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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

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.