PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

4.5. EXAMPLES 63
1
((Q−P 2 )/2)2 can take all possible independent integer values (K +1) and (m−J) respectively.
We find from (4.3.23) that the inverses of the relevant half-BPS partition functions are:
φ e (τ; a 0 , c 0 ) = η(2τ) 24 , φ m (τ; b 0 , d 0 ) = η(2τ) 24 . (4.5.3.16)
The factor of 2 in the argument of η is due to the fact that Q ′2 /2 = (d 0 Q−b 0 P ) 2 /2 = (Q−P ) 2 /4
and P ′2 /2 = (−c 0 Q + a 0 P ) 2 /2 = (Q + P ) 2 /4 entering in (4.3.23) are twice the usual integer
normalized combinations 1 8 (Q ± P )2 . This gives, from (4.3.14), (4.3.22) and (4.5.3.3)
̂Φ(ˇρ, ˇσ, ˇv) ∼ { (ˇρ s − ˇσ s ) 2 η((ˇρ s + ˇσ s − 2ˇv s )/4) 24 η((ˇρ s + ˇσ s + 2ˇv s )/4) 24 + O((ˇρ s − ˇσ s ) 4 ) } ,
(4.5.3.17)
near
(
ˇρ s
)
≃ ˇσ s . Since η(2τ) transforms covariantly under τ → (ατ + 1 β)/(2γτ + δ) with
2
α β
∈ SL(2, Z), both φ
γ δ
e and φ m have full SL(2, Z) symmetry. Using (4.3.29), (4.3.30)
and (4.5.3.8) to represent them as symplectic transformations on the variables (ˇρ s , ˇσ s , ˇv s ) we
get the following two sets of symplectic matrices:
1
2
⎛
α 1 + 1 α 1 − 1 2β 1 2β 1
⎜ α 1 − 1 α 1 + 1 2β 1 2β 1
⎝ γ 1 /2 γ 1 /2 δ 1 + 1 δ 1 − 1
γ 1 /2 γ 1 /2 δ 1 − 1 δ 1 + 1
⎞ ⎛
p 1 + 1 −p 1 + 1 2q 1 −2q 1
⎟
⎠ , 1 ⎜ −p 1 + 1 p 1 + 1 −2q 1 2q 1
2 ⎝ r 1 /2 −r 1 /2 s 1 + 1 −s 1 + 1
−r 1 /2 r 1 /2 −s 1 + 1 s 1 + 1
⎞
⎟
⎠ ,
α 1 , β 1 , γ 1 , δ 1 , p 1 , q 1 , r 1 , s 1 ∈ Z, α 1 δ 1 − β 1 γ 1 = p 1 s 1 − q 1 r 1 = 1 . (4.5.3.18)
Next we turn to the wall corresponding to the decay (Q, P ) → (Q − P, 0) + (P, P ). This
corresponds to the choice
( ) ( )
a0 b 0 1 1
= , (4.5.3.19)
c 0 d 0 0 1
and is associated with the zero of ̂Φ at
ˇσ + ˇv = 0 . (4.5.3.20)
Since (Q−P ) 2 /8 = m−J and P 2 /4 = K −J can take independent integer values, we should be
able to use (4.3.9), (4.3.10). The behaviour of ̂Φ near this zero is however somewhat ambiguous
since one of the decay products – the state carrying charge (Q−P, 0) – is not a primitive dyon.
As a result the index associated with this state is ambiguous. 11 Nevertheless if we go ahead
11 For half-BPS states in N = 2 supersymmetric theories a modification of the wall crossing formula for
such non-primitive decays has been suggested in [54]. It is not clear a priori how to modify it for the decays
of quarter BPS dyons in N = 4 supersymmetric string theories. In §4.6 we shall propose a formula for the
partition function of the states being studied in this section and examine it to find what the modification should
be.