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 R^2n T^2n correspond to Sp (n) -instantons over R^2n (T^2n) ^*. 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 X^4n, 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 -23 (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.