[论文解读] Anomalies of non-invertible self-duality symmetries: fractionalization and gauging

Andrea Antinucci, Francesco Benini+3 more|arXiv (Cornell University)|Aug 22, 2023

Molecular spectroscopy and chirality被引用 24

一句话总结

本文提出一个针对二维与四维中不可逆自对偶对称性异常的两步阻碍框架，通过 Symmetry TFT，将对偶不变量、等变换化、以及对称性分数化与量化阻碍（gauging 的障碍）联系起来。

ABSTRACT

We study anomalies of non-invertible duality symmetries in both 2d and 4d, employing the tool of the Symmetry TFT. In the 2d case we rephrase the known obstruction theory for the Tambara-Yamagami fusion category in a way easily generalizable to higher dimensions. In both cases we find two obstructions to gauging duality defects. The first obstruction requires the existence of a duality-invariant Lagrangian algebra in a certain Dijkgraaf-Witten theory in one dimension more. In particular, intrinsically non-invertible (a.k.a. group theoretical) duality symmetries are necessarily anomalous. The second obstruction requires the vanishing of a pure anomaly for the invertible duality symmetry. This however depends on further data. In 2d this is specified by a choice of equivariantization for the duality-invariant Lagrangian algebra. We propose and verify that this is equivalent to a choice of symmetry fractionalization for the invertible duality symmetry. The latter formulation has a natural generalization to 4d and allows us to give a compact characterization of the anomaly. We comment on various possible applications of our results to self-dual theories.

研究动机与目标

• Motivate a unified understanding of ’t Hooft anomalies for non-invertible self-duality symmetries in d = 2 and d = 4.
• Describe how the Symmetry TFT encodes obstructions to gauging duality defects.
• Identify two concrete obstructions: existence of a G-invariant Lagrangian algebra and vanishing of a pure anomaly for the invertible duality symmetry.
• Relate equivariantization data to symmetry fractionalization and potential anomaly cancellation.
• Provide concrete checks in Tambara-Yamagami categories and higher-dimensional cases.

提出的方法

• Adopt the Symmetry TFT framework to translate anomalies into boundary condition obstructions.
• Use Dijkgraaf-Witten theories as bulk models for duality defects and analyze gauging via Lagrangian algebras.
• Introduce a two-obstruction scheme: (i) existence of a G-invariant Lagrangian algebra in DW theory, (ii) vanishing of a cubic/DM twist determined by equivariantization data.
• Employ equivariantization (lavor) to encode symmetry fractionalization and its effect on anomalies.
• Generalize the analysis from 2d Tambara-Yamagami categories to 4d duality defects and their symmetry TFTs.

实验结果

研究问题

• RQ1 What obstructions prevent gauging of non-invertible duality defects in 2d and 4d?
• RQ2 How does the Symmetry TFT encode the anomaly data for duality defects via Lagrangian algebras and equivariantization?
• RQ3 How does symmetry fractionalization influence the cancellation or manifestation of anomalies in these settings?
• RQ4 Can the obstruction framework reproduce known TY-category anomalies and extend to higher dimensions?
• RQ5 What is the relationship between duality invariance, Lagrangian algebras, and Dijkgraaf-Witten twists in the bulk?

主要发现

• Two obstructions to gauging duality defects are identified: a G-invariant Lagrangian algebra in the bulk DW theory (duality symmetry is non-intrinsic and intrinsically anomalous if absent) and the vanishing of a pure anomaly for the invertible duality symmetry (dependent on equivariantization data).
• In 2d, the second obstruction is tied to a choice of equivariantization that encodes symmetry fractionalization for the invertible duality symmetry.
• The framework reproduces known TY-category anomaly results in 2d and extends the obstruction analysis to 4d duality defects.
• The bulk analysis involves DW theories, SPT twists, and the interplay of Rep(G) gauging with Lagrangian algebras to yield Neumann boundary conditions that diagnose anomalies.
• Equivariantization data (lavor) can shift the SPT phase Y and thus affect the cubic anomaly, providing a mechanism for anomaly cancellation in some cases.