PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

4.5. EXAMPLES 59
uncorrelated and can take arbitrary integer values, it follows from (4.3.22) that at this zero ̂Φ
goes as
̂Φ(ˇρ, ˇσ s , ˇv s ) ∝ (2ˇv s + ˇσ s ) 2 φ m
(ˇρ + 1 ) ( ) 1
4 ˇσ s + ˇv s ; 1, 1 φ e
4 ˇσ s; 1, 0 +O ( (2ˇv s + ˇσ s ) 4) . (4.5.2.29)
φ m (τ; 1, 1) denotes the partition function of half-BPS states carrying charges (P, P ), with τ
being conjugate to the variable P ′2 /2 = P 2 /2. Thus we have φ m (τ; 1, 1) = (η(τ)) 24 . On the
other hand (φ e (τ; 1, 0)) −1 is the partition function of half BPS states carrying charges (Q ′ , 0) =
(Q − P, 0) with τ being conjugate to Q ′2 /2 = (Q − P ) 2 /2. Since (Q − P ) 2 /2 = (−2m + K + J)
can take arbitrary integer values, the corresponding partition function is also given by η(τ) −24 .
Thus we have near (ˇσ s + 2ˇv s ) = 0
̂Φ(ˇρ, ˇσ s , ˇv s ) ∝
{
(2ˇv s + ˇσ s ) 2 η
(
ˇρ + 1 4 ˇσ s + ˇv s
) 24
η
̂Φ given in (4.5.2.7) can be shown to satisfy this property.
( ) 24 ˇσs
+ O((2ˇv s + ˇσ s ) )}
4
4
(4.5.2.30)
φ m (τ; 1, 1) transforms as a modular form of weight 12 under τ → (α 1 τ + β 1 )/(γ 1 τ + δ 1 )
with α 1 , β 1 , γ 1 , δ 1 ∈ Z, α 1 δ 1 −β 1 γ 1 = 1. On the other hand φ e (τ; 1, 0) transforms as a modular
form of weight 12 under τ → (p 1 τ + q 1 )/(r 1 τ + s 1 ) with p 1 , q 1 , r 1 , s 1 ∈ Z, p 1 s 1 − q 1 r 1 = 1.
Using (4.3.29), (4.3.30) and (4.5.2.14) we see that the the action of these transformations on
the variables (ˇρ, ˇσ s , ˇv s ) may be represented by the symplectic matrices
⎛
⎞ ⎛
α 1 (α 1 − 1)/2 β 1 0
⎜ 0 1 0 0
⎟
⎝ γ 1 γ 1 /2 δ 1 0 ⎠ ,
γ 1 /2 γ 1 /4 (δ 1 − 1)/2 1
⎞
1 (1 − p 1 )/2 q 1 −2q 1
⎜ 0 p 1 −2q 1 4q 1
⎝ 0 0 1 0
0 r 1 /4 (1 − s 1 )/2 s 1
⎟
⎠ .
(4.5.2.31)
By comparing with the matrices given in (4.5.2.16) we see however that the transformations
(4.5.2.31) generates symmetries of the full partition function only after we impose the additional
constraints
r 1 , γ 1 ∈ 4 Z . (4.5.2.32)
Thus here we encounter a case where only a subset of the symmetries of the partition function
near a pole is lifted to a full symmetry of the partition function. By examining the
details carefully one discovers that in this case the pole comes from the first term in (4.5.2.7).
Whereas this term displays the full symmetry given in (4.5.2.31), requiring that the other term
also transforms covariantly under this symmetry generates the additional restrictions given in
(4.5.2.32).
Finally we turn to the constraint from black hole entropy. As in §4.5.1, in this case we
have g(τ) = η(τ) 24 in (4.4.4). Thus (4.4.6) takes the form
̂Φ(ˇρ, ˇσ, ˇv) ∝ (2v − ρ − σ) 10 {v 2 η(ρ) 24 η(σ) 24 + O(v 4 )} , (4.5.2.33)