  The Nappi-Witten model is a Wess-Zumino-Witten model in which the target
space is the nonreductive Heisenberg group $H_4$. We consider the
representation theory underlying this conformal field theory. Specifically, we
study the category of weight modules, with finite-dimensional weight spaces,
over the associated affine vertex operator algebra $\mathsf{H}_4$. In
particular, we classify the irreducible $\mathsf{H}_4$-modules in this category
and compute their characters. We moreover observe that this category is
nonsemisimple, suggesting that the Nappi-Witten model is a logarithmic
conformal field theory.
