PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

4.1. THE DYON PARTITION FUNCTION 39
4.1 The dyon partition function
Let us consider a particular N = 4 supersymmetric string theory in four dimensions with a
total of r U(1) gauge fields including the six graviphotons. The electric and magnetic charges
in this theory are represented by r dimensional vectors Q and P , and there is a T-duality
invariant metric L of signature (6, r − 6) that can be used to define the inner product of the
charges. Let us consider an (infinite) set B of dyon charge vectors (Q, P ) with the property
that if two different members of the set have the same values of Q 2 ≡ Q T LQ, P 2 ≡ P T LP and
Q·P ≡ Q T LP then there must exist a T-duality transformation that relates the two members.
In other words if there are T-duality invariants other than Q 2 , P 2 and Q·P then for all members
of the set B with a given set of values of (Q 2 , P 2 , Q · P ) these other T-duality invariants must
have the same values. We shall generate such a set B by beginning with a family A of charge
vectors (Q, P ) labelled by three integers such that the triplet (Q 2 , P 2 , Q · P ) are independent
linear functions of these three integers, and then define B to be the set of all (Q, P ) which
are in the T-duality orbit of the set A. Such a set B automatically satisfies the restriction
mentioned above since given two elements of B with the same values of (Q 2 , P 2 , Q · P ), each
will be related by a T-duality transformation to the unique element of A with these values of
(Q 2 , P 2 , Q · P ). An example of such a set A can be found in eqs.(4.5.1.3), (4.5.1.4).
Our object of interest is the index d(Q, P ), measuring the number of bosonic supermultiplets
minus the number of fermionic supermultiplets of quarter BPS dyons carrying charges
(Q, P ) ∈ B. Typically the index, besides depending on (Q, P ), also depends of the domain in
which the asymptotic moduli lie. These domains are bounded by walls of marginal stability associated
with decays of the form (Q, P ) → (αQ+βP, γQ+δP )+((1−α)Q−βP, −γQ+(1−δ)P )
for appropriate values of (α, β, γ, δ) associated with the quantization conditions [42,43,60]. For
fixed values of the other moduli these walls describe circles or straight lines in the axion-dilaton
moduli space labelled by the complex parameter τ [42, 44]. We denote by ⃗c the collection of
(α i , β i , γ i , δ i ) bordering a particular domain in the moduli space; inside any such domain the
index remains unchanged. It has been shown in [44] that the parameters ⃗c labelling a domain
remain invariant under a simultaneous T-duality transformation on the charges and the
moduli. Since d(Q, P ) must be invariant under simultaneous T-duality transformation on the
charges and the moduli, we can conclude that for a given ⃗c the index d(Q, P ) for (Q, P ) ∈ B
will be a function only of the T-duality invariants (Q 2 , P 2 , Q · P ). We shall express this as
f(Q 2 , P 2 , Q · P,⃗c).
Let us now introduce the partition function
1
̂Φ(ˇρ, ˇσ, ˇv) ≡
∑
Q 2 ,P 2 ,Q·P
(−1) Q·P +1 f(Q 2 , P 2 , Q · P ;⃗c 0 )e iπ(ˇσQ2 +ˇρP 2 +2ˇvQ·P ) . (4.1.1)
where ⃗c 0 denotes some specific domain in the moduli space bounded by a set of walls of marginal
stability. The sum runs over allowed values of Q 2 , P 2 and Q · P for the dyons belonging to the
set B. The factor of (−1) Q·P +1 has been included for convenience. ̂Φ so defined is expected to