On 2-form gauge models of topological phases
Clement Delcamp (Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada, Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, Garching, 85748, Germany); Apoorv Tiwari (Department of Physics, University of Zurich, Winterthurerstrasse 190, Zurich, 8057, Switzerland)
We explore 2-form topological gauge theories in (3+1)d. These theories can be constructed as sigma models with target space the second classifying space B 2 G of the symmetry group G, and they are classified by cohomology classes of B 2 G. For finite symmetry groups, 2-form topological theories have a natural lattice interpretation, which we use to construct a lattice Hamiltonian model in (3+1)d that is exactly solvable. This construction relies on the introduction of a cohomology, dubbed 2-form cohomology, of algebraic cocycles that are identified with the simplicial cocycles of B 2 G as provided by the so-called W -construction of Eilenberg-MacLane spaces. We show algebraically and geometrically how a 2-form 4-cocycle reduces to the associator and the braiding isomorphisms of a premodular category of G-graded vector spaces. This is used to show the correspondence between our 2-form gauge model and the Walker-Wang model.
