We propose a new discrete model-the twisted quantum double model-of 2D topological phases based on a finite group G and a 3-cocycle α over G. The detailed properties of the ground states are studied, and we find that the ground-state subspace can be characterized in terms of the twisted quantum double D α (G) of G. When α is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group G, in which the elementary excitations are known to be classified by the quantum double D(G) of G. Our model can be viewed as a Hamiltonian extension of the Dijkgraaf-Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string-net 6j symbols to the corresponding 3-cocycles. The paper is presented in a way such that it is accessible to a wide range of physicists.

We construct in the K matrix formalism concrete examples of symmetry enriched topological phases, namely intrinsically topological phases with global symmetries. We focus on the Abelian and non-chiral topological phases and demonstrate by our examples how the interplay between the global symmetry and the fusion algebra of the anyons of a topologically ordered system determines the existence of gapless edge modes protected by the symmetry and that a (quasi)-group structure can be defined among these phases. Our examples include phases that display charge fractionalization and more exotic non-local anyon exchange under global symmetry that correspond to general group extensions of the global symmetry group. CONTENTSA. Phases in SEP (Z 4 × Z 4 , Z 2 ) with W g = 1 21B. Some useful matrices 22 C. From SPT to Topological phases 22References 23

We study the stability, propagation and interactions of braid states in models of quantum gravity in which the states are four-valent spin networks embedded in a topological three manifold and the evolution moves are given by the dual Pachner moves. There are results for both the framed and unframed case. We study simple braids made up of two nodes which share three edges, which are possibly braided and twisted. We find three classes of such braids, those which both interact and propagate, those that only propagate, and the majority that do neither.Comment: 34 pages, 30 figures, typos corrected, 2 references added, to match the version accepted for publication in Nucl. Phys.

We propose an exactly solvable lattice Hamiltonian model of topological phases in 3+1 dimensions, based on a generic finite group G and a 4-cocycle ω over G. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the 3-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasi-excitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the SL(3, Z) generators as the modular S and T matrices of the ground states, which yield a set of topological quantum numbers classified by ω and quantities derived from ω. Our model fulfills a Hamiltonian extension of the 3 + 1-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group G. This work is presented to be accessible for a wide range of physicists and mathematicians.