Symmetry Breaking and Link Homologies I

TL;DR

This paper introduces a new homotopy type invariant for links derived from symmetry-breaking in principal G-connections, which leads to novel link homologies and spectral sequences connecting to existing theories.

Contribution

It constructs the Strict Broken Symmetries homotopy type as a link invariant and explores its applications to twisted cohomology theories and known link homologies.

Findings

sB(w) is independent of presentation and satisfies Markov properties.

sB(L) yields spectral sequences converging to link cohomology.

Universal twists recover sl(n)-link homologies.

Abstract

Given a compact connected Lie group G endowed with root datum, and an element w in the corresponding Artin braid group for G, we describe a filtered G-equivariant stable homotopy type, up to a notion of quasi-equivalence. We call this homotopy type Strict Broken Symmetries, sB(w). As the name suggests, sB(w) is constructed from the stack of principal G-connections on a circle, whose symmetry is broken between consecutive arcs in a manner prescribed by a presentation of w. We show that sB(w) is independent of the choice of presentation of w, and also satisfies Markov type properties. Specializing to the case of the unitary group G = U(r), these properties imply that sB(w) is an invariant of the link L obtained by closing the r-stranded braid w. As such, we denote it by sB(L). In the follow up to this article, we will show that the construction of strict broken symmetries allows us to incorporate twistings. Under suitable conditions, U(r)-equivariant (twisted) cohomology theories EU(r) applied to sB(L) give rise to a spectral sequence of link invariants converging to the cohomology of the direct limit of the filtration of sB(L). For instance we describe Triply-graded link homology as the E2-term in the spectral sequence one obtains on applying Borel-equivariant singular cohomology HU(r) to sB(L). We also study a universal twist of HU(r). Here one recovers sl(n)-link homologies for any value of n (depending on the choice of specialization of the universal twist).