Symmetry Breaking and Link Homologies III

Abstract: In the first part of this paper, we constructed a filtered U(r)-equivariant stable homotopy type called the spectrum of strict broken symmetries sB(L) of links L given by closing a braid with r strands. We further showed that evaluating this spectrum on suitable U(r)-equivariant cohomology theories gives rise to a spectral sequence of link invariants, converging to the homology of the limiting spectrum. In this followup, we fix a positive integer n and apply a version of an equivariant K-theory known as Dominant K-theory, which is built from level n representations of the loop group of U(r). The E_2-term of the spectral sequence appears to be a deformation of sl(n)-link homology, and has the property that its value on the unlink is the Grothendieck group of level n-representations of the loop group of U(1). Seen in contrast to the standard interpretation of sl(n)-link homology using the fundamental representation of the quantum group U_q(sl(n)), suggests a level-rank duality at play.