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Density and unitarity of the Burau representation from a non-semisimple TQFT

Part of: Locally compact groups and their algebras Groups and algebras in quantum theory Miscellaneous applications of functional analysis Differential topology

Published online by Cambridge University Press:  28 July 2025

Nathan Geer ,
Aaron Lauda Open the ORCID record for Aaron Lauda [Opens in a new window] ,
Bertrand Patureau-Mirand  and
Joshua Sussan
Nathan Geer
Affiliation:
Mathematics and Statistics, Utah State University , Logan, UT 84322, United States e-mail: nathan.geer@gmail.com
Aaron Lauda
Affiliation:
Department of Mathematics and Department of Physics, University of Southern California , Los Angeles, CA 90089, United States e-mail: lauda@usc.edu
Bertrand Patureau-Mirand
Affiliation:
LMBA, Université de Bretagne Sud , Vannes, France e-mail: bertrand.patureau@univ-ubs.fr
Joshua Sussan*
Affiliation:
Department of Mathematics, CUNY Medgar Evers , New York City, NY 11225, United States
*
e-mail: jsussan@mec.cuny.edu
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Abstract

We study the density of the Burau representation from the perspective of a non-semisimple topological quantum field theory (TQFT) at a fourth root of unity. This gives a TQFT construction of Squier’s Hermitian form on the Burau representation with possibly mixed signature. We prove that the image of the braid group in the space of possibly indefinite unitary representations is dense. We also argue for the potential applications of non-semisimple TQFTs toward topological quantum computation.

Keywords

Braid groupsBurau representationunitaritydensity

MSC classification

Primary: 81R50: Quantum groups and related algebraic methods 57R56: Topological quantum field theories 22D10: Unitary representations of locally compact groups 46N50: Applications in quantum physics

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Type
Article
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Canadian Journal of Mathematics , First View , pp. 1 - 29
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

N.G. is partially supported by NSF grant DMS-2104497. A.D.L. is partially supported by NSF grants DMS-1902092 and DMS-2200419, the Army Research Office W911NF-20-1-0075, and the Simons Foundation collaboration grant on New Structures in Low-dimensional topology. J.S. is partially supported by a Simons Foundation Travel Support Grant and PSC CUNY Enhanced Award 66685-00 54. Computations associated with this project were conducted utilizing the Center for Advanced Research Computing (CARC) at the University of Southern California.

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