A link invariant from the symplectic geometry of nilpotent slices

Abstract

We define an invariant of oriented links in $S^3$ using the symplectic geometry of certain spaces that arise naturally in Lie theory. More specifically, we present a knot as the closure of a braid that, in turn, we view as a loop in configuration space. Fix an affine subspace ${\mathcal{S}}_m$ of the Lie algebra ${\mathfrak{sl}}_{2m}(\mathbb{C})$ which is a transverse slice to the adjoint action at a nilpotent matrix with two equal Jordan blocks. The adjoint quotient map restricted to ${\mathcal{S}}_m$ gives rise to a symplectic fibre bundle over configuration space. An inductive argument constructs a distinguished Lagrangian submanifold $L_{\wp {}_{\pm }}$ of a fibre ${\mathcal{Y}}_{m,t_0}$ of this fibre bundle; we regard the braid $\beta$ as a symplectic automorphism of the fibre and apply Lagrangian Floer cohomology to $L_{\wp {}_{\pm }}$ and $\beta (L_{\wp {}_{\pm }})$ inside ${\mathcal{Y}}_{m,t_0}$. The main theorem asserts that this group is invariant under the Markov moves and hence defines an oriented link invariant. We conjecture that this invariant coincides with Khovanov's combinatorially defined link homology theory, after collapsing the bigrading of the latter to a single grading