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"Localization of unitary braid group representations", Eric Rowell, Zhenghan Wang, 2012

Reviewed October 7, 2023

Citation: Rowell, Eric C., and Zhenghan Wang. "Localization of unitary braid group representations." Communications in Mathematical Physics 311 (2012): 595-615.

Web: https://arxiv.org/abs/1009.0241

Tags: Property-F, Modular-tensor-categories

This paper introduces an important concept of localization of braid group representations. The intuition is that the sequence of braid group representations attached to an TQFTs should be local if and only if its states can be described by local behaviors around each particle, i.e., can be decomposed as a tensor product of vector spaces, one for each particle, and for which braiding acts by identical two-qudit matrices for every pair of particles braided. Essentially no TQFTs are local by this strict definition. If you weaken things up a bit (i.e. only force the braid group reps to be embaddable into such a local family) you still only get a restrictive subclass of TQFTs. Conjecturally, these are the weakly integral TQFTs.

Local braid group representations (conjecturally) always have finite image, so this localization perspective could be a great approach towards proving the Property F conjecture.

Local braid group representations are also nice because they can be defined using only linear algebra, and give a nice playing group for TQC. This made them amenable to a braid-group-representation-centered introduction to TQC:

> Delaney, Colleen, Eric C. Rowell, and Zhenghan Wang. "Local unitary representations of the braid group and their applications to quantum computing." Revista Colombiana de Matemáticas 50.2 (2016): 211-276.

Generalizations of this concept and further work is found in

> Galindo, César, Seung-Moon Hong, and Eric C. Rowell. "Generalized and quasi-localizations of braid group representations." International Mathematics Research Notices 2013.3 (2013): 693-731.

NOTE: This notion of localization is not to be confused with the earlier notation of localization of modular functor, where one implements a TQFT with a local Hamiltonian:

>Freedman, Michael H. "Quantum computation and the localization of modular functors." Foundations of Computational Mathematics 1 (2001): 183-204.