$\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras -- II

TL;DR

This paper analyzes the decomposition of Siegel modular forms related to BKM Lie superalgebras, extending previous work to new cases and exploring the algebraic structure for different orbifold parameters.

Contribution

It introduces a decomposition approach of Siegel modular forms into sub-algebra characters, including new cases for N=5, expanding understanding of associated Lie superalgebras.

Findings

Decomposition of modular forms into $\\hat{sl(2)}$ characters

Identification of new algebraic structures for N=5,6

Extension of previous work on Umbral moonshine cases

Abstract

The square-root of Siegel modular forms of CHL Z_N orbifolds of type II compactifications are denominator formulae for some Borcherds-Kac-Moody Lie superalgebras for N=1,2,3,4. We study the decomposition of these Siegel modular forms in terms of characters of two sub-algebras: one is a $\widehat{sl(2)}$ and the second is a Borcherds extension of the $\widehat{sl(2)}$. This is a continuation of our previous work where we studied the case of Siegel modular forms appearing in the context of Umbral moonshine. This situation is more intricate and provides us with a new example (for N=5) that did not appear in that case. We restrict our analysis to the first N terms in the expansion as a first attempt at deconstructing the Siegel modular forms and unravelling the structure of potentially new Lie algebras that occur for N=5,6.