Abstract: I will describe joint work with Ciprian Manolescu on
constructing an analogue of instanton Floer homology replacing the
group SU(2) by SL(2,C). Having failed to do so using the standard
Floer theoretic tools of gauge theory and symplectic topology, we
turned to sheaf theory to produce an invariant. After describing our
approach, I will discuss some features of this theory that are
expected to be visible from a Floer-theoretic point of view, but that
we cannot currently access.