Anyons From Discrete Group Elements
Chapter in Topological Quantum, Oxford University Press (OUP) (2023) 265-274
Abstract:
Abstract Given a discrete group, we consider planar diagrams where each edge is labeled with an element of the group, and vertices must obey the rule that multiplication of the edges incident on the vertex (in order) must yield the identity. The possible F-matrices for such a diagram are discussed and are known as 3-cocycles from the formalism of group cohomology. We discuss a simple example of building anyons from the ZN group. Noncommutative groups can be considered although they cannot describe a braided theory.Appendix: Kac and Other Resources for TQFTs
Chapter in Topological Quantum, Oxford University Press (OUP) (2023) 567-572
Abstract:
Abstract A number of tools for determining properties of TQFTs are described. In particular we outline the use of the computer program 鈥淜ac鈥. We also give pointers to other useful resources.Appendix: Some Mathematical Basics
Chapter in Topological Quantum, Oxford University Press (OUP) (2023) 573-584
Abstract:
Abstract This chapter is included to introduce some basic mathematical concepts. This includes basic notions about manifolds, basic ideas about groups, group theory, Lie groups, Lie algebras, representations, and the topological idea of a fundamental group.Bosons and Fermions from Group Representations: Rep( G )
Chapter in Topological Quantum, Oxford University Press (OUP) (2023) 275-286
Abstract:
Abstract We consider building planar diagram algebras by labeling edges with group representations and vertices follow the fusion rules of the group representations. We review using character tables to deduce fusion rules and quantum dimensions. The groups S3 and Q8 (quaternions) are used as simple examples. We show how F-matrices can be constructed using generalized Clebsch-Gordan coefficients. We deduce the possible braidings for such theories and conclude that only bosons or fermions are possible. We reconsider the possibility of parastatistics and argue why, given the structure of diagram algebras, only bosons and fermions are possible in 3+1 dimensions.Braiding Diagrams
Chapter in Topological Quantum, Oxford University Press (OUP) (2023) 169-176