Title: Gauge Theory on Hyperkahler Manifolds and 3-Sasakian Links

Abstract: Manifolds with special holonomy — such as Calabi-Yau, hyperkahler, G2, and Spin(7)-manifolds — are challenging objects to study.  In recent years, geometers have proposed various gauge-theoretic PDE (known as “instanton equations”) on such spaces, partly in an effort to define enumerative invariants.  While much work has been done in the Calabi-Yau, G2, and Spin(7) settings, the hyperkahler (HK) case has thus far received less attention.

In this talk, we discuss a natural class of Yang-Mills connections over hyperkahler 4n-manifolds X, known as Sp(n)-instantons, that generalize ASD instantons.  To model their conical singularities, we relate Sp(n)-instantons over HK cones to contact instantons over their links, and establish some dimensional reductions.  We then prove a “Lewis-type theorem” on asymptotically conical (AC) hyperkahler manifolds X to the following effect: If X admits AC Sp(n)-instantons, then any Hermitian Yang-Mills connection over X that decays sufficiently rapidly is necessarily an Sp(n)-instanton.  This is joint work with Emily Windes (Oregon).