PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

4.5. EXAMPLES 71
On the other hand the electric partition function can be calculated by analyzing the untwisted
sector BPS spectrum of the fundamental heterotic string [64–67]. After taking into account the
fact that we are computing the partition function of odd Q 2 /2 states only, and the ˇσ → ˇσ s /2
replacement, the result is
φ −1
e (ˇσ) = 1 2 (ψ e(ˇσ s ) − ψ e (ˇσ s + 1)) ,
[ 1 (
ψ e (ˇσ s ) = 8 η(ˇσ s /2) −24 ϑ2 (ˇσ s ) 8 + ϑ 3 (ˇσ s ) 8 + ϑ 4 (ˇσ s ) 8) ]
− ϑ 3 (ˇσ s /2) 4 ϑ 4 (ˇσ s /2) 4 .
2
(4.5.5.12)
In (4.5.5.12) ψ e describes the partition function before projecting on to the odd Q 2 /2 sector [65].
φ m (ˇρ s ) given( in (4.5.5.11) ) transforms as a modular form of weight 8 under ˇρ s → (αˇρ s +
α β
β)/(γ ˇρ s + δ) with ∈ Γ
γ δ 0 (2) with a multiplier (−1) β . On the other hand φ e (ˇσ) given in
(4.5.5.12) ( can)
be shown to transform as a modular form of weight 8 under ˇσ s → (pˇσ s +q)/(rˇσ s +
p q
s) for ∈ Γ
r s 0 (2), with a multiplier (−1) q . These duality symmetries correspond to the
symplectic transformations
⎛
⎞ ⎛
⎞
α 0 β 0
1 0 0 0
⎜ 0 1 0 0
⎟
⎝ γ 0 δ 0 ⎠ and ⎜ 0 p 0 q
⎟
⎝ 0 0 1 0 ⎠ ,
0 0 0 1
0 r 0 s
acting on (ˇρ s , ˇσ s , ˇv s ).
αδ − βγ = 1, ps − qr = 1, α, β, δ, p, q, s ∈ Z, γ, r ∈ 2 Z , (4.5.5.13)
Since for the set B the ( S-duality ) group ( is Γ(2), ) in this case there is another wall of marginal
a0 b
stability, associated with
0 1 1
= , which cannot be related to the previous wall
c 0 d 0 0 1
by an S-duality transformation. This corresponds to the decay (Q, P ) → (Q − P, 0) + (P, P )
and controls the behaviour of ̂Φ near ˇv + ˇσ = 0. As usual we first need to determine if there
are any subtleties of the type mentioned below eq.(4.3.11). Eq.(4.5.5.1) ⎛ shows that for ⎞ a given
−(2K + 1)
(Q−P ) 2 = 4(K +J −m) there is an infinite family of (Q−P ) = ⎜ (2m − 2J + 1)/2
⎟
⎝ −1 ⎠ labelled
−2
⎛
⎞
1 0 0 K
by 2K. However all of these can be related by T-duality transformation ⎜ 0 1 −K 0
⎟
⎝ 0 0 1 0 ⎠
0 0 0 1