Symmetry Breaking and Link Homologies I

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) constructed from the stack of pincipal G-connections on a circle, whose holonomy is broken between consecutive sectors. 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). The construction of strict broken symmetries also allows us to incorporate twistings. Applying suitable U(r)-equivariant (twisted) cohomology theories E to sB(L) gives rise to a spectral sequence of link invariants converging to the cohomology of the limiting value of the filtration. In followup articles, we offer two examples of such theories. In the first example, we study a universal twist of Borel-equivariant singular cohomology. The E_2-term in this case appears to recover sl(n) link homologies for any value of n (depending on the choice of specialization of the universal twist). We also show that Triply-graded link homology corresponds to the trivial twist. In the next example, we apply a version of an equivariant K-theory known as Dominant K-theory, which can be interpreted as twisted U(r)-equivariant K-theory built from level n representations of the loop group of U(r). In this case, the E_2-term recovers a deformation of sl(n)-link homology, and has the property that its value on the unknot is the Grothendieck group of level n-representations of the loop group of U(1).Comment: Substantial change in notation to resolve some previous notational conflict. Minor corrections made and exposition vastly improved based on feedbac