We propose an exactly solvable Hamiltonian for topological phases in 3 + 1 dimensions utilising ideas from higher lattice gauge theory, where the gauge symmetry is given by a finite 2-group. We explicitly show that the model is a Hamiltonian realisation of Yetter's homotopy 2-type topological quantum field theory whereby the groundstate projector of the model defined on the manifold M 3 is given by the partition function of the underlying topological quantum field theory for M 3 × [0, 1]. We show that this result holds in any dimension and illustrate it by computing the ground state degeneracy for a selection of spatial manifolds and 2-groups. As an application we show that a subset of our model is dual to a class of Abelian Walker-Wang models describing 3 + 1 dimensional topological insulators.Contents arXiv:1606.06639v2 [cond-mat.str-el]

We define the thin fundamental Gray 3-groupoid S3(M ) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps S3(M ) → C(H), where H is a 2-crossed module of Lie groups and C(H) is the Gray 3groupoid naturally constructed from H. As an application, we define Wilson 3-sphere observables.

Abstract. We define the thin fundamental categorical group P 2 (M, * ) of a based smooth manifold (M, * ) as the categorical group whose objects are rank-1 homotopy classes of based loops on M and whose morphisms are rank-2 homotopy classes of homotopies between based loops on M . Here two maps are rank-n homotopic, when the rank of the differential of the homotopy between them equals n. Let C(G) be a Lie categorical group coming from a Lie crossed module G = (∂ : E → G, ). We construct categorical holonomies, defined to be smooth morphisms P 2 (M, * ) → C(G), by using a notion of categorical connections, being a pair (ω, m), where ω is a connection 1-form on P , a principal G bundle over M , and m is a 2-form on P with values in the Lie algebra of E, with the pair (ω, m) satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.

We generalize the BF theory action to the case of a general Lie crossed module (∂ : H → G, ⊲), where G and H are non-abelian Lie groups. Our construction requires the existence of G-invariant non-degenerate bilinear forms on the Lie algebras of G and H and we show that there are many examples of such Lie crossed modules by using the construction of crossed modules provided by short chain complexes of vector spaces. We also generalize this construction to an arbitrary chain complex of vector spaces, of finite type. We construct two gauge-invariant actions for 2-flat and fake-flat 2-connections with auxiliary fields. The first action is of the same type as the BFCG action introduced by Girelli, Pfeiffer and Popescu for a special class of Lie crossed modules, where H is abelian. The second action is an extended BFCG action which contains an additional auxiliary field. However, these two actions are related by a field redefinition. We also construct a three-parameter deformation of the extended BFCG action, which we believe to be relevant for the construction of non-trivial invariants of knotted surfaces embedded in the four-sphere.