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

TL;DR

This paper introduces a new duality-invariant distance measure for 2D conformal field theories based on a formula for the Petersson inner product involving Eisenstein series, with applications to evaluating distances in Narain moduli space.

Contribution

It derives a convergent sum formula for the Petersson inner product of Eisenstein series and explores its use as a duality-invariant distance measure on 2D CFTs.

Findings

Derived a convergent sum formula for the Petersson inner product.

Applied the formula to evaluate distances in Narain moduli space.

Demonstrated utility of the inner product as a distance measure for 2D CFTs.

Abstract

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.