  We prove that in the limit of large dimension, the distribution of the
logarithm of the characteristic polynomial of a generalized Wigner matrix
converges to a log-correlated field. In particular, this shows that the
limiting joint fluctuations of the eigenvalues are also log-correlated. Our
argument mirrors that of \cite{BouMod2019}, which is in turn based on the
three-step argument of \cite{ErdPecRmSchYau2010,ErdSchYau2011Uni}, but applies
to a wider class of models, and at the edge of the spectrum. We rely on (i) the
results in the Gaussian cases, special cases of the results in
\cite{BouModPai2021}, (ii) the local laws of \cite{ErdYauYin2012}(iii) the
observable \cite{Bou2020} introduced and its analysis of the stochastic
advection equation this observable satisfies, and (iv) the argument for a
central limit theorem on mesoscopic scales in \cite{LanLopSos2021}. For the
proof, we also establish a Wegner estimate and local law down to the
microscopic scale, both at the edge of the spectrum.
