Gapped boundaries of 3D bosonic topological orders from mapping class group representations

Topologically ordered (TO) phases display interesting bulk-boundary relationships: a striking example of this is a chiral edge mode of a fractional quantum Hall phase. Non-chiral TOs, by contrast, allow gapped boundary conditions. Gapped boundaries of 2D TOs have been classified in terms of Lagrangian algebras. A similar systematic understanding in 3D has not been developed. In this work, we take steps in that direction. We consider a Dijkgraaf-Witten gauge theory defined on the spacetime T³ x S¹, where spatial coordinates form a 3-torus T³. The ground states of this 3D TO form a representation of the mapping class group of T³, isomorphic to SL(3,Z). In the quasiparticle basis, the eigenvectors stabilized by the S and T matrices – generators of SL(3,Z) – correspond to vector-valued partition functions of gapped boundary states. We demonstrate this for the simplest example, the 3D Z₂ TO, and find that our method reproduces the known rough and smooth boundaries. Further, we consider Z₂×Z₂ TO and find 5 classes of gapped boundaries. We relate our results to gapped phases of 2D symmetric quantum models via the Symmetry-TO correspondence. We propose a way to systematically identify symmetry-enforced gaplessness of 2D bosonic theories.