SU(2) character varieties of tangles: From Casson’s invariant to Fukaya categories and Khovanov homology

In 1985 Casson introduced an invariant of a homology 3-sphere as a signed count of its SU(2) representations; the sign is determined by the auxillary choice of a Heegaard splitting. Shortly thereafter, Taubes gave a gauge theory definition of Casson’s invariant, as the Poincaré-Hopf index of the Chern-Simons function, which requires no splitting, but instead relies on holonomy perturbations near a link in the 3-manifold. This simplification permitted Floer to construct a ``categorification’’ of Casson’s invariant called instanton homology, a construction generalized to the (3,1)+1 cobordism category by Kronheimer-Mrowka in a series of influential articles. In parallel, Wehrheim-Woodward introduced quilted Floer homology and Floer field theory to bridge the gap between the instanton-Floer theory and Lagragian Floer theory, providing a path towards a counterpart of bordered HF theory for instanton homology via the Fukaya categories of the SU(2) character varieties of 2-manifolds, equipped with their Atiyah-Bott-Goldman symplectic forms.

In these lectures I will attempt to explain aspects of the previous paragraph from the classical perspective of knot theory, fundamental groups of 2 and 3-manifolds, and the spherical geometry of SU(2). The required background is basic knot and 3-manifold theory (Wirtinger presentation, Seifert-Van Kampen theorem, incompressible tori) and familiarity with the geometry of the quaternions. I will explain how certain low-dimensional moduli spaces can be used to avoid the techical difficulties of working with J-holomorphic polygons, as well as how a relationship between the Fukaya category of a certain character variety with Khovanov homology arises from this perspective.