SGGTC this Friday: Nitu Kitchloo
Oleg Lazarev
Dear all,
This Friday Nitu Kitchloo from Johns Hopkins will give a talk at 1:00 pm in Math 407. We will meet at 11:35 in the lobby before the talk and go out for lunch.
Best, Oleg
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Nitu Kitchloo: Symmetry Breaking and 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 called strict Broken Symmetries, sBSy(w). As the name suggests, sBSy(w) is constructed from pincipal G-connections on a circle, whose holonomy is broken between consecutive sectors in a manner prescribed by a presentation of w. Specializing to the case of the unitary group G=U(r), we show 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 as terms in a spectral sequence obtained on applying suitable U(r)-equivariant cohomology theories E to sBSy(L). We offer two examples of such theories. In the first example, we take E to be Borel-equivariant singular cohomology. In this case, one recovers an unreduced,integral form of the Triply-graded link homology as the E_2-term. In the next example, we apply a version of an equivariant K-theory known as Dominant K-theory, which is 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).
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 called strict Broken Symmetries, sBSy(w). As the name suggests, sBSy(w) is constructed from pincipal G-connections on a circle, whose holonomy is broken between consecutive sectors in a manner prescribed by a presentation of w. Specializing to the case of the unitary group G=U(r), we show 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 as terms in a spectral sequence obtained on applying suitable U(r)-equivariant cohomology theories E to sBSy(L). We offer two examples of such theories. In the first example, we take E to be Borel-equivariant singular cohomology. In this case, one recovers an unreduced,integral form of the Triply-graded link homology as the E_2-term. In the next example, we apply a version of an equivariant K-theory known as Dominant K-theory, which is 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).