  We revisit the proposal that the ensemble average over free boson CFTs in two
dimensions - parameterized by Narain's moduli space - is dual to an exotic
theory of gravity in three dimensions dubbed $U(1)$ gravity. We consider
flavored partition functions, where the usual genus $g$ partition function is
weighted by Wilson lines coupled to the conserved $U(1)$ currents of these
theories. These flavored partition functions obey a heat equation which relates
deformations of the Riemann surface moduli to those of the chemical potentials
which measure these $U(1)$ charges. This allows us to derive a Siegel-Weil
formula which computes the average of these flavored partition functions. The
result takes the form of a "sum over geometries," albeit with modifications
relative to the unflavored case.
