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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.

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.

Abstract:

Abstract We continue the development of diagrammatics now considering diagrams living in 3 dimensions. We generalize the discussion of the R-matrix for braiding (non-identical) particles around each other, and discuss the Hexagon equations which provide a consistency condition for the R-matrices. As with the pentagon, Hexagon solutions are also rigid.

Abstract:

Abstract Continuing our exploration of the detailed structure of anyon theories, we explain how one can describe the same Hilbert space with different basis sets. Changing between these bases is done with the so-called F-matrix which is a gauge dependent quantity. The F-matrices satisfy a set of consistency equations known as the pentagon equations which are “rigid” up to Gauge transformations (i.e., solutions are discrete and cannot be deformed small amounts to produce other solutions). We again use Fibonacci and Ising anyons as simple examples.

Abstract:

Abstract We give a brief introduction to Chern-Simons theory starting with Abelian Chern-Simons theory and making the connection to the charge-flux model. We discuss the vacuum partition function and the idea of a manifold invariant. We then move on to discuss Nonabelian Chern-Simons theory and the idea of Wilson loop operations, and Witten’s famous connection between knot invariants and Chern-Simons theory. The Chern-Simons theory is a paradigm for topological quantum field theories or TQFTs.