Symmetry Breaking and Link Homologies II

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 that converges to the cohomology of the limiting spectrum. In this followup, we apply Borel equivariant singular cohomology to our construction. We show that the E_2-term is isomorphic to an unreduced, integral form of triply-graded link homology, whose value on the unknot is a free graded algebra on two generators in degrees 2 and 1 resp. More precisely, we show that the E_1-term of the spectral sequence is isomorphic to the Hochschild-homology complex of Soergel bimodules that was shown by M. Khovanov to compute triply-graded link homology. We also set up the theory that allows for twisting equivariant cohomologies by adjoint-equivariant local systems on U(r). This allows us to twist Borel equivariant cohomology by a power series p(x) with coefficients given by formal variables, and no constant term. Based on computations, we speculate that the specialization to the case p(x) = x^n, gives rise to a spectral sequence whose E_2-term is isomorphic to sl(n)-link homology. In particular, p(x) can can be viewed as the universal differential of a potential that has no linear termComment: Substantial change in notation to resolve some previous notational conflict. Minor corrections made and exposition vastly improved based on feedbac