PHYS08200604018 Shamik Banerjee - Homi Bhabha National ...

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where Λ is a large positive number, I(z) denotes the imaginary part of z,
Q 2 R = Q T (M + L)Q, P 2 R = P T (M + L)P, Q R · P R = Q T (M + L)P , (4.0.12)
τ ≡ τ 1 +iτ 2 denotes the asymptotic value of the axion-dilaton moduli which belong to the
gravity multiplet and M is the asymptotic value of the symmetric matrix valued moduli
field of the matter multiplet satisfying MLM T = L. The choice (4.0.11) of course is not
unique since we can deform the contour without changing the result for the index as long
as we do not cross a pole of the partition function.
Independent of the above analysis, the change in the index across a wall of marginal
stability can be computed using the wall crossing formula [44, 49–54]. This tells us that
as we cross a wall of marginal stability associated with the decay (Q, P ) → (Q 1 , P 1 ) +
(Q 2 , P 2 ), the index jumps by an amount 4
(−1) Q 1·P 2 −Q 2·P 1 +1 (Q 1 · P 2 − Q 2 · P 1 ) d h (Q 1 , P 1 )d h (Q 2 , P 2 ) (4.0.13)
up to a sign, where d h (Q, P ) denotes the index of half-BPS states carrying charge
(Q, P ). For the decay described in (4.0.9) the relevant half-BPS indices are of the form
d h (a 0 M 0 , c 0 M 0 ) and d h (b 0 N 0 , d 0 N 0 ) where M 0 ≡ d 0 Q − b 0 P and N 0 ≡ −c 0 Q + a 0 P . T-
duality invariance implies that – modulo some subtleties discussed below eqs.(4.3.11) –
the dependence of d h (a 0 M 0 , c 0 M 0 ) and d h (b 0 N 0 , d 0 N 0 ) on M 0 and N 0 must come via the
combinations M 2 0 and N 2 0 respectively. We now define
e πiτN2 0 dh (b 0 N 0 , d 0 N 0 ) ,
φ e (τ; a 0 , c 0 ) ≡ ∑ e πiτM2 0 dh (a 0 M 0 , c 0 M 0 ), φ m (τ; b 0 , d 0 ) ≡ ∑
M0
2 N0
2 (4.0.14)
where the sums are over the sets of (M0 2 , N0 2 ) values which arise in the possible decays of
the dyons in the set B via (4.0.9). Then (4.0.13) agrees with the residue of the partition
function at the pole (4.0.10) if we assume that ̂Φ has a double zero at (4.0.10) where it
behaves as
̂Φ(ˇρ, ˇσ, ˇv) ∝ ˇv ′2 φ e (ˇσ ′ ; a 0 , c 0 ) φ m (ˇρ ′ ; b 0 , d 0 ) , ˇv ′ ≡ ˇρc 0 d 0 + ˇσa 0 b 0 + ˇv(a 0 d 0 + b 0 c 0 ),
ˇσ ′ ≡ c 2 0 ˇρ + a 2 0ˇσ + 2a 0 c 0ˇv, ˇρ ′ ≡ d 2 0 ˇρ + b 2 0ˇσ + 2b 0 d 0ˇv . (4.0.15)
Since for any given system the allowed values of (a 0 , b 0 , c 0 , d 0 ) can be found from charge
quantization laws, (4.0.15) gives us information about the locations of the zeroes on ̂Φ
and its behaviour at these zeroes in terms of the spectrum of half-BPS states in the
theory.
4 Eq.(4.0.13) holds only if the dyons (Q 1 , P 1 ) and (Q 2 , P 2 ) are primitive. As will be discussed later, this
formula gets modified for non-primitive decay.