Twisted gauge theories in 3D Walker-Wang models

Understand

Three dimensional gauge theories with a discrete gauge group can emerge from spin models as a gapped topological phase with fractional point excitations (gauge charge) and loop excitations (gauge flux).

• It is known that 3D gauge theories can be "twisted", in the sense that the gauge flux loops can have nontrivial braiding statistics among themselves and such twisted gauge theories are realized in models discovered by Dijkgraaf and Witten.
• A different framework to systematically construct three dimensional topological phases was proposed by Walker and Wang and a series of examples have been studied.
• Can the Walker Wang construction be used to realize the topological order in twisted gauge theories? This is not immediately clear because the Walker-Wang construction is based on a loop condensation picture while the Dijkgraaf-Witten theory is based on a membrane condensation picture.