On Sp(n)-Instantons and the Fourier-Mukai Transform of Complex Lagrangians

The real Fourier-Mukai (RFM) transform relates calibrated graphs to so-called "deformed instantons" on Hermitian line bundles. We show that under the RFM transform, complex Lagrangian graphs in R²ⁿ × T²ⁿ correspond to Sp(n)-instantons over R²ⁿ × (T²ⁿ)^*. In other words, the deformed Sp(n)-instanton equation coincides with the usual Sp(n)-instanton equation. Motivated by this observation, we study Sp(n)-instantons on hyperkahler manifolds X⁴ⁿ, with an emphasis on conical singularities. First, when $X = C(M)$ is a hyperkahler cone, we relate Sp(n)-instantons on X to tri-contact instantons on the 3-Sasakian link M and consider various dimensional reductions. Second, when X is an asymptotically conical (AC) hyperkahler manifold of rate ν ≤ -2/3(2n+1), we prove a Lewis-type theorem to the following effect: If the set of AC Sp(n)-instantons is non-empty, then every AC Hermitian Yang-Mills connection over X with sufficiently fast decay at infinity is an Sp(n)-instanton.