On the Imaginary Simple Roots of the Borcherds Algebra $g_{II_{9,1}}$

TL;DR

This paper investigates the imaginary simple roots of the Borcherds algebra $g_{II_{9,1}}$, confirming the conjecture for roots with norm ≥ -8, but showing it fails for roots with norm -10 and beyond, using a modified denominator formula.

Contribution

It introduces an independent test for the conjecture on imaginary simple roots and develops an efficient method using a modified denominator formula.

Findings

Conjecture holds for roots with norm ≥ -8

Fails for roots with norm -10 and beyond

Provides detailed multiplicities up to norm -24

Abstract

In a recent paper (hep-th/9703084) it was conjectured that the imaginary simple roots of the Borcherds algebra $g_{II_{9,1}}$ at level 1 are its only ones. We here propose an independent test of this conjecture, establishing its validity for all roots of norm $\geq -8$. However, the conjecture fails for roots of norm -10 and beyond, as we show by computing the simple multiplicities down to norm -24, which turn out to be remakably small in comparison with the corresponding $E_{10}$ multiplicities. Our derivation is based on a modified denominator formula combining the denominator formulas for $E_{10}$ and $g_{II_{9,1}}$, and provides an efficient method for determining the imaginary simple roots. In addition, we compute the $E_{10}$ multiplicities of all roots up to height 231, including levels up to $\ell =6$ and norms -42.