Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases (1912.02817v1)

Abstract: We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.

An Overview of the Paper "Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases"

Introduction

The paper under review explores the intricate notion of fusion category symmetries in 1+1D quantum field theories (QFTs) and the 't Hooft anomalies associated with such symmetries. A paramount aspect of this work is the classification of gapped phases stabilized by these symmetries and the implications for theories exhibiting these exotic symmetries. The significant departure from traditional group symmetries, notably the absence of invertibility in the topological defect lines, underpins the complexity addressed by the fusion category framework.

Main Contributions

1. Fusion Categories and 't Hooft Anomalies: The authors delve into the environment where symmetries of QFTs are not defined by invertible transformations but instead by fusion categories. They develop an approach to track symmetries through renormalization group (RG) flows via the algebra of topological operators that form a fusion category. This setup extends the classical understanding of 't Hooft anomalies by including non-group-like symmetries.
2. Anomaly In-Flow and Turaev-Viro Model: The paper introduces a mechanism to paper anomalies through the lens of anomaly in-flow, connecting 1+1D theories to 2+1D Turaev-Viro models. By modeling a boundary condition within the Turaev-Viro formalism, the authors propose how fusion category symmetries can be gauged on the boundary of such models.
3. Classification of Gapped Phases: A cornerstone of the paper is the classification of gapped phases under fusion category symmetries. In particular, they categorize these phases using module categories over the fusion category and subsequently propose conditions, such as the existence of fiber functors, for anomaly-free phases.
4. Applications to Known Models: By applying their theoretical framework, the authors provide a rigorous analysis of several known conformal field theories (CFTs), including the Ising model and its dualities. They elucidate the impact of respective symmetries and anomalies, particularly focusing on Tambara-Yamagami categories, which illustrate self-dualities under distinct gauging operations.

Analytical Framework

The paper constructs its theoretical infrastructure on the algebraic structures provided by fusion categories—primarily focusing on scenarios where dimensions of certain elements may not be integral. By employing Turaev-Viro/Levin-Wen models, it distinguishes between symmetry-preserving and symmetry-breaking phases and illustrates how the anomaly in-flow technique governs the possible phase diagram transformations.

An instrumental part of the discussion involves the construction of gapped boundary conditions and the exploration of when these conditions can be deemed topologically invariant. The paper demonstrates that establishing a non-degenerate, symmetry-preserving boundary is equivalent to finding a fiber functor for the associated fusion category.

Results and Implications

The insightful analysis reveals several important phenomena:

• The necessity of non-integral quantum dimensions for the presence of anomalies, reinforcing the idea that certain fusion categories are inherently anomalous.
• Identification of self-dual SPT phases and their constructions via duality twisted sectors, offering new insights into how symmetry fractionalization can manifest in quantum theories.
• Clarification of gauge theory analogs where non-invertible magnetic symmetries are understood in higher dimensions.

The implications of this paper are profound in both theoretical and practical spheres. The expansion of symmetry-related concepts in QFTs presented here suggests that many known phenomena could be revisited through the lens of fusion categories, potentially unveiling novel topological phases and quantum orders.

Future Directions

The authors propose several avenues for future research, such as classifying higher-dimensional TQFTs with fusion category symmetries, understanding MPO symmetry realizations on lattices, and refining anomaly in-flow mechanisms for more complex topological settings. Moreover, the exploration of non-invertible categories in the landscape of condensed matter physics and higher-energy theoretical models holds the promise of enriching both foundational knowledge and applicability.

In conclusion, the paper successfully positions fusion categories as a versatile framework for capturing unconventional symmetries in QFTs, potentially reshaping the theoretical landscape by providing tools to explore novel quantum phenomena.