PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

32 CHAPTER 4. GENERALITIES OF QUARTER BPS DYON PARTITION FUNCTION
Q 2 , P 2 and Q · P :
d(Q, P ) = f(Q 2 , P 2 , Q · P ) . (4.0.1)
Note that for (4.0.1) to hold it is necessary to choose B in the way we have described. In
particular if B had contained two elements with same Q 2 , P 2 and Q · P but not related
by a T-duality transformation, then d(Q, P ) can be different for these two elements
and (4.0.1) will not hold. Even when B is chosen according to the prescription given
above, eq.(4.0.1) cannot be strictly correct since d(Q, P ) could depend on the asymptotic
moduli besides the charges and one can construct more general T-duality invariants
using these moduli and the charges. Indeed, even though the index is not expected
to change under a continuous change in the moduli, it could jump across the walls of
marginal stability giving d(Q, P ) a dependence on the moduli. The reason that we
can still write eq.(4.0.1) is that it is possible to label the different domains bounded
by the walls of marginal stability by a set of discrete parameters ⃗c such that T-duality
transformation does not change the parameters ⃗c [24, 44, 48]. Physically, if a domain is
bounded by n walls of marginal stability, with the ith wall associated with the decay
(Q, P ) → (α i Q + β i P, γ i Q + δ i P ) + ((1 − α i )Q − β i P, −γ i Q + (1 − δ i )P ), then ⃗c is
the collection of the numbers {(α i , β i , γ i , δ i ); 1 ≤ i ≤ n}. Due to T-duality invariance
of ⃗c, d(Q, P ) inside a given domain labelled by ⃗c will be invariant under a T-duality
transformation on the charges only and will have the form (4.0.1). For different ⃗c the
function f will be different, ı.e. f has a hidden ⃗c dependence. We now define the dyon
partition function associated with the set B to be 2
1
̂Φ(ˇρ, ˇσ, ˇv) ≡
∑
Q 2 ,P 2 ,Q·P
(−1) Q·P +1 f(Q 2 , P 2 , Q · P )e iπ(ˇσQ2 +ˇρP 2 +2ˇvQ·P ) , (4.0.2)
where the sum runs over all the distinct triplets (Q 2 , P 2 , Q · P ) which are present in the
set B. This relation can be inverted as
∫
f(Q 2 , P 2 , Q · P ) ∝ (−1) Q·P +1 dˇρdˇσdˇv e −iπ(ˇσQ2 +ˇρP 2 +2ˇvQ·P ) 1
̂Φ(ˇρ, ˇσ, ˇv) , (4.0.3)
C
where C denotes an appropriate three dimensional subspace of the complex (ˇρ, ˇσ, ˇv) space.
Along the ‘contour’ C the imaginary parts of ˇρ, ˇσ and ˇv are fixed at values where the
sum in (4.0.2) converges, and the real parts of ˇρ, ˇσ and ˇv vary over an appropriate unit
cell determined by the quantization laws of Q 2 , P 2 and Q · P inside the set B.
In all known cases the function f in different domains ⃗c is given by (4.0.3) with identical
integrand, but the integration contour C depends on the choice of ⃗c. Put another way,
the same function 1/̂Φ(ˇρ, ˇσ, ˇv) admits different Fourier expansion in different regions in
the complex (ˇρ, ˇσ, ˇv) space, since a Fourier expansion that is convergent in one region
2 For N = 4 supersymmetric Z N orbifolds reviewed in [24] the function ̂Φ is related to the function ˜Φ of [24]
by the relation ̂Φ(ˇρ, ˇσ, ˇv) = ˜Φ(˜ρ, ˜σ, ṽ) with (˜ρ, ˜σ, ṽ) = (ˇσ/N, N ˇρ, ˇv).