[Paper Review] 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 chiralityChemistry24 citations

TL;DR

The paper develops a two-obstruction framework for anomalies of non-invertible self-duality symmetries in 2d and 4d via Symmetry TFT, relating duality invariants, equivariantization, and symmetry fractionalization to obstruction to gauging.

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.

Motivation & Objective

• 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.

Proposed method

• 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 (lavored by ermi-like data) 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.

Experimental results

Research questions

• 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?

Key findings

• 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.