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. 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.
References
External links "A New Kind of Symmetry Shakes Up Physics" by Kevin Hartnett, Quanta Magazine "Non-Invertible Symmetries and their Representations", video lecture by Sahand Seifnashri at Institute for Advanced Study