PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

88CHAPTER 5. PARTITION FUNCTIONS OF TORSION > 1 DYONS IN HETEROTIC STRIN
This allows us to determine the partition function of all other sets of ( torsion ) r dyons ( from)
the
a b 1 1
partition function given in (5.0.4). In particular for r = 2, choosing = we
c d 0 1
recover the dyon partition function proposed in [71].
We shall now show that the proposed partition function (5.0.4) satisfies various consistency
tests described in [71].
S-duality invariance: We shall first verify the required S-duality invariance of the partition
function described ( in eq.(5.0.8). ) Using Sp(2, Z) invariance of Φ 10 (x, y, z), and that b is a
a b
multiple of r for ∈ Γ
c d
0 (r) one can show that
(
Φ 10 ˇρ ′ , s 2ˇσ ′ + k¯s , 2 sˇv′ + l¯s
)
= Φ 10
(ˇρ, s 2ˇσ )
+ k′
, sˇv + , (5.0.11)
¯s
2
l′¯s
where
Thus
¯s
∑
2 −1
k=0
∑¯s−1
l=0
k ′ = kd 2 − 2cdlr ∈ Z, l ′ = (ad + bc)l − bdk/r ∈ Z . (5.0.12)
(
Φ 10 ˇρ ′ , s 2ˇσ ′ + k¯s l¯s) −1 ¯s , ∑
2 −1
2 sˇv′ + =
and we have the required relation (5.0.8).
k ′ =0
∑¯s−1
l ′ =0
(
Φ 10 ˇρ, s 2ˇσ ) −1
+ k′
, sˇv + , (5.0.13)
¯s
2
l′¯s
Wall crossing formula: We shall now verify that (5.0.7) is consistent with the wall crossing
formula. As in [71] we shall only consider the decay into a pair of half-BPS dyons [42–44, 60]
(Q, P ) → (Q 1 , P 1 ) + (Q 2 , P 2 ) , (5.0.14)
(Q 1 , P 1 ) = (αQ + βP, γQ + δP ), (Q 2 , P 2 ) = (δQ − βP, −γQ + αP ) , (5.0.15)
αδ = βγ, α + δ = 1 . (5.0.16)
Since any lattice vector lying in the plane of Q and P can be expressed as a linear combination
of Q/r and P with integer coefficients, we must have β, δ, α ∈ Z, γ ∈ Z/r. Thus we can
write γ = γ ′ /K, where K ∈ Z, K|r and gcd(γ ′ , K)=1. The condition αδ = βγ together with
the integrality of α, β, δ now tells us that β must be of the form Kβ ′ with β ′ ∈ Z. Thus we
have
β = K β ′ , γ = γ′
K , K, α, δ, β′ , γ ′ ∈ Z, K|r, gcd(γ ′ , K) = 1 . (5.0.17)
Using eqs.(5.0.16), (5.0.17) we have
α + δ = 1, αδ = β ′ γ ′ , α, β ′ , γ ′ , δ ∈ Z . (5.0.18)
The analysis of [44] now shows that we can find a ′ , b ′ , c ′ , d ′ such that
α = a ′ d ′ , β ′ = −a ′ b ′ , γ ′ = c ′ d ′ , δ = −b ′ c ′ , a ′ , b ′ , c ′ , d ′ ∈ Z, a ′ d ′ − b ′ c ′ = 1 . (5.0.19)