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Abstract:

Abstract The mathematical structure known as a Quantum Group can be thought of as a deformation of a classical Lie group. The (deformed) representation theory of quantum groups gives a natural way to generate data (F-matrices, and R-matrices) for anyon theories. In many cases, this approach gives data equivalent to Chern-Simons theory based on the corresponding Lie group.

Abstract:

Abstract Topologically ordered matter is famously robust to small perturbations. If a Hamiltonian has a topologically ordered (TQFT) ground state with a gap, then adding a small pertrubation (on the scale of the gap) to the Hamiltonian cannot change its TQFT properties. We explore this robustness with the example of the toric code. We show that the robustness is guaranteed by the fact that the toric code has a protected code space. We further discuss how the properties of the quasiparticles are also unchanged under perturbations of the Hamiltonian. These properties might be unsurprising, given the rigidity of the properties of TQFTs. We define the notion of topological order and give a definition of a topological phase of matter.

Abstract:

Abstract The diagram algebras that we have developed are closely related to knot invariants. To make this connection we want the algebras to be “isotopy invariant”, menaing that the diagrams can be freely deformed as long as strands are not cut. To achieve this we change the diagram normalization to use so-called “Isotopy Normalization”, where a particle loop is weighted by the quantum dimension. This almost achieves isotopy invariance of certain diagrams, except for a possible sign known as the Frobenius-Schur indicator. We carefully discuss this sign, its origin, physical meaning, and how one nonetheless achieves isotopy invariant diagrams. Having handled this sign, we achieve what essentially amounts to a knot invariant — an isotopy invariant mapping from a labeled knot or link diagram to a complex number output. Even given this success it may still be the case that diagrams with branching may incur phases from deforming an up-branching to a down-branching. We briefly discuss other properties of unitary fusion categories including the spherical and pivotal properties.

Abstract:

Abstract A brief discussion of how and why Chern-Simons theory was pursued in the context of quantum gravity.