Bosonic topological phases of matter: bulk-boundary correspondence, SPT invariants and gauging

TL;DR

The paper develops a bulk–boundary framework to classify bosonic SPT phases with onsite symmetry $G$, using invertible TQFTs and background $G$-gauge-field probes to define topological invariants via $\mathcal{Z}^q[N,A]=e^{iI^q[N,A]}$ and their gauged DW counterparts. It demonstrates how SPT invariants arise from generators of cobordism groups $\Omega^{SO}_{d+1}(BG)$ and how gauging leads to DW theories, with ungauging achieved by proliferating dual symmetry defects; the boundary theories exhibit $G$-t’Hooft anomalies that are cancelled by the bulk, allowing a consistent bulk–boundary coupling. The work provides explicit bulk actions for $2+1d$ and $3+1d$ SPTs across cocycle types I–IV, computes invariants on lens spaces, $T^3$, and $T^4$, and constructs $G$-characters from boundary data to reproduce bulk topological data. It also analyzes a mixed $U(1)$ and $\Z_2^{T/R}$ anomaly in $2+1d$ and proposes bulk terms in $3+1d$ that realize corresponding SPTs with these symmetries. Overall, the results establish a robust correspondence between bulk topological response, gauged boundary theories, and boundary anomalies, providing concrete tools to extract bulk invariants from edge data and to classify higher-dimensional SPTs with abelian onsite symmetries.

Abstract

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 field theory which upon coupling to a background flat $G$ gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines 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 spoils 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 cancelled 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 $G$-characters that generate topological data for the bulk topological gauge theory. Finally, we study a $2+1d$ quantum field theory with a mixed $\mathbb{Z}_2^{T/R} \times U(1)$ anomaly where $\mathbb{Z}_2^{T/R}$ 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 $U(1)\times \mathbb{Z}_{2}^{T,R}$.

Figures (4)

• Figure 1: Triangulation of a three-torus containing one 0-simplex, three 1-simplices, three 2-simplices and six 3-simplices.
• Figure 2: A triangulation of $T^2$ with flux $a,b\in G$ along the two cycles. Dijkgraaf Witten theory labelled by $H^{2}_{\text{group}}(G,U(1))$ associates the $U(1)$ phase $c(a,b)/c(b,a)$ to this assohnment $A$.
• Figure 3: Configuration of a flat $\mathbb{Z}_n$ gauge field on a 3-simplex. $a,b,c\in \mathbb{Z}_n$.
• Figure 4: Triangulation of a three-torus containing one 0-simplex, three 1-simplices, three 2-simplices and six 3-simplices.