Description
I will explain the basic structure of non-invertible symmetries and dualities using various quantum
spin chains. I will first show how Kramers-Wannier duality provides a canonical example. The key
to finding more general analogs is to define a Hamiltonian using a fusion category; most wellknown quantum spin chains can be defined in such a fashion. Non-invertible symmetries and dualities are then built in from the start. Topological defects provide a key tool, as their action implements the symmetry/duality.
I won’t assume any prior knowledge of duality or categories. It would be useful to understand a little about the quantum Ising Hamiltonian in advance; the level of Wikipedia should be sufficient
(search “transverse-field Ising model”). Most of the material will come from the pair of papers I
wrote with Aasen and Mong (1601.07185 and 2008.08598), but I’ll be presenting it from a somewhat different starting point. Thus there’s no requirement to read them in advance, although of
course a glance wouldn’t hurt.