Abstract
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 B2G of the symmetry group G, and they are classified by cohomology classes of B2G. 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 B2G 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.
Highlights
Over the last several decades, quantum field theories have emerged as the central language in which modern theoretical physics is formulated
We first provide further detail regarding the interplay between the 2-form cohomology group H4(G[2], U(1)) and abelian braided monoidal categories, we study to which extent our 2-form gauge model is related to the Walker-Wang model
Gauge and higher gauge models of topological phases of matter have been under intense investigation in the past years, one reason being that they seem to encapsulate most of the known models displaying non-trivial topological order in (3+1)d
Summary
Over the last several decades, quantum field theories have emerged as the central language in which modern theoretical physics is formulated. Throughout this manuscript, we focus most of our attention on (3+1)d topological sigma models with the second classifying space B2G as the target space where G is a finite abelian group, or equivalently discrete (3+1)d 2-form topological lattice gauge theories. As explained above, such higher form gauge theories arise naturally from a mathematical point of view. When the input data of the premodular category is a finite abelian group and a quadratic form, the Walker-Wang model provides a Hamiltonian realization of a 2-form gauge theory that describes the topological order mentioned above. In appendix F we propose explicit expressions of q-form topological actions using the language of Deligne-Beilinson cohomology
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