qabelianization for line defects
Connections for Women: Holomorphic Differentials in Mathematics and Physics August 15, 2019  August 16, 2019
Location: SLMath: Eisenbud Auditorium
04Yan
I will talk about some joint work with Andrew Neitzke, where we introduce a new "invariant" (with possible wallcrossing) for framed links in a threemanifold M=C \times R with C being an oriented surface. This invariant, denoted as F(L) for a framed link L, is valued in the GL(1) skein algebra of another threemanifold M'=C' \times R, where C' is an Nfold cover of C.
Under various special limits, F(L) turns into more familiar objects. When L is contained in a 3ball in M, F(L) reproduces certain onevariable limit of the HOMFLY polynomial of L. When the projection of L to C has no crossings and the homology class of L is nontrivial, F(L) becomes a generating function encoding the spectrum of framed BPS states associated with certain halfBPS line defect in a 4d N=2 supersymmetric theory. In general, F(L) is a "hybrid" of the above two quantities.
The construction of F(L) is realized via a homomorphism from the GL(N) skein algebra of M to the GL(1) skein algebra of M'. In my talk I will first review the notion of skein algebras. Then I will describe this homomorphism for the special case of N=2, followed by some examples.
Notes

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04Yan
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