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Knot invariants and new weight systems from general 3D TFTs
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
2012 (English)In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 62, no 2, 242-271 p.Article in journal (Refereed) Published
Abstract [en]

We introduce and study the Wilson loops in general 3D topological field theories (TFTs), and show that the expectation value of Wilson loops also gives knot invariants as in the Chern-Simons theory. We study the TFTs within the Batalin-Vilkovisky (BV) and the 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 (co)cycles of certain extended graph complex (extended from Kontsevich's 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.

Place, publisher, year, edition, pages
2012. Vol. 62, no 2, 242-271 p.
Keyword [en]
Knot invariants, Topological field theory, BV formalism, Wilson loops
National Category
Physical Sciences
Identifiers
URN: urn:nbn:se:uu:diva-172057DOI: 10.1016/j.geomphys.2011.10.008ISI: 000301314900010OAI: oai:DiVA.org:uu-172057DiVA: diva2:513343
Available from: 2012-04-02 Created: 2012-04-01 Last updated: 2017-12-07Bibliographically approved

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Qiu, JianZabzine, Maxim

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