Non-invertible Symmetries Induce Weak Hopf Algebras and Extended Systems Via Tannaka-Krein Duality

The fundamental nature of symmetry plays a crucial role in physics, yet conventional understanding largely focuses on symmetries that are perfectly reversible, or invertible. Shadi Ali Ahmad from New York University, Marc S. Klinger from the California Institute of Technology, and Yifan Wang, also from New York University, now explore the more subtle world of non-invertible symmetries, revealing how these less-understood principles shape physical systems. Their work establishes a deep connection between mathematical frameworks describing these symmetries, demonstrating that a categorical symmetry induces an algebraic symmetry with potentially broader implications. This research introduces a novel method for analysing how these non-invertible symmetries break down, identifying a key quantity that measures the extent of this breakdown and ultimately offering new insights into the behaviour of complex systems, particularly in areas like topological and conformal field theories where such symmetries arise naturally as defects or boundary conditions.

Topological Phases, Symmetry, and Quantum Field Theory

This compilation details advanced theoretical physics research, focusing on topological quantum field theory, conformal field theory, and non-invertible symmetries, alongside operator algebras, entanglement, condensed matter physics, and holography. Core theoretical frameworks like topological and conformal field theories are highlighted, alongside the mathematical foundations provided by operator algebras, particularly Von Neumann algebras. A recurring theme is entanglement entropy, used to probe quantum states and reconstruct geometric properties. A dominant focus lies on non-invertible symmetries and the process of gauging these symmetries, which involves consistently incorporating them into physical theories.

These symmetries are often associated with topological defects, such as defect lines, which play a crucial role in understanding their properties. The references also explore 2-group symmetries, a more refined type relevant to non-invertible cases. Mathematical tools like modular categories, orbifolds, fusion rules, and relative entropy are central to these investigations. Connections to condensed matter physics are evident through references to Levin-Wen models, topological phases, and Kitaev models. The bibliography also explores emergent geometry and holography, including the AdS/CFT correspondence and the reconstruction of geometry from entanglement. Recent trends emphasize non-invertible symmetries, the connection between entanglement and geometry, defect TQFTs, and the challenges of gauging non-invertible symmetries. In summary, this collection represents a cutting-edge snapshot of research exploring the interplay between quantum information, topology, and emergent spacetime, with a notable emphasis on non-invertible symmetries.

Fusion Categories and Non-Invertible Symmetry Reconstruction

Scientists have established a precise connection between fusion categories, which describe symmetries, and weak Hopf algebras, which encode algebraic symmetries, through a process known as Tannaka-Krein duality. This work demonstrates that a fusion categorical symmetry induces an algebraic symmetry, but this reconstruction is not unique, leading to an extended system relative to the original categorical one. Researchers developed a method for analysing symmetry breaking patterns in these non-invertible symmetries, utilizing a conditional expectation analogous to group averaging, and identified an index of this expectation as a key information theoretic quantity. Experiments reveal that the entropic order parameter, a measure of symmetry breaking, is bounded by a value dependent on the symmetry category, specifically by log(r|C|), where ‘r’ represents a total quantum dimension.

This bound refines previous understandings of symmetry breaking and incorporates the effects of choices made during the algebraic reconstruction. The team demonstrated that anomalous symmetries produce greater diversity in symmetry breaking, as measured by the relative entropy S(ψ|ψsym). Furthermore, the study establishes a link between 2-dimensional quantum field theories and these symmetries through strip algebras, also known as annular algebras, which are C∗weak Hopf algebras reconstructed from the fusion category. These algebras naturally arise when studying quantum field theories with boundaries, encoding superselection sectors, and play a role in defining entanglement entropy by factorizing the Hilbert space. The research confirms that the entropic order parameter remains non-negative and is zero only when the system is fully symmetric, providing a faithful diagnostic for symmetry breaking and quantifying the extent of that breaking. This work lays the foundation for investigating higher-dimensional symmetries through layered fusion categories and higher weak Hopf algebras.

Categorical Symmetries and Weak Hopf Algebra Mapping

This research establishes a novel connection between algebraic and categorical descriptions of symmetry, particularly focusing on non-invertible symmetries. Scientists demonstrate how a fusion categorical symmetry induces an algebraic symmetry encoded within a weak Hopf algebra. Importantly, this mapping is not unique, leading to an extended algebraic symmetry compared to the original categorical one. The team developed a method for analysing how these non-invertible symmetries break down, introducing a conditional expectation that functions as an analogue to group averaging for standard symmetries.

They identify an index related to this expectation as a key quantity determining the extent of symmetry breaking. The researchers highlight that ambiguities arising from the non-unique mapping between categories and algebras result in distinct patterns of symmetry breaking compared to those observed with invertible symmetries. They illustrate their approach using examples from topological and conformal field theories, interpreting non-invertible symmetries as defect operators and boundary conditions. The authors acknowledge that the non-uniqueness of the categorical reconstruction introduces complexities in fully characterizing symmetry breaking patterns, and further investigation is needed to explore these nuances. Future research directions include applying this framework to a wider range of physical systems and exploring the implications for renormalization group flows and duality relationships.