On \(\textrm{Sp}(n)\)-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 \(\mathbb {R}^{2n} \times T^{2n}\) correspond to \(\textrm{Sp}(n)\)-instantons over \(\mathbb {R}^{2n} \times (T^{2n})^*\). In other words, the deformed \(\textrm{Sp}(n)\)-instanton equation coincides with the usual \(\textrm{Sp}(n)\)-instanton equation. Motivated by this observation, we study \(\textrm{Sp}(n)\)-instantons on hyperkähler manifolds \(X^{4n}\), with an emphasis on conical singularities. First, when \(X = C(M)\) is a hyperkähler cone, we relate \(\textrm{Sp}(n)\)-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) hyperkähler manifold of rate \(\nu \le -\frac{2}{3}(2n+1)\), we prove a Lewis-type theorem to the following effect: If the set of AC \(\textrm{Sp}(n)\)-instantons is non-empty, then every AC Hermitian Yang–Mills connection over X with sufficiently fast decay at infinity is an \(\textrm{Sp}(n)\)-instanton.