[论文解读] Line Defect Quantum Numbers & Anomalies

T. Daniel Brennan, Clay Córdova+1 more|arXiv (Cornell University)|Jun 30, 2022

Physics of Superconductivity and Magnetism被引用 21

一句话总结

该论文建立了一个框架，将线缺陷的对称性分数化与 ’t Hooft 奇点联系起来，使用 Maxwell theory 推导通过 RG 流动得到的非阿贝尔规范理论的异常。它将该方法应用于具备不同费米子表示的 SU(2) 规范理论，揭示混合异常和引力异常。

ABSTRACT

We explore the connection between the global symmetry quantum numbers of line defects and 't Hooft anomalies. Relative to local (point) operators, line defects may transform projectively under both internal and spacetime symmetries. This phenomenon is known as symmetry fractionalization, and in general it signals the presence of certain discrete 't Hooft anomalies. We describe this in detail in the context of free Maxwell theory in four dimensions. This understanding allows us to deduce the 't Hooft anomalies of non-Abelian gauge theories with renormalization group flows into Maxwell theory by analyzing the fractional quantum numbers of dynamical magnetic monopoles. We illustrate this method in $SU(2)$ gauge theories with matter fermions in diverse representations of the gauge group. For adjoint matter, we uncover a mixed anomaly involving the 0-form and 1-form symmetries, extending previous results. For $SU(2)$ QCD with fundamental fermions, the 't Hooft anomaly for the 0-form symmetries that is encoded by the fractionalization patterns of lines in the Maxwell phase is a consequence of the familiar perturbative (triangle) anomaly.

研究动机与目标

• Motivate and formalize the connection between symmetry fractionalization of line defects and discrete ’t Hooft anomalies.
• Describe how Maxwell theory in four dimensions encodes 1-form and 0-form symmetry data and their anomalies.
• Demonstrate how RG flows to Maxwell theory determine ’t Hooft anomalies of non-Abelian gauge theories with fermions in various representations.
• Analyze specific SU(2) gauge theories to identify the corresponding line defect quantum numbers and anomalies.
• Extend the framework to understand gravitational and mixed anomalies arising in these contexts.

提出的方法

• Review line defects and 1-form symmetries in Maxwell theory and their coupling to background fields.
• Introduce background 2-form fields B_e^{(2)} and B_m^{(2)} and the associated anomaly inflow action with A=i/2π ∫_{M_5} B_m^{(2)} ∧ dB_e^{(2)}.
• Restrict to discrete 1-form symmetry by using flat B^{(2)} fields and apply the Bockstein β_N to define discrete anomalies.
• Explain symmetry fractionalization for a connected 0-form symmetry G^{(0)} and how line defects can carry projective representations.
• Describe how RG flows triggered by Yukawa couplings and adjoint Higgs fields connect non-Abelian theories to Maxwell theory, enabling anomaly inference from IR line charges.
• Compute and interpret specific anomaly structures for SU(2) theories with fundamental and adjoint fermions, including gravitational and mixed anomalies.

实验结果

研究问题

• RQ1 How do line defect quantum numbers under G^{(0)} reflect ’t Hooft anomalies in the bulk theory?
• RQ2 What discrete anomalies arise from 1-form symmetries when Maxwell theory is coupled to background fields?
• RQ3 Can RG flows to Maxwell theory determine the ’t Hooft anomalies of non-Abelian gauge theories with various fermion representations?
• RQ4 What are the specific anomaly structures for SU(2) gauge theories with fundamental or adjoint fermions, and how do they relate to known perturbative anomalies?
• RQ5 How does symmetry fractionalization behave in the presence of 2-group obstructions and what does this imply for anomaly matching?

主要发现

• Maxwell theory exhibits an ’t Hooft anomaly between electric and magnetic 1-form symmetries, realizable via inflow from a 5D action with B_m^{(2)} ∧ dB_e^{(2)}.
• Discrete Z_N subgroups of the 1-form symmetry yield a reduced anomaly captured by β_N(b_e^{(2)}) ∪ b_m^{(2)} (and symmetrically).
• If Wilson and ’t Hooft lines transform projectively under SO(3), the model implements a 2-form obstruction that signals a mixed or gravitational anomaly.
• For SU(2) gauge theories with matter, the authors derive specific ’t Hooft anomalies for G^{(0)} in cases with fundamental and adjoint fermions, including a mixed anomaly in adjoint cases.
• In certain RG-flow contexts, anomalies of the IR Maxwell phase reproduce the perturbative chiral anomalies of the UV flavor symmetries, establishing a practical anomaly-detection method via line defect fractionalization.
• The framework extends to all-fermion electrodynamics and relates line defect fractionalization to discrete gravitational anomalies under appropriate manifold conditions.