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

Abstract

We analyze $2+1d$ and $3+1d$ Bosonic Symmetry Protected Topological (SPT)\nphases of matter protected by onsite symmetry group $G$ by using dual bulk and\nboundary approaches. In the bulk we study an effective field theory which upon\ncoupling to a background flat $G$ gauge field furnishes a purely topological\nresponse theory. The response action evaluated on certain manifolds, with\nappropriate choice of background gauge field, defines a set of SPT topological\ninvariants. Further, SPTs can be gauged by summing over all isomorphism classes\nof flat $G$ gauge fields to obtain Dijkgraaf-Witten topological $G$ gauge\ntheories. These topological gauge theories can be ungauged by first introducing\nand then proliferating defects that spoils the gauge symmetry. This mechanism\nis related to anyon condensation in $2+1d$ and condensing bosonic gauge charges\nin $3+1d$. In the dual boundary approach, we study $1+1d$ and $2+1d$ quantum\nfield theories that have $G$ 't-Hooft anomalies that can be precisely cancelled\nby (the response theory of) the corresponding bulk SPT. We show how to\nconstruct/compute topological invariants for the bulk SPTs directly from the\nboundary theories. Further we sum over boundary partition functions with\ndifferent background gauge fields to construct $G$-characters that generate\ntopological data for the bulk topological gauge theory. Finally, we study a\n$2+1d$ quantum field theory with a mixed $\\mathbb{Z}_2^{T/R} \\times U(1)$\nanomaly where $\\mathbb{Z}_2^{T/R}$ is time-reversal/reflection symmetry, and\nthe $U(1)$ could be a 0-form or 1-form symmetry depending on the choice of time\nreversal/reflection action. We briefly discuss the bulk effective action and\ntopological response for a theory in $3+1d$ that cancels this anomaly. This\nsignals the existence of SPTs in $3+1d$ protected by 0,1-form $U(1)\\times\n\\mathbb{Z}_{2}^{T,R}$.\n