Abstract: I will describe a physically-motivated conjectural approach (developed with Shamit Kachru and Arnav Tripathy) to the construction of hyperkahler deformations of $(\mathbb{R}^{4-r} \times T^r)/\Gamma$ with $0\le r \le 4$, where $\Gamma$ is a finite group, as well as recent work with Arnav Tripathy which proves some of these conjectures. This generalizes Kronheimer’s construction of ALE manifolds in the $r=0$ case, but for $r\not=0$ it involves an infinite-dimensional hyperkahler quotient. I will explain how to reinterpret this as a gauge theory problem on the dual torus, where one is now interested in a moduli space of singular equivariant instantons. Many results from Kronheimer’s work generalize to this setting, and conversely we obtain some novel results which apply to the ALE setting. I will explain how the $r\ge 2$ case interacts with open string mirror symmetry, enumerative geometry, and another construction of hyperkahler manifolds. Finally, I will specialize to the $r=1$ case, the rigorous study of which we have recently completed, and which serves as a model for many of the phenomena we expect to find at larger values of $r$.
 
virtual (zoom): Virtual: http://berkeley.zoom.us/j/93328405860?pwd=Um1GbHBCSUJMdUlWWnd0ZVMxQmwwdz09