Abstract
We construct a lattice model based on a crossed module of possibly non-abelian finite groups. It generalizes known topological quantum field theories, but in contrast to these models admits local physical excitations. Its degrees of freedom are defined on links and plaquettes, while gauge transformations are based on vertices and links of the underlying lattice. We specify the Hilbert space, define basic observables (including the Hamiltonian) and initiate a discussion on the model’s phase diagram. The constructed model reduces in appropriate limits to topological theories with symmetries described by groups and crossed modules, lattice Yang-Mills theory and 2-form electrodynamics. We conclude by reviewing classifying spaces of crossed modules, with an emphasis on the direct relation between their geometry and properties of gauge theories under consideration.
Highlights
Robust against local perturbations, and have been suggested to possess potential to be used in fault-tolerant quantum computation [11, 12]
The constructed model reduces in appropriate limits to topological theories with symmetries described by groups and crossed modules, lattice Yang-Mills theory and 2-form electrodynamics
A popular framework for description of topological order is that of Topological Quantum Field Theories (TQFTs) [13, 14]
Summary
Homotopy classes of (parametrized) paths in a topological space form a structure very similar to a group, since they can be composed in a way which is associative and admits multiplicative inverses. A lattice gauge field may be specified by giving a homomorphism π1(X1; ∗) → G for some ∗ ∈ X0 and the values of ge for edges e from any maximal tree T These data can be chosen independently because there are no relations between generators of π1(X1; ∗) and elements of T. It admits a decomposition with exactly one cell in every dimension up to 2 — see figure 8 In this case the groupoid π1(X1; X0) has one object ∗ and one generator e. If f is -connected, the element γ f does not depend on the choice of γ In this case any other allowed path takes the form γ = γ (∂f )n for some n and ∂f f = f , by the second Peiffer identity. The fact that γ φf is independent of the choice of γ relies on the fake flatness constraint
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