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

Published online by Cambridge University Press:  28 July 2025

Nathan Geer
Affiliation:
Mathematics and Statistics, https://ror.org/00h6set76 Utah State University , Logan, UT 84322, United States e-mail: nathan.geer@gmail.com
Aaron Lauda
Affiliation:
Department of Mathematics and Department of Physics, https://ror.org/03taz7m60 University of Southern California , Los Angeles, CA 90089, United States e-mail: lauda@usc.edu
Bertrand Patureau-Mirand
Affiliation:
LMBA, https://ror.org/04ed7fw48 Université de Bretagne Sud , Vannes, France e-mail: bertrand.patureau@univ-ubs.fr
Joshua Sussan*
Affiliation:
Department of Mathematics, https://ror.org/03we2aj97 CUNY Medgar Evers , New York City, NY 11225, United States

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.

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Article
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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