Moduli of stable rank two bundles with amplec 1 on K3 surfaces

A strong coupling test of S-duality

“…(−1) s exp(2πis 2 /2N ) 24 How to compute such sums is explained in [49], as was pointed out to us by Dick Gross. 25 An odd lattice is simply a lattice such that v · v is odd for some v. We will see in §4 that at least for N = 2, this equation for c is true identically and not just mod 4.…”

“…We would like to compute A N (L) for an arbitrary self-dual lattice L and verify (3.20). 24 It is easy to prove the following properties for A N (L):…”

“…Solutions of this kind have ǫ = +1 (since ∂h i /∂y a = ∂s a /∂u i and the determinant in (2.12) is positive). If these are the only solutions, then (2.11) reduces 24) with N the total number of solutions of the original equation. Thus if all solutions of the extended system are at y i = 0, then the number of solutions of the extended system, weighted by sign, is the same as the total number of solutions of the original system.…”

“…we rely on constructions by Mukai and others [23][24][25]; for CP 2 we use formulas of Yoshioka and Klyachko [26][27][28]; for the case of blowing up a point in a Kähler manifold we use a formula of Yoshioka [26]; and for ALE spaces we use formulas of Nakajima [29]. All of these formulas give functions with modular properties.…”

“…If E is an SU (2) instanton bundle on M = K3 with instanton number k, one can seek [23][24] a convenient description of E by finding a line bundle L on K3 such that the index of the ∂ operator coupled to L −1 ⊗ E is 1. 26 It will then be generically true that 26 These matters were explained to us by P. Kronheimer.…”