Anomalies of Global Symmetries on the Lattice

Symmetries play a crucial role in both quantum field theory and quantum many-body systems on the lattice. One key topological feature that a global symmetry can have is a "('t Hooft) anomaly". But how exactly should an anomaly be defined? In this talk, I give a precise definition of (invertible) symmetries and anomalies in lattice systems, which makes it clear that an anomaly is a property of the way the symmetry acts, without reference to any Hamiltonian. Then, I will discuss progress towards the general classification based on this definition. I uncover a potentially surprising result: lattice anomalies are *not* in one-to-one correspondence with 't Hooft anomalies of QFTs in the continuum. However, there is a map from lattice anomalies to QFT anomalies, but the map in general is neither injective nor surjective. I will show that lattice anomalies have a number of interesting consequences in their own right, including constraints on commuting projector Hamiltonians and many-body localization (MBL). The techniques I describe to characterize anomalies in lattice systems could potentially be extended to give new perspectives on anomalies in QFTs.