PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

92CHAPTER 5. PARTITION FUNCTIONS OF TORSION > 1 DYONS IN HETEROTIC STRIN
Now according to (5.0.20) the pair of integers (a ′ K, c ′ ) are relatively prime, and the pair of
integers (b ′ K, d ′ ) are also relatively prime. Thus using S-duality invariance we can write
(
Q1
d h , P )
1
L 1 L 1
(
Q2
d h , P )
2
L 2 L 2
= d h (a ′ KQ ′ , c ′ Q ′ ) = d h (Q ′ , 0),
= d h (b ′ KP ′ , d ′ P ′ ) = d h (P ′ , 0) . (5.0.47)
Using (5.0.44), (5.0.46) and (5.0.47) we can now express (5.0.43) as
(∆d(Q, P )) s0 = (−1) Q 1·P 2 −Q 2·P 1 +1 g(s 0 ) 1 (
Q1
(Q 1 · P 2 − Q 2 · P 1 ) d h , P ) (
1 Q2
d h , P )
2
.
s 0 L 1 L 1 L 2 L 2
(5.0.48)
For a given decay K is fixed but L 1 and L 2 can vary over all the factors of N 1 and N 2 . Thus
the total change in the index is obtained by summing over all possible values of s 0 of the form
KL 1 L 2 . Thus gives
∆d(Q, P )) = (−1) Q 1·P 2 −Q 2·P 1 +1 (Q 1 · P 2 − Q 2 · P 1 )
∑ ∑
(
1 Q1
g(KL 1 L 2 ) d h , P ) (
1 Q2
d h , P )
2
. (5.0.49)
KL 1 L 2 L 1 L 1 L 2 L 2
L 2 |N 2
L 1 |N 1
We can now fix the form of the function g(s) by considering a decay where the decay products
are primitive, ı.e. N 1 = N 2 = 1. In this case we have L 1 = L 2 = 1, and (5.0.49) takes the form 1
∆d(Q, P )) = (−1) Q 1·P 2 −Q 2·P 1 +1 (Q 1 · P 2 − Q 2 · P 1 ) g(K) 1 K d h(Q 1 , P 1 ) d h (Q 2 , P 2 ) . (5.0.50)
In order that this agrees with the standard wall crossing formula for primitive decay [44, 49–
54,73] we must have g(K) = K. Since this result should hold for all K|r we see that we must
set g(s) = s as given in (5.0.5). Using this we can simplify the wall crossing formula (5.0.49)
for generic non-primitive decays to the form given in (5.0.6).
Black hole entropy: In order to reproduce the leading contribution to the black hole entropy
in the limit of large charges, the partition function must have a pole at [7]
ˇρˇσ − ˇv 2 + ˇv = 0 . (5.0.51)
1 In this argument we have implicitly assumed that for a given K, it is possible to find integers a ′ , b ′ , c ′ , d ′
satisfying a ′ d ′ − b ′ c ′ = 1, gcd(c ′ , K) = gcd(d ′ , K) = gcd(a ′ , ¯K) = gcd(b ′ , ¯K) = 1, so that (5.0.20) holds and we
( have N 1 ) = N 2 ( = 1 according ) ( to (5.0.22). ) If either K or ¯K is odd then this assumption holds with the choice
a
′
b ′ 2 1 1 1
c ′ d ′ = or . If K and
1 1 1 2
¯K are both even then we cannot find a ′ , b ′ , c ′ and d ′ satisfying
all the requirements since ( in order ) ( to satisfy ) a ′ d ′ − b ′ c ′ = 1 at least one of them must be even. However in
a
′
b
this case if we choose
′ 2 1
c ′ d ′ = then we satisfy (5.0.20) and have N
1 1
1 = 1 and N 2 = 2 according
to (5.0.22). We can now demand that the wall crossing formula in this case agrees with the one derived in [71]
for decays where one of the decay products is twice a primitive vector. This gives g(K) = K even for K, ¯K
both even.