Understanding Non-Invertible Symmetries in Higher Dimensions Using Topological Defects

Symmetry is a fundamental concept of physics that describes how the laws of physics remain unchanged under certain transformations. Generalized symmetries are an extension of this concept, which, in recent times, has been applied to the analysis of quantum field theories. Among these are non-invertible symmetries which do not have inverse elements and cannot be undone, unlike traditional symmetries. However, they are less understood in higher dimensions than in two dimensions.

Generalized symmetries have been discovered by identifying the relations between symmetries and topological defects. These defects are like disruptions or “wrinkles” in the fabric of a physical system. They cannot be removed or smoothed out easily without creating a defect elsewhere. An interesting approach to understanding non-invertible symmetries is to construct topological defects.

Now, in a new study published in the Progress of Theoretical and Experimental Physics, researchers have, for the first time, presented concrete examples of non-invertible symmetries in four dimensions (4D) using this approach.

Specifically, the researchers constructed topological defects associated with the Kramers-Wannier-Wegner (KWW) duality in the 4D pure Z₂ lattice gauge theory, similar to those in the two-dimensional Ising model. They discovered that these defects were non-invertible. Additionally, they constructed Z₂ symmetry defects as well as defect junctions between these defects and the KWW duality defects. They derived the crossing relations and using these, calculated the expectation values for some of the defects.

Having accelerated the study of non-invertible symmetries in four-dimensional theories, this study was honored with the Outstanding Paper Award by the Physical Society of Japan. It marks a significant step towards understanding higher-dimensional non-invertible symmetries and the fundamental nature of the universe.