• Corpus ID: 204734622

Symmetry Breaking and Link Homologies I

@inproceedings{Kitchloo2019SymmetryBA,
  title={Symmetry Breaking and Link Homologies I},
  author={Nitu Kitchloo},
  year={2019}
}

• N. Kitchloo
• Published 14 October 2019
• Mathematics

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, sBSy(w). As the name suggests, sBSy(w) is constructed from the stack of pincipal G-connections on a circle, whose holonomy is broken between consecutive sectors in a manner prescribed by a presentation of w. We show that sBSy(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 sBSy(w) is an invariant of the link L obtained by closing the r-stranded braid w. As such, we denote it by sBSy(L). We may therefore obtain (group valued) link homology theories on applying suitable U(r)-equivariant cohomology theories EU(r) to sBSy(L). More precisely, one has spectral sequences induced by the filtration, converging to E∗U(r)(sBSy∞(L)), where sBSy∞(L) is the direct limit of the filtration. In the followup articles to this article [2, 3], we offer two examples of such theories. In the first example, we take EU(r) to be Borelequivariant singular cohomology HU(r). In this case, one recovers an unreduced, integral form of the Triply-graded link homology as the E2-term, with the value on the unknot being the free graded algebra Z[x] ⊗ Λ(y), where x and y are generators in degree 2 and 1 respectively. In the next example, we apply a version of an equivariant K-theory nKU(r) known as Dominant K-theory, which is built from level n representations of the loop group of U(r). In this case, the E2-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), given by Z[x]/(x − 1).