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 Knot Invariants and New Weight Systems from General 3D TFTs


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We introduce and study the Wilson loops in a general 3D topological field theories TFTs, and show that the expectation value of Wilson loops also gives knot invariants as in Chern-Simons theory. We study the TFTs within the Batalin-Vilkovisky BV and Alexandrov-Kontsevich-Schwarz-Zaboronsky AKSZ framework, and the Ward identities of these theories imply that the expectation value of the Wilson loop is a pairing of two dual constructions of cocycles of certain extended graph complex extended from Kontsevichs graph complex to accommodate the Wilson loop. We also prove that there is an isomorphism between the same complex and certain extended Chevalley-Eilenberg complex of Hamiltonian vector fields. This isomorphism allows us to generalize the Lie algebra weight system for knots to weight systems associated with any homological vector field and its representations. As an example we construct knot invariants using holomorphic vector bundle over hyperKahler manifolds.



Author: Jian Qiu; Maxim Zabzine

Source: https://archive.org/



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