⎟
⎝ 1 ⎠ , m, K, J ∈ Z . (4.5.5.1)
−2
0
Since w ′ = −2 for this state, it represents an untwisted sector state. For this state we have
Q 2 = −2(2m + 1), P 2 = 2(2K + 1), Q · P = −2J . (4.5.5.2)
12 This is the symmetry refered to as g 3 (1, 0) in [12] in a different representation.