non-invertible symmetry

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In physics, a non-invertible symmetry is a symmetry of a quantum field theory that is not described by a group, and which in particular does not have an inverse.

Non-invertible symmetries were first studied in 2-dimensional conformal field theory, where fusion categories govern the fusion rules, rather than a group.{{cite arXiv|last=Schafer-Nameki|first=Sakura|author-link=Sakura Schafer-Nameki|date=2023|title=ICTP Lectures on (Non-)Invertible Generalized Symmetries|eprint=2305.18296|class=hep-th}}{{Cite arXiv |eprint=2308.00747 |last1=Shao |first1=Shu-Heng |title=What's Done Cannot be Undone: TASI Lectures on Non-Invertible Symmetries |date=2023 |class=hep-th }}

Four-dimensional examples of non-invertible symmetries can be obtained from Maxwell theory with topological theta term, via a combination of its SL(2,Z) duality and a discrete subgroup of its electric or magnetic 1-form symmetry.{{cite arXiv|last=Sela|first=Orr|date=2024|title=Emergent non-invertible symmetries in N=4 Super-Yang-Mills theory|eprint=2401.05032|class=hep-th}}