Higher $d$ Eisenstein Series and a Duality-Invariant Distance Measure

The Petersson inner product is a natural inner product on the space of modular invariant functions. We derive a formula, written as a convergent sum over elementary functions, for the inner product $E_s(G,B)$ of the real analytic Eisenstein series $E_s(\tau, \bar{\tau})$ and a general point in Narain moduli space. We also discuss the utility of the Petersson inner product as a distance measure on the space of 2d CFTs, and apply our procedure to evaluate this distance in various examples.