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 which arise naturally in Lie theory. More specifically, we present a knot as the closure of a braid, which in turn we view as a loop in configuration space. Fix an affine subspace Sm of the Lie algebra sl2m(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 Sm gives rise to a symplectic fibre bundle over configuration space. An inductive argument constructs a distinguished Lagrangian submanifold L℘ ± of a fibre Ym,t 0 of this fibre bundle; we regard the braid β as a symplectic automorphism of the fibre, and apply Lagrangian Floer cohomology to L℘ ± and β(L℘ ± ) inside Ym,t 0 . The main theorem asserts that this group is invariant under the Markov moves, hence defines an oriented link invariant. We conjecture that this invariant co-incides with Khovanov's combinatorially defined link homology theory, after collapsing the bigrading of the latter to a single grading.

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

“…(Such affine braid group actions have played a central role in constructions of knot homology, both in mathematics and physics, cf. [54][55][56][57][58][59].) In the 2d reductions of 3d gauge theories that we study in section 7, two commuting braid-group actions will appear.…”

“…where the index 'a' labels generators of the Cartan subalgebra of the gauge group G. The ring relations are quantized tô 54) and the quantum exponentials is generated from an identity vector |N L , which satisfiesN…”

Gauge theory and knot homologies

“…In particular, when topological reduction of the gauge theory gives A-model, the branes B and B ′ are represented by Lagrangian submanifolds in M. This leads to a construction of link homologies via symplectic geometry, as in [32,33,34].…”

The symplectic topology of some rational homology balls

“…Due to the existence of many exact Lagrangian spheres in S n , this hypersurface has been instrumental in constructing many interesting examples in symplectic geometry (see [24], [28], [41], [36]). We will recall some generalities about S n , and we refer the reader to loc.…”

“…The latter has a unique Stein structure and its symplectic topology is well-studied (see [24], [28], [41], [36]). …”