Team Constructs (3+1)-Dimensional Topological Quantum Field Theories Realizing Anomalies of Finite Symmetries

The quest to understand anomalies, inconsistencies arising in physical theories, drives much of modern physics, and recent work by Arun Debray, Weicheng Ye, and Matthew Yu from the University of Kentucky establishes a powerful new method for building quantum field theories that exhibit specific anomalous behaviour. These researchers develop a systematic framework for constructing (3+1)-dimensional topological quantum field theories, crucial for modelling systems ranging from particle physics to condensed matter materials. Their approach extends existing symmetry-extension techniques to encompass fermionic systems, leveraging advances in the classification of anomalous theories and a refined method for calculating key mathematical properties. By uniting supercohomology, cobordism theory, and higher-categorical structures, the team provides a concrete route to designing quantum field theories with precisely controlled anomalies, forging a conceptual link between mathematical consistency and physical realisation.

Anomaly Inflow and Dimensional Reduction of TQFTs

Researchers have developed a systematic framework for constructing (3+1)-dimensional topological quantum field theories that exhibit anomalies. These theories arise from a mathematical process called dimensional reduction, where higher-dimensional (4+1)-dimensional theories with a symmetry group G are reduced to lower-dimensional (3+1)-dimensional theories with a subgroup H. The approach centres on understanding how anomalies on the boundary of the higher-dimensional theory constrain the allowed anomalies in the lower-dimensional theory, enabling the construction of consistent anomalous theories. The core of the method involves classifying the ways anomalies can be cancelled through this inflow process, providing a complete description of the allowed anomaly patterns in the (3+1)-dimensional theory.

Scientists demonstrate that the anomaly structure is determined by the choice of subgroup H within G and the corresponding representation theory of the symmetry group. This provides a powerful tool for building specific anomalous theories with desired properties, offering a systematic way to explore the landscape of possible quantum field theories. The team shows that this framework establishes a connection between the anomaly structure of the (3+1)-dimensional theory and the properties of the (4+1)-dimensional parent theory, providing new insights into the relationship between different dimensions and symmetry structures.

Topological quantum field theories (TQFTs) that realise specified anomalies of finite symmetries appear in gauge theories with fermions or fermionic lattice systems. The approach generalises the Wang, Wen, Witten symmetry-extension construction to the fermionic setting, building on recent advances in the study of fermionic TQFTs and related homotopy theory. By integrating supercohomology and cobordism methods, the research establishes a framework for constructing and analysing TQFTs with fermionic symmetries.

Fermionic Anomalies in Four-Dimensional TQFTs

Scientists have developed a systematic framework for constructing four-dimensional topological quantum field theories (TQFTs) that exhibit specific anomalies associated with finite symmetries, relevant to gauge theories containing fermions or fermionic lattice systems. This work extends the established Wang-Wen-Witten symmetry-extension construction to encompass fermionic systems, building upon recent advances in the study of fermionic TQFTs and related homotopy theory. The team leverages a categorical classification of anomalous TQFTs in four dimensions and further develops a method for accelerating computations of supercohomology groups, closely mirroring techniques from cobordism theory. The research integrates supercohomology and cobordism methods within a recently established categorical framework of fusion 2-categories, providing a concrete route to constructing fermionic TQFTs with specified anomalies.

Scientists demonstrate the construction of these TQFTs by explicitly providing a procedure for trivializing anomalies through symmetry extension, effectively enlarging the symmetry group to accommodate the anomaly. The team utilizes supercohomology, a mathematical tool for classifying TQFTs, and describes fermionic symmetries using a bosonic symmetry group, a class in first cohomology, and a class in second cohomology. This framework allows for a precise description of the symmetry and its associated anomaly, offering new insights into the relationship between symmetry, anomaly, and topological order in physical systems.

Topological Foundations For Anomalies And Physics

This extensive list of references focuses heavily on algebraic topology, differential topology, mathematical physics, and condensed matter physics. The following breakdown categorises the key themes and highlights interconnected areas. I. Core Algebraic and Differential Topology (Foundation) This section represents the fundamental tools and concepts used throughout the bibliography. The works of Eilenberg and MacLane are essential for understanding homology and cohomology theories, which underpin much of the research.

Atiyah’s work on Riemann surfaces and spin structures provides a classic connection between differential geometry, topology, and spin structures, crucial for understanding anomalies and string theory. Conner and Floyd, and Gilkey delve into specific aspects of bordism theory, particularly with involutions and spin structures. Bordism is a powerful tool for classifying manifolds and understanding their properties, with a focus on involutions being important for understanding symmetry and related anomalies. The Stacks Project is a modern, collaborative online resource for algebraic geometry and related fields, serving as a go-to reference for definitions and concepts.

II. Anomalies, Symmetry, and Mathematical Physics This is a major theme, connecting topology to physics. Cordova, Ohmori, Shao, and Yan focus on decorated defects and their anomalies. Davighi and Lohitsiri explore anomalies, particularly in the context of 2-groups and cobordism, developing a systematic way to understand anomalies using topological methods. Debray is a central figure, with work at the intersection of bordism, anomalies, symmetry breaking, and string theory.

Papers with Krulewski, Liu, Pacheco-Tallaj, and Thorngren suggest a focus on understanding the interplay between symmetry, defects, and anomaly matching. Freed and Hopkins’ work on the consistency of M-theory applies topological methods to understand the consistency of M-theory on non-orientable manifolds. Gitler, Mahowald, and Milgram deal with immersion problems and cohomology operations. Guo, Ohmori, Putrov, Wan, and Wang use cobordism to study interacting symmetric/crystalline orders. Lawson provides a comment on the Atiyah-Hirzebruch spectral sequence, a powerful tool in K-theory, which is related to anomalies.

Mahowald and Bruner work on the existence of certain maps in stable homotopy, relevant to computations in K-theory and potentially anomaly calculations. III. Condensed Matter Physics and Topological Phases This section focuses on applying topological ideas to understand materials with novel properties. Behrens, Hill, Hopkins, and Mahowald work on the Adams spectral sequence, a tool for computing stable homotopy groups, which can be used to classify topological phases of matter. Zhang, Wang, Yang, Qi.