Symmetry Breaking and Link Homologies II

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. Evaluating this filtered spectrum on suitable U(r)-equivariant cohomology theories gives rise to a spectral sequence of link invariants that converges to the cohomology of the limiting spectrum. In this paper, we evaluate our construction on Borel equivariant singular cohomology HU(r). We show that the E_2-term of the spectral sequence is isomorphic to an unreduced verision of triply-graded link homology. More precisely, we show that the E_1-term of the spectral sequence is isomorphic to the Hochschild-homology of Soergel bimodules that was shown by M. Khovanov in to compute triply-graded link homology. We also set up the theory that allows for evaluating our construction on equivariant cohomologies that are twisted by adjoint-equivariant local systems on U(r). This allows us to apply Borel equivariant cohomology twisted by a universal power series p(x) with coefficients given by formal variables, and no constant term. The specialization of p(x) to x^n gives rise to a link homology that has recently been shown by T. Mej{\i}a Gomez to be isomorphic to sl(n)-link homology. In other words, the power series p(x) can be interpreted as the (differential of the) universal potential function with no linear term, in the language of matrix factorizations.