Gauge Theory on Hyperkähler Manifolds: Sp(n)-Instantons

I will describe recent progress in understanging instanton equations on hyperk¨ahler manifolds. This project is motivated by the observation that the deformed Sp(n)-instanton equation arising from the real Fourier-Mukai transform of a complex Lagrangian graph coincides with the classical Sp(n)-instanton equation. Results include new relationships between gauge-theoretic equations on hyperK¨ahler cones, their 3-Sasakian links, and various dimensional reductions as well as a compact and non-compact Lewis-type theorem. Lastly, I will present explicit examples of Sp(2)-instantons, Hermitian Yang–Mills connections, and Spin(7)-instantons in the cohomogeneity-one setting of the Calabi manifold $T^\ast\mathbb{C}P^2$. This talk is based on joint work with Jesse Madnick and Izar Alonso:

• Preprint 2025: arxiv.org/abs/2508.17119, Gauge theory on $T^\ast\mathbb{C}P^2$: explicit Sp(2)-instantons, HYM connections, and Spin(7)-instantons (I Alonso, J Madnick, EA Windes)
  
• Communications in Mathematical Physics, 2025, On Sp(n)-Instantons and the Fourier–Mukai Transform of Complex Lagrangians (J Madnick, EA Windes)