Lattice Gauge Theory Reveals Higher Symmetries Encoded in Crossed Group Structures

The fundamental symmetries governing particle physics extend beyond the familiar rotations and translations, and researchers increasingly investigate more complex, higher-form symmetries. Anton Kapustin and Lev Spodyneiko, both from the California Institute of Technology, along with their colleagues, explore these symmetries within the framework of lattice gauge theory, a powerful tool for understanding the strong force. Their work reveals a deep connection between these higher symmetries and ‘t Hooft anomalies, subtle quantum effects that constrain theoretical models. By demonstrating how these symmetries arise from restrictions of symmetry transformations within specific regions of space, the team proposes a novel mathematical structure, crossed n-cubes of groups, to model these phenomena, offering new insights into the underlying principles of quantum field theory and potentially guiding the development of more accurate physical models

Restrictions are naturally packaged into a higher group structure. For example, in two-dimensional gauge theories, this information is encoded in a crossed square of groups, an algebraic model of a 3-group. This work proposes that higher groups appear in lattice models and quantum field theory as crossed n-cubes of groups, using a nonabelian version of the ˇCech construction. This approach builds upon established concepts in homotopy theory and algebraic topology to provide a framework for understanding systems with complex symmetries and interactions, potentially revealing new avenues for theoretical development.

Higher Symmetries and Topological Matter Classification

This document presents a detailed exploration of advanced concepts in mathematics and physics, specifically related to symmetry, topology, and quantum field theory. The central theme is the investigation of symmetries beyond traditional group theory, including 2-group symmetries and higher-dimensional algebraic structures, connecting these to the classification and understanding of topological phases of matter, phases characterized by robust properties insensitive to local disturbances. A crucial concept is that of anomalies, inconsistencies in quantum field theory arising from symmetries, and how these anomalies restrict possible symmetries and phases. The work utilizes crossed modules and 2-groups as algebraic tools to describe these higher symmetries, relying heavily on concepts from algebraic topology, such as homotopy groups, to classify and understand them.

The ideas are applied to both continuous quantum field theory and discrete lattice systems, bridging these two approaches. The document begins by introducing the importance of higher symmetries in understanding topological phases of matter, highlighting the limitations of traditional symmetry concepts. It then lays out the mathematical framework, detailing crossed modules, 2-groups, and homotopy groups, before connecting this framework to physics, demonstrating how higher symmetries can classify topological phases and how anomalies restrict possible symmetries. Applications to quantum field theory and lattice systems are explored, and specific examples illustrate these concepts.

The work demonstrates that traditional group theory is insufficient to capture all forms of symmetry in physical systems. Crossed modules and 2-groups provide a powerful framework for describing these more subtle symmetries, leading to topological invariants that characterize different phases of matter. Anomalies play a crucial role in restricting possible symmetries and phases, bridging mathematics (algebraic topology, group theory) and physics (condensed matter, quantum field theory).

Higher Symmetries and ‘t Hooft Anomalies Revealed

Researchers have developed a novel framework for understanding symmetries in quantum systems, moving beyond traditional approaches to explore more complex, higher-form symmetries. This work establishes a deep connection between algebraic structures called “crossed n-cubes” and the symmetries present in lattice gauge theories and quantum field theories. The core idea revolves around examining how symmetries act not on the entire system at once, but on localized regions within it, revealing subtle constraints and anomalies that would otherwise remain hidden. The team demonstrates that these higher symmetries are fundamentally linked to the presence of ‘t Hooft anomalies, obstructions to making a symmetry truly local.

By analyzing how symmetry transformations behave when restricted to different regions of space, researchers can pinpoint these anomalies and characterize the symmetry structure of the system, framing anomalies as inconsistencies arising when attempting to separate the global symmetry action into localized components. Specifically, the research reveals that these symmetries and anomalies are encoded within the crossed n-cubes, algebraic objects that capture the relationships between symmetry transformations in different regions. For a two-dimensional gauge theory, this manifests as a “crossed square of groups,” a structure that allows the extraction of topological information, including homotopy groups, which describe the fundamental properties of the symmetry. In one example, a gauge theory with modified constraints exhibits anomalous behaviour, consistent with expectations from other theoretical approaches. Importantly, the framework extends beyond simple cases, offering a pathway to analyze higher-dimensional systems and more complex symmetry structures, potentially bridging the gap between mathematical structures and physical symmetries in quantum field theories.

Higher Symmetries and Anomalies in Gauge Theories

This research investigates higher-form symmetries within lattice gauge theories, employing tools from homotopy and operator algebras to reveal connections between symmetry and anomalies. The work demonstrates that these symmetries, and the anomalies associated with them, arise from considering how symmetry transformations act on localized regions of space, packaging the relevant data into higher groups, specifically crossed n-cubes of groups, providing an algebraic framework for understanding topological properties and interactions between symmetries. The researchers successfully applied this framework to several gauge theories, including Z2 and Zn gauge theories, both with standard and modified Gauss law constraints. They showed how the presence or absence of anomalies in these theories can be understood through the algebraic structure of the associated higher groups, confirming results previously obtained using Euclidean methods. Notably, the analysis of a modified Gauss law theory with fermionic vortex excitations verified the existence of an anomalous 1-form symmetry, aligning with expectations from other approaches.