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While the topological order in two dimensions has been studied extensivelysince the discover of the integer and fractional quantum Hall systems,topological states in 3 spatial dimensions are much less understood. In thispaper, we propose a general formalism for constructing a large class ofthree-dimensional topological states by stacking layers of 2D topologicalstates and introducing coupling between them. Using this construction,different types of topological states can be obtained, including those withonly surface topological order and no bulk topological quasiparticles, andthose with topological order both in the bulk and at the surface. For bothclasses of states we study its generic properties and present several explicitexamples. As an interesting consequence of this construction, we obtain examplesystems with nontrivial braiding statistics between string excitations. Inaddition to studying the string-string braiding in the example system, wepropose a generic topological field theory description which can capture bothstring-particle and string-string braiding statistics. Lastly, we provide aproof of a general identity for Abelian string statistics, and discuss anexample system with non-Abelian strings.

Layer construction of 3D topological states and string braiding statistics