Adding Flavor to the Narain Ensemble

TL;DR

This paper explores the average behavior of flavored partition functions in 2D free boson CFTs within the Narain ensemble, revealing a new Siegel-Weil formula that relates geometric and charge deformations, advancing the understanding of their dual gravity theories.

Contribution

It introduces a novel approach to compute the average flavored partition functions using a heat equation and derives a new Siegel-Weil formula for these quantities.

Findings

Derived a heat equation relating moduli and chemical potentials.

Established a Siegel-Weil formula for flavored partition functions.

Presented a modified sum over geometries for the flavored case.

Abstract

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.