Twisted gauge theories in 3D Walker-Wang models

Abstract

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. In this paper, we show that the answer to this question is Yes, by presenting an explicit construction of the Walker Wang models which realize both the twisted and untwisted gauge theories with gauge group ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$. We identify the topological order of the models by performing modular transformations on the ground state wave functions and show that the modular matrices exactly match those for the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ gauge theories. By relating the Walker-Wang construction to the Dijkgraaf-Witten construction, our result opens up a way to study twisted gauge theories with fermonic charges, and correspondingly strongly interacting fermionic symmetry protected topological phases and their surface states, through exactly solvable models.

23 pages

References in corpus (7)

• Characterizing topological order by studying the ground states of an infinite cylinder
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Anomalies of discrete symmetries in three dimensions and group cohomology
• Twisted Gauge Theory Model of Topological Phases in Three Dimensions
• Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
• Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding
• Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations

Cited by in corpus (1)

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