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

TL;DR

This paper explores the relationship between complex Lagrangian graphs and Sp(n)-instantons via the Fourier-Mukai transform, and studies Sp(n)-instantons on hyperkähler manifolds with singularities, establishing links to tri-contact instantons and decay properties.

Contribution

It demonstrates that complex Lagrangian graphs correspond to Sp(n)-instantons under the Fourier-Mukai transform and analyzes Sp(n)-instantons on hyperkähler manifolds with conical singularities and decay conditions.

Findings

Complex Lagrangian graphs correspond to Sp(n)-instantons via RFM transform.

Sp(n)-instantons on hyperkähler cones relate to tri-contact instantons.

Decay conditions imply Hermitian Yang-Mills connections are Sp(n)-instantons.

Abstract

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 $R^{2n} \times T^{2n}$ correspond to Sp($n$)-instantons over $R^{2n} \times (T^{2n})^*$. 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 $\nu \leq -\frac{2}{3}(2n+1)$, 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.