Modified Villain formulation of the abelian Chern-Simons theory

I will discuss a construction of the U(1) level-k Chern-Simons theory on the lattice which has a correct level quantisation, and a correct 1-form global symmetry with the expected ’t Hooft anomaly. The crucial component of the construction is the so-called modified Villain construction of abelian gauge theories. The theory has a peculiar subsystem symmetry which projects out all naive Wilson loops. However the correct topological charges of the 1-form symmetry are not Wilson loops, but strips, or framed Wilson loops. This is as one expects from a lattice regularisation of the CS theory.