Bosonic topological phases of matter: Bulk-boundary correspondence, symmetry protected topological invariants, and gauging

摘要

We analyze 2 + 1d and 3 + 1d bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group G by using dual bulk and boundary approaches. In the bulk, we study an effective held theory, which upon coupling to a background flat G gauge held furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, delines a set of SPT topological invariants. Further. SPTs can be gauged by summing over all isomorphism classes of flat G gauge fields to obtain Dijkgraaf-Witten topological G gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in 2 + 1d and condensing bosonic gauge charges in 3 + 1d. In the dual boundary approach, we study 1 + 1d and 2 + 1d quantum field theories that have G 't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct C characters that generate topological data for the bulk topological gauge theory. Finally, we study a 2 + 1d quantum field theory with a mixed Z_2~(T/R) × U(1) anomaly where Z_2~(T/R) is time-reversal/reflection symmetry, and the U(1) could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in 3 + 1d that cancels this anomaly. This signals the existence of SPTs in 3 + 1d protected by 0,1-form U(1) × Z_2~(T,R).