PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

may not be convergent in another region. The coefficients of expansion in these different
regions in the complex (ˇρ, ˇσ, ˇv) plane may then be regarded as the index f(Q 2 , P 2 , Q · P )
in different domains in the asymptotic moduli space labelled by ⃗c. We shall assume that
this result holds for all sets of dyons in all N = 4 string theories.
2. Consequences of S-duality symmetry: We now consider the effect of an S-duality
transformation on the set B. A generic S-duality transformation will take an element
of B to outside B, – we denote by H the subgroup of the S-duality group that leaves
B invariant. This is the subgroup relevant for constraining the dyon partition function
associated with the set B. Since a generic element of H takes us from one domain
bounded by walls of marginal stability to another such domain, it relates the function
f for one choice of ⃗c to the function f for another choice of ⃗c. However since we have
assumed that the dyon partition function 1/̂Φ is independent of the domain label ⃗c, we
can use invariance under H to constrain the form of ̂Φ. In particular ( one finds ) that an
a b
S-duality symmetry of the form (Q, P ) → (aQ + bP, cQ + dP ) with ∈ H gives
c d
the following constraint on ̂Φ:
̂Φ(ˇρ, ˇσ, ˇv) = ̂Φ(d 2 ˇρ + b 2ˇσ + 2bdˇv, c 2 ˇρ + a 2ˇσ + 2acˇv, cdˇρ + abˇσ + (ad + bc)ˇv) . (4.0.4)
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Defining
we can express (4.0.4) as
( )
ˇρ ˇv ˇΩ = , (4.0.5)
ˇv ˇσ
where
̂Φ((AˇΩ + B)(C ˇΩ + D) −1 ) = (det(C ˇΩ + D)) k ̂Φ(Ω) , (4.0.6)
⎛
( )
A B
= ⎜
C D ⎝
d b 0 0
c a 0 0
0 0 a −c
0 0 −b d
and k is as yet undermined since det(CΩ + D) = 1.
⎞
⎟
⎠ , (4.0.7)
Besides this symmetry, quantization of Q 2 , P 2 and Q · P within the set B also gives rise
to some translational symmetries of ̂Φ of the form ̂Φ(ˇρ, ˇσ, ˇv) = ̂Φ(ˇρ + a 1 , ˇσ + a 2 , ˇv + a 3 )
with a 1 , a 2 , a 3 taking values in an appropriate set. These can also be expressed as (4.0.6)
with
⎛
⎞
( )
1 0 a 1 a 3
A B
= ⎜ 0 1 a 3 a 2
⎟
C D ⎝ 0 0 1 0 ⎠ . (4.0.8)
0 0 0 1