Automorphic correction of the hyperbolic Kac-Moody algebra $E_{10}$

TL;DR

This paper explores automorphic correction of the hyperbolic Kac-Moody algebra E10 using Borcherds products and discusses broader implications for Lorentzian Kac-Moody algebras.

Contribution

It introduces a method for automorphic correction of E10 via Borcherds products and clarifies aspects of automorphic correction for Lorentzian Kac-Moody algebras.

Findings

Automorphic correction of E10 achieved using Borcherds product.

Provides heuristic reasons for universal automorphic correction in Lorentzian Kac-Moody algebras.

Clarifies theoretical aspects of automorphic correction for these algebras.

Abstract

In this paper we study automorphic correction of the hyperbolic Kac-Moody algebra $E_{10}$, using the Borcherds product for O(10,2) attached to a weakly holomorphic modular form of weight -4 for $SL_2(\mathbb Z)$. We also clarify some aspects of automorphic correction for Lorentzian Kac-Moody algebras and give heuristic reasons for the expectation that every Lorentzian Kac-Moody algebra has an automorphic correction.