Topological Aspects of the $c=2$ Model in the Continuum and on the Lattice

We discuss the continuum two-dimensional conformal field theory, whose target space is a two-torus. In addition to its standard zero-form internal symmetries, the model also has a $-1$-form symmetry associated with its $\theta$-parameter, which is also known as a background $B$-field. These symmetries combine into a higher group with anomalies. A spatially winding $B$-field generically breaks the spatial translation symmetry completely, and reduces the continuous target-space (internal) momentum symmetry to a (projectively realized) discrete subgroup. When $B$ varies linearly, a discrete translation subgroup survives, and it combines with the internal symmetry into a non-Abelian group, while the other translations survive as non-invertible symmetries. Even though the model lacks the standard spatial translation symmetry, the non-invertible translation symmetry explains why it is solvable via a Fourier decomposition. We then consider a lattice version of this system, using its modified Villain formulation. Surprisingly, all these subtleties in the symmetries of the continuum model exist on a finite lattice. Finally, we discuss various generalizations of this model, demonstrating that the phenomena we uncovered are more broadly valid.