Kavli Affiliate: Jeffrey A. Harvey
| First 5 Authors: Harry Fosbinder-Elkins, Jeffrey A. Harvey, , ,
| Summary:
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.
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