Modular invariance groups and defect McKay-Thompson series

TL;DR

This paper investigates the modular invariance groups of defect McKay-Thompson series related to the Moonshine module, revealing many are not genus zero, thus extending understanding of their symmetry properties beyond classical Hauptmoduln.

Contribution

The authors construct new defect McKay-Thompson series, analyze their modular invariance groups, and demonstrate that many are not genus zero, challenging previous assumptions about their symmetry groups.

Findings

Many invariance groups are not genus zero

Constructed several new defect McKay-Thompson series

Analyzed modular properties and invariance groups

Abstract

It has been known since 1992 that the McKay-Thompson series $T_g(q)$ of the Moonshine module form Hauptmoduln for genus zero subgroups of $SL(2, \mathbb{R})$. In 2021, Lin and Shao constructed a series analogous to the McKay-Thompson series (a twined partition function of the Monster CFT), but using a non-invertible topological defect rather than an element of the Monster group $\mathcal{M}$. This "defect McKay-Thompson series" was found to be invariant under a genus zero subgroup of $SL(2, \mathbb{R})$, but was shown not to be the Hauptmodul of the subgroup. Nevertheless, one might wonder if a weaker version of Borcherds' theorem holds for non-invertible defects: perhaps defect McKay-Thompson series enjoy genus zero invariance groups in $SL(2, \mathbb{R})$, whether or not they are Hauptmoduln for those groups. Using the decompositions of the monster stress tensor found in Bae et al. (2021), we construct several new defect McKay-Thompson series, study their modular properties, and determine their invariance groups in $SL(2, \mathbb{R})$. We discover that many of the invariance groups are not genus zero.