模不变群与缺陷McKay-Thompson级数
Modular invariance groups and defect McKay-Thompson series
AI总结：
本文构造新的缺陷McKay-Thompson级数，研究其模性质并确定不变群，发现许多不变群非亏格零，从而弱化Borcherds定理对非可逆缺陷的适用性。
摘要：
自1992年以来已知，Moonshine模的McKay-Thompson级数$T_g(q)$构成$SL(2, \mathbb{R})$的亏格零子群的Hauptmoduln。2021年，Lin和Shao构造了一个类似于McKay-Thompson级数的级数（Monster CFT的twined配分函数），但使用的是非可逆拓扑缺陷，而非Monster群$\mathcal{M}$的元素。这个“缺陷McKay-Thompson级数”被发现对$SL(2, \mathbb{R})$的一个亏格零子群不变，但被证明不是该子群的Hauptmodul。尽管如此，人们可能会想，Borcherds定理的一个较弱版本是否对非可逆缺陷成立：也许缺陷McKay-Thompson级数在$SL(2, \mathbb{R})$中享有亏格零不变群，无论它们是否是这些群的Hauptmoduln。利用Bae等人（2021）中发现的Monster应力张量的分解，我们构造了几个新的缺陷McKay-Thompson级数，研究它们的模性质，并确定它们在$SL(2, \mathbb{R})$中的不变群。我们发现许多不变群不是亏格零的。
英文摘要：
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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