[论文解读] Anomalies of Non-Invertible Symmetries in (3+1)d

Clay Córdova, Po-Shen Hsin+1 more|arXiv (Cornell University)|Aug 22, 2023

Atomic and Subatomic Physics Research被引用 14

一句话总结

本文通过在4+1d中使用对称性 TQFTs，分析非可逆的3+1d对称性的异常，聚焦于对偶性样缺陷和更高Frobenius-Schur型数据，以判定此类对称性何时能够在能隙相中实现。

ABSTRACT

Anomalies of global symmetries are important tools for understanding the dynamics of quantum systems. We investigate anomalies of non-invertible symmetries in 3+1d using 4+1d bulk topological quantum field theories given by Abelian two-form gauge theories, with a 0-form permutation symmetry. Gauging the 0-form symmetry gives the 4+1d "inflow" symmetry topological field theory for the non-invertible symmetry. We find a two levels of anomalies: (1) the bulk may fail to have an appropriate set of loop excitations which can condense to trivialize the boundary dynamics, and (2) the "Frobenius-Schur indicator" of the non-invertible symmetry (generalizing the Frobenius-Schur indicator of 1+1d fusion categories) may be incompatible with trivial boundary dynamics. As a consequence we derive conditions for non-invertible symmetries in 3+1d to be compatible with symmetric gapped phases, and invertible gapped phases. Along the way, we see that the defects characterizing $\mathbb{Z}_{4}$ ordinary symmetry host worldvolume theories with time-reversal symmetry $\mathsf{T}$ obeying the algebra $\mathsf{T}^{2}=C$ or $\mathsf{T}^{2}=(-1)^{F}C,$ with $C$ a unitary charge conjugation symmetry. We classify the anomalies of this symmetry algebra in 2+1d and further use these ideas to construct 2+1d topological orders with non-invertible time-reversal symmetry that permutes anyons. As a concrete realization of our general discussion, we construct new lattice Hamiltonian models in 3+1d with non-invertible symmetry, and constrain their dynamics.

研究动机与目标

• Motivate and define anomalies of non-invertible symmetries in 3+1d using symmetry TQFTs in one higher dimension.
• Classify and diagnose first and second level obstructions to realizing symmetric gapped phases.
• Relate anomalies to 4+1d abelian 2-form gauge theories, Lagrangian subgroups, and 4+1d SPTs with Frobenius-Schur-type data ω and ωf.
• Provide concrete lattice model realizations and discuss ST^n gauging and duality defects.
• Generalize KW-type duality analyses to higher dimensions and connect to 2+1d FS indicators.
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• A Non-invertible fusion rules from Aut(K) symmetry A.1 Example: Z_N two-form gauge theory B Generalized Frobenius Schur-indicator and statistics C Trivializing the anomaly by symmetry extension

提出的方法

• Frame non-invertible 3+1d symmetries via symmetry TQFTs in 4+1d obtained by gauging a 0-form symmetry of an Abelian 2-form gauge theory.
• Study Lagrangian subgroups of Z_N 2-form gauge theories to realize gapped boundaries compatible with a given Aut(K) action.
• Analyze the polarization constraint ensuring the Lagrangian subgroup intersections with a canonical polarization are trivial, governing anomaly freedom.
• Investigate Kramers-Wannier-like non-invertible symmetry ST^n gauging and classify admissible N for duality-invariant boundaries.
• Incorporate the 4+1d SPT (ω, ω_f) data as decorating domain walls, leading to a second level obstruction to anomaly cancellation.
• Decorate domain walls with T^2=C or T^2=(-1)^F C SPTs and cancel endpoints via duality defect anomalies to determine when the symmetry is anomaly-free.

实验结果

研究问题

• RQ1 What are the criteria for the first level obstruction (Lagrangian subgroups) to allow a gapped, symmetry-preserving boundary for 3+1d non-invertible symmetries?
• RQ2 How do ω and ω_f (the 4+1d SPT/Frobenius-Schur indicators) affect the second level obstruction and anomaly cancellation for non-invertible T symmetries in 3+1d?
• RQ3 In which cases (values of N and n) can ST^n gauging yield anomaly-free duality-invariant boundaries?
• RQ4 How do decorated domain walls and symmetry extensions cancel anomalies in 3+1d non-invertible symmetries?
• RQ5 What lattice realizations illustrate ST^n gauging and non-invertible fusion rules in 3+1d?

主要发现

• For trivial ω or ω_f, duality defects can form anomaly-free non-invertible symmetries given a duality-invariant magnetic Lagrangian subgroup.
• When ω or ω_f is nontrivial, the symmetry is always anomalous for odd N, but can be anomaly-free or anomalous for even N depending on ω classifications.
• ω describes 4+1d Z_4 SPTs decorated domain walls ending on a boundary with endpoints hosting 2+1d T^2=C anomalies; cancellation with duality defect anomalies can occur for even N, not for odd N.
• ω_f describes 4+1d Z_4 SPTs decorated domain walls ending on a boundary with T^2=(-1)^F C anomalies; cancellation occurs for even N and even ω_f classes, leading to anomaly-free cases.
• The framework reproduces known 1+1d results via a higher-dimensional, decorated domain wall perspective and extends to general KW duality-like symmetries in any spacetime dimension.
• Concrete lattice models for Z_N 1-form symmetry theories and ST^n gauging illustrate phases and fusion/condensation structures, including the ST^n domain walls and their boundaries