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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.

Abstract:

Abstract We explicitly work out the details of some simple examples anyon theories. We start with a fusion ring and using consistency conditions we build F-matrices, then the possible braidings (R-matrices). We consider Z2 fusion rules, Fibonacci fusion rules and Z3 fusion rules as simple cases. We consider Ising fusion rules, obtaining eight possible (modular) unitary braided theories. We then describe several more abelian theories, including the toric code, and general ZN anyons. These can be extended to describe all possible abelian braided theories in terms of a limited number of so-called “prime” theories.

Abstract:

Abstract Manifolds can be decomposed into simplicies (triangles in 2D or tetrahedra in 3D). TQFTs can be constructed as state sums over discrete quantum numbers on the on these discretized manifolds. Such sums appear like partition functions, statistical mechanics sums of Boltzmann weights. In order to for these sums to yield manifold invariants the sum must be independent on the particular simplicial decomposition (or triangulation) of the manifold. The so-called “Pachner Moves” describe all possible changes of the decomposition, so a sum which is unchanged under Pachner moves gives a manifold invariant. One such state sum is the the Turaev-Viro state sum, which takes as an input a (spherical) fusion category (F matrices satisfying the pentagon equation) and then allows one to assign a (scalar) manifold invariant to a 3D manifold. The corresponding TQFT is known as the quantum double, or Drinfeld double of the fusion category. Much of the study of the Turaev-Viro model has been in the context of so-called spin-network models of quantum gravity. A very similar state sum TQFT is the Dijkgraaf-Witten model, which (in 3D) takes as an input a group and a 3-cocyle (acting as F-matrices). The Dijkgraaf-Witten model is generalizable to any dimension.

Abstract:

Abstract We refine our notion of a topoogical phase of matter to account for symmetry protection. We describe SPT (symmetry protected topological or trivial) phases and give some history of this idea. We focus on the case of on-site symmetries. The case of one-dimensional SPTs with on-site symmetry is discussed. For understanding two-dimensional SPTs we focus on the case of Z2 symmetry and argue that there are two possible Z2 paramagnetic phases, i.e., gapped phases where the Z2 symmetry is unbroken. These two phases are in one-to-one correspondence with the two possible Z2 loop gases. We argue that with any discrete group symmetry, we generally classify SPT phases with the possible 3-cocyles. We briefly discuss the idea of symmetry defects and explain that such particles are the excitations of the corresponding twisted Kitaev quantum double model of the same cocyle. We briefly discuss the edge modes of SPT phases. Finally we argue that the SPT phases can be considered as the “ungauging” of the corresponding Dijkgraaf-Witten model.