Adding flavor to the Narain ensemble

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.

Automated summary

We revisit the proposal that the ensemble average over free boson CFTs in two dimensions is dual to an exotic theory of gravity in three dimensions dubbed U(1) gravity. Flavored partition functions, weighted by Wilson lines coupled to U(1) currents, fulfill a heat equation relating Riemann surface moduli deformations to chemical potential deformations. This leads to a Siegel-Weil formula that computes the average of these flavored partition functions.