PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

60 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
where (ˇρ, ˇσ, ˇv) and (ρ, σ, v) are related via (4.4.7). ̂Φ(ˇρ, ˇσ, ˇv) given in (4.5.2.6) can be shown
to satisfy this property. In fact the relevent pole of ̂Φ −1 comes from the first term on the right
hand side of (4.5.2.6). The location of the zeroes of Φ 10 are given in (4.6.8), and it follows from
this that the second term does not have a pole at v = 0.
4.5.3 Dyons of torsion 2 in heterotic string theory on T 6
We consider again heterotic string theory on T 6 and take the set A to consist of charge vectors
of the form ⎛ ⎞ ⎛ ⎞
1
2K + 1
Q = ⎜ 2m + 1
⎟
⎝ 1 ⎠ , P = ⎜ 2J + 1
⎟
⎝ 1 ⎠ , m, K, J ∈ Z . (4.5.3.1)
1
−1
This has
Q 2 = 4(m + 1), P 2 = 4(K − J), Q · P = 2(K + J − m + 1) . (4.5.3.2)
Furthermore gcd{Q i P j − Q j P i }=2. Thus we have a family of charge vectors with torsion 2. It
was shown in [48,63] that for r = 2 there are three T-duality orbits for given (Q 2 , P 2 , Q·P ) – in
the first Q is twice a primitive lattice vector, in the second P is twice a primitive lattice vector
and in the third both Q and P are primitive but Q ± P are twice primitive lattice vectors. The
dyon charges given in (4.5.3.1) are clearly of the third kind. In the notation of [48] the discrete
T-duality invariants of these charges are (r 1 = 1, r 2 = 1, r 3 = 2, u 1 = 1). Note that as we vary
m, J and K, Q 2 /2 and P 2 /2 take all possible even values and Q · P takes all possible values
subject to the restriction that Q ± P are twice primitive lattice vectors. The latter condition
requires Q · P to be even and Q · P − 1 2 Q2 − 1 2 P 2 to be a multiple of four. It now follows from
the result of [48,63] that the T-duality orbit B of the set A consists of all the pairs (Q, P ) with
(r 1 = 1, r 2 = 1, r 3 = 2, u 1 = 1) and even values of Q 2 /2, P 2 /2.
Since Q 2 /2, P 2 /2 and Q · P are all even and Q 2 + P 2 + 2Q · P is a multiple of 8, it is
natural to introduce new charge vectors and variables
so that we have
Q s ≡ Q/2, P s ≡ P/2, ˇρ s ≡ 4ˇρ, ˇσ s ≡ 4ˇσ, ˇv s ≡ 4ˇv , (4.5.3.3)
1
2 Q2 s = 1 2 (m + 1), 1
2 P s 2 = 1 2 (K − J), Q s · P s = 1 (K + J − m + 1) , (4.5.3.4)
2
quantized in half integer units subject to the constraint that
1
2 Q2 s + 1 2 P 2 s + Q s · P s = K + 1 , (4.5.3.5)