PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

54 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
4.5.2 Dyons with unit torsion and even Q 2 /2 in heterotic on T 6
We now consider again heterotic string theory on T 6 , but choose the set A to be collection of
(Q, P ) of the form:
⎛ ⎞ ⎛ ⎞
0
K
Q = ⎜ 2m
⎟
⎝ 0 ⎠ , P = ⎜ J
⎟
⎝ 1 ⎠ , m, K, J ∈ Z . (4.5.2.1)
−1
0
This has
Q 2 = −4m, P 2 = 2K, Q · P = −J . (4.5.2.2)
We note that all the charge vectors have Q 2 /2 even. Since Q 2 is T-duality invariant, any other
charge vector which can be obtained from this one by a T-duality transformation has Q 2 /2
even. Thus the set B now consists of charge vectors which have even Q 2 /2 and arbitrary integer
values of P 2 /2 and Q · P . Since this set B is a subset of charges for which the spectrum was
analyzed in §4.5.1 we do not expect to derive any new results. Nevertheless we have chosen
this example as this will serve as a useful guide to our analysis in later sections.
We first note that the quantization conditions of Q 2 , P 2 and Q · P imply the following
periods of the partition function:
(ˇρ, ˇσ, ˇv) → (ˇρ + a 1 , ˇσ + a 2 , ˇv + a 3 ), a 1 ∈ Z, a 2 ∈ 1 2 Z, a 3 ∈ Z . (4.5.2.3)
The period along ˇσ is not an integer. We can remedy this by defining
Q s = Q/2 , ˇσ s = 4ˇσ, ˇv s = 2ˇv . (4.5.2.4)
so that Q 2 s/2 and Q s · P are now quantized in half integer units. The periods (ã 1 , ã 2 , ã 3 ) of the
variables (ˇρ, ˇσ s , ˇv s ) conjugate to (P 2 s /2, Q 2 s/2, Q s · P s ) are now integers, given by,
ã 1 ∈ Z, ã 2 ∈ 2 Z, ã 3 ∈ 2 Z . (4.5.2.5)
The dyon partition function in this case can be easily calculated from the one for §4.5.1 by
taking into account the evenness of Q 2 /2. This amounts to adding to the original partition
function another term where ˇσ is shifted by 1/2. Thus we have
1
̂Φ(ˇρ, ˇσ, ˇv) = 1 2
(
)
1
Φ 10 (ˇρ, ˇσ, ˇv) + 1
Φ 10 (ˇρ, ˇσ + 1, ˇv) , (4.5.2.6)
2
or, in terms of the rescaled variables,
1
̂Φ(ˇρ, ˇσ, ˇv) = 1 (
)
1
2 Φ 10 (ˇρ, 1 ˇσ 4 s, 1 ˇv 2 s) + 1
Φ 10 (ˇρ, 1 ˇσ 4 s + 1, 1 ˇv . (4.5.2.7)
2 2 s)