Symmetry Breaking and Link Homologies II

TL;DR

This paper constructs a filtered homotopy type for links, connecting it to triply-graded link homology and sl(n)-link homology through equivariant cohomology evaluations and spectral sequences.

Contribution

It introduces a new filtered homotopy type for links and relates it to existing link homologies via spectral sequences and equivariant cohomology evaluations.

Findings

E_2-term is isomorphic to unreduced triply-graded link homology

E_1-term corresponds to Hochschild-homology of Soergel bimodules

Specialization yields sl(n)-link homology

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.