PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

66 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
subspace of charges given in (4.5.1.1), now the momentum n ′ along S 1 is quantized in units of
1/2 whereas the Kaluza-Klein monopole charge N ′ along S 1 is quantized in units of 2 [15]. We
shall take the set A to be consisting of charge vectors of the form
For this state we have
⎛ ⎞
0
Q = ⎜ m/2
⎟
⎝ 0 ⎠ ,
−1
⎛
P = ⎜
⎝
K
J
1
0
⎞
⎟
⎠ , m, K, J ∈ Z . (4.5.4.1)
Q 2 = −m, P 2 = 2K, Q · P = −J . (4.5.4.2)
As usual we denote by B the set of all (Q, P ) which are related to the ones given in (4.5.4.2)
by a T-duality transformation. Since Q 2 /2, P 2 /2 and Q · P are quantized in units of 1/2, 1
and 1 respectively, ̂Φ satisfies the periodicity conditions (4.2.16) with
a 1 ∈ Z, a 2 ∈ 2 Z, a 3 ∈ Z . (4.5.4.3)
Comparison of (4.5.4.1) and (4.5.1.1) shows that the winding charge −w ′ along S 1 is 1 for this
state. Thus it represents a twisted sector state.
Our next task is to determine the subgroup of the S-duality group that leaves the set
( B invariant. ) In this case the full S-duality group is Γ 0 (2), generated by matrices of the form
a b
with a, b, d ∈ Z, c ∈ 2 Z, ad − bc = 1. It was shown in [24] that the set B is
c d
closed under the full S-duality group. Thus the full S-duality group must be a symmetry of
the partition function.
We now turn to the constraints from the wall crossing formula. Consider first the wall
associated with the decay (Q, P ) → (Q, 0) + (0, P ), – this in fact is the only case we need
to analyze since all the walls are related to this one by S-duality transformation [44]. First
of all note from (4.5.4.1), (4.5.4.2) that for a given Q 2 = −m the charge vector Q ∈ A is
fixed uniquely. Thus the index of half-BPS states with charge (Q, 0) can be regarded as a
function of Q 2 . On the other hand for a given P 2 = 2K there is a family of P ∈ A labelled
by J, but these can be transformed ⎛ to the⎞vector corresponding to J = 0 by the T-duality
1 0 0 J
transformation matrix [24] ⎜ 0 1 −J 0
⎟
⎝ 0 0 1 0 ⎠ . Thus the index of the charge vector (0, P ) can
0 0 0 1
also be expressed as a function of P 2 . Finally we see from (4.5.4.2) that the allowed values of
Q 2 and P 2 are uncorrelated. Thus we can use eqs.(4.3.9), (4.3.10) to extract the behaviour of
̂Φ near ˇv = 0. The electric partition function can be calculated by examining the spectrum of
twisted sector states in the heterotic string theory [64–67]. On the other hand the magnetic
partition function can be calculated by examing the spectrum of D1-D5 system in a dual type