[Paper Review] Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology

Anton Kapustin|arXiv (Cornell University)|Mar 6, 2014

Topological Materials and PhenomenaPhysics and Astronomy13 references177 citations

TL;DR

This paper proposes that bosonic Symmetry Protected Topological (SPT) phases with finite internal symmetry groups are classified by the Pontryagin-dual of the torsion subgroup of the degree-$d$ oriented bordism group of the classifying space $BG_0$, refining the group cohomology classification. It shows that for $d \leq 6$, the cobordism classification is strictly finer than group cohomology, capturing previously missed SPT phases—especially in $d=4$—and connects this to anomalies in lower dimensions via anomaly inflow.

ABSTRACT

We propose that Symmetry Protected Topological Phases with a finite symmetry group G are classified by cobordism groups of the classifying space of G. This provides an explanation for the recent discovery of bosonic SPT phases which do not fit into the group cohomology classification. We discuss the connection of the cobordism classification of SPT phases to gauge and gravitational anomalies in various dimensions.

Motivation & Objective

• To resolve the failure of group cohomology to classify certain 4D bosonic SPT phases with time-reversal symmetry.
• To provide a refined classification of interacting bosonic SPT phases using oriented cobordism groups with $U(1)$ coefficients.
• To clarify the relationship between SPT phases and anomalies in lower dimensions via anomaly inflow.
• To extend the classification to systems with time-reversal symmetry by introducing twisted cobordism with $\rho$-twisted coefficients.
• To demonstrate that higher Stiefel-Whitney classes ($w_2^2$, $w_4$) can support new SPT phases not captured by group cohomology.

Proposed method

• Classify SPT phases using the Pontryagin-dual of the torsion subgroup of the oriented bordism group $\Omega_{SO,d}(BG_0)$, denoted $\Omega^{d}_{SO}(BG_0, U(1)) / \mathrm{im}\, e$.
• Introduce twisted cobordism groups $\Omega^{d}_{SO}(BG, U(1)^\rho)$ for symmetry groups $G$ with time-reversal elements, where $\rho: G \to \mathbb{Z}_2$ defines the twist.
• Use the Thom homomorphism to relate group cohomology $H^d(BG, U(1))$ to the cobordism classification, showing the map is neither injective nor surjective.
• Analyze anomalies via anomaly inflow: SPT phases in $d$ dimensions correspond to 'anomalous' theories in $d-1$ dimensions with nontrivial partition functions.
• Construct topological actions using Stiefel-Whitney classes $w_2$, $w_3$, $w_4$ to describe SPT phases, particularly in $d=4$ and $d=5$.
• Verify consistency via boundary anomaly cancellation: the bulk action variation is canceled by a boundary action coupling to $w_2$ and $w_3$.

Experimental results

Research questions

• RQ1 Why do certain 4D bosonic SPT phases with ${\mathbb{Z}}_2^T$ symmetry fail to be classified by group cohomology?
• RQ2 How can cobordism groups provide a more complete classification of SPT phases than group cohomology?
• RQ3 What is the role of higher Stiefel-Whitney classes in constructing topological actions for SPT phases?
• RQ4 How are SPT phases related to anomalies in lower-dimensional theories via anomaly inflow?
• RQ5 Under what conditions does the cobordism classification refine or differ from the group cohomology classification?

Key findings

• The cobordism classification captures 4D SPT phases not described by group cohomology, such as those involving $w_2^2$ or $w_4$.
• For $d \leq 6$, the cobordism classification is strictly finer than group cohomology, with the map from cohomology to cobordism being neither injective nor surjective.
• In $d=4$, new SPT phases arise from nontrivial combinations of Stiefel-Whitney classes that are not captured by $H^4(BG, U(1))$.
• The classification of SPT phases with time-reversal symmetry requires twisted cobordism groups $\Omega^{d}_{SO}(BG, U(1)^\rho)$, which are pure torsion and thus do not require quotienting by real coefficients.
• The partition function of the proposed 4D SPT phase on $\mathbb{C}\mathbb{P}^2$ is predicted to be $-1$, distinguishing it from trivial phases.
• Boundary anomalies in $d=3$ and $d=4$ are canceled by coupling to $\mathbb{Z}_2$ topological gauge theories with actions involving $w_2$ and $w_3$.