Topological terms in abelian lattice field theories

Understand

In this contribution we revisit the lattice discretization of the topological charge for abelian lattice field theories.

• The construction departs from an initially non-compact discretization of the gauge fields and after absorbing $2\pi$ shifts of the gauge fields leads to a generalized Villain action that also includes the topological term.
• The topological charge in two, as well as in four dimensions can be expressed in terms of only the integer-valued Villain variables.
• We test various properties of the topological charge and in particular analyze the index theorem in two dimensions and discuss the Witten effect in 4-d.