Temperly–Lieb Algebra and Jones–Kauffman Anyons

Steven H. Simon

Published Sep 29, 2023• 0 citations

Abstract

Abstract Ideas similar to that of the Kauffman bracket introduced in chapter 2 can be used to generate valid anyon theories. We begin with a planar diagram algebra defined only by the value of a loop (the loop weight). One wants to build new particle types by grouping together multiple strands of loops. However, to do so one needs so-called Jones-Wenzl projectors to orthogonalize between species resulting in the so-called Temperly-Lieb Algebra. We demonstrate this construction by building a planar diagram algebra corresponding to the Z2 loop gas (bosons, fermions, semions), and then Ising Anyons. We show how this generalizes to a much bigger set of possible theories. We show how these theories can be made unitary, and how the F-matrices may be calculated. Finally we discuss how the R-matrices can be simply obtained by returning to the Kauffman bracket rules.

DOI: 10.1093/oso/9780198886723.003.0022

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