A sheaf‐theoretic SL(2,C) Floer homology for knots

Abstract

Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three‐manifolds, similar to the three‐manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat connections with fixed holonomy around the meridian of the knot. Thus, our invariant is a sheaf‐theoretic SL(2,C) analogue of the singular knot instanton homology of Kronheimer and Mrowka. We prove that for two‐bridge and torus knots, the SL(2,C) invariant is determined by the l‐degree of the trueÂ‐polynomial. However, this is not true in general, as can be shown by considering connected sums of knots.

Geometric Unification of Higgs Bundle Vacua

“…The details in deriving such a flow and properly treating the gauge ambiguity of W (A) is quite involved, even in this "simple" example, so we refer the reader to section (5.2) of [32] for details. Defining a complexified Floer theory is of deep mathematical interest and it would be intriguing to explore the recent work of [33] and [34] to derive more examples of these half-BPS domain walls in 4D N = 1 systems (see also [32]).…”