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We revisit the proposal that the ensemble average over free boson CFTs in twodimensions - parameterized by Narain's moduli space - is dual to an exotictheory of gravity in three dimensions dubbed $U(1)$ gravity. We considerflavored partition functions, where the usual genus $g$ partition function isweighted by Wilson lines coupled to the conserved $U(1)$ currents of thesetheories. These flavored partition functions obey a heat equation which relatesdeformations of the Riemann surface moduli to those of the chemical potentialswhich measure these $U(1)$ charges. This allows us to derive a Siegel-Weilformula which computes the average of these flavored partition functions. Theresult takes the form of a "sum over geometries," albeit with modificationsrelative to the unflavored case.

Adding Flavor to the Narain Ensemble