Toward the group completion of the Burau representation
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Abstract:
Following Boardman-Vogt, McDuff, Segal, and others, we construct a monoidal topological groupoid or space of finite subsets of the plane, and interpret the Burau representation of knot theory as a topological quantum field theory defined on it. Its determinant or writhe is an invertible braided monoidal TQFT which group completes to define a Hopkins-Mahowald model for integral homology as an $E_2$ Thom spectrum. We use these ideas to construct an infinite cyclic (Alexander) cover for the space of finite subsets of $\mathbb {C}$, and we argue that the TQFT defined by Burau is closely related to the SU(2)-valued Wess-Zumino-Witten model for string theory on $\mathbb {R}^3_+$.References
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Additional Information
- Jack Morava
- Affiliation: Department of Mathematics, The Johns Hopkins University, Baltimore, Maryland 21218
- MR Author ID: 217965
- Email: jack@math.jhu.edu
- Dale Rolfsen
- Affiliation: Pacific Institute for the Mathematical Sciences and Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
- Email: rolfsen@math.ubc.ca
- Received by editor(s): July 30, 2019
- Received by editor(s) in revised form: April 12, 2022, July 14, 2022, and August 9, 2022
- Published electronically: November 16, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 1845-1865
- MSC (2020): Primary 55U40; Secondary 20F36
- DOI: https://doi.org/10.1090/tran/8796
- MathSciNet review: 4549693