Topological nonlinear σ -model, higher gauge theory, and a systematic construction of 3+1D topological orders for boson systems

Abstract

A discrete non-linear σ-model is obtained by triangulate both the space-time M d+1 and the target space K. If the path integral is given by the sum of all the complex homomorphisms φ : M d+1 → K, with an partition function that is independent of space-time triangulation, then the corresponding non-linear σ-model will be called topological non-linear σ-model which is exactly soluble. Those exactly soluble models suggest that phase transitions induced by fluctuations with no topological defects (i.e. fluctuations described by homomorphisms φ) usually produce a topologically ordered state and are topological phase transitions, while phase transitions induced by fluctuations with all the topological defects give rise to trivial product states and are not topological phase transitions. If K is a space with only non-trivial first homotopy group G which is finite, those topological nonlinear σ-models can realize all 3+1D bosonic topological orders without emergent fermions, which are described by Dijkgraaf-Witten theory with gauge group π1(K) = G. Here, we show that the 3+1D bosonic topological orders with emergent fermions can be realized by topological non-linear σ-models with π1(K) = finite groups, π2(K) = Z2, and πn>2(K) = 0. A subset of those topological non-linear σ-models corresponds to 2-gauge theories, which realize and classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections. The classification of 3+1D bosonic topological orders may correspond to a classification of unitary fully dualizable fully extended topological quantum field theories in 4-dimensions. CONTENTS 7 C. Topological non-linear σ-models 8 D. Labeling simplices in a complex 8 III. Dijkgraaf-Witten gauge theories from topological non-linear σ-models 9 A. Lattice gauge theories from topological non-linear σ-models 9 B. Classification of exactly soluble 1-gauge theories 11 IV. 2-gauge theories from topological non-linear σ-models 11 A. 2-groups 11 B. 2-gauge theories 13 C. 2-group cocycles 14 D. Cohomology of 2-group 15 V. Pure 2-gauge theory of 2-gauge-group B(Π2, 2) 16 A. Pure 2-group and pure 2-gauge theory 16 B. Pure 2-gauge theory in 3+1D 17 1. n = odd case 17 2. n = even case 17 3. Properties and duality relations 17 VI. 3+1D 2-gauge theory of 2-gauge-group B(G b , Z f 2 ) 18 A. The Lagrangian and space-time path integral 18 B. The equivalence between [k0,ē2(ā),n3(ā),ν4(ā)]'s 18 C. 2-gauge transformations in the cocycle σ-model 19 D. The vanishing of the partition function 19 E. The pointlike and stringlike excitations in the 2-gauge theory 20 VII. Classify and realize 3+1D EF1 topological orders by 2-gauge theories of 2-gauge-group B(G b , Z f 2 ) 21 VIII. Realize 3+1D EF2 topological orders by topological non-linear σ-models 21 A. Construction of topological non-linear σ-models 21 B. The canonical boundary of topological non-linear σ-models 26 IX. Summary 26 A. Space-time complex, cochains, and cocycles 27 B. Lyndon-Hochschild-Serre spectral sequence 30 C. Partition f...

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