Kavli Affiliate: Jeffrey 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 $SL(2, mathbbR)$. 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 $mathcalM$. This "defect McKay-Thompson series" was found to be invariant under a genus zero subgroup of $SL(2, mathbbR)$, 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 $SL(2, mathbbR)$, 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 $SL(2, mathbbR)$. We discover that many of the invariance groups are not genus zero.
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