论文标题
拓扑Mathieu Moonshine
Topological Mathieu Moonshine
论文作者
论文摘要
我们探索了$ tmf^\ bullet [\ frac12] $的Atiyah-hirzebruch光谱序列 - 分类空间的共同体$ bm_ {24} $是最大的Mathieu $ m_ {24} $的$ bm_ {24} $，由a级$ω\ in h^4 {bm_ z {24};我们的探索包括$ M_ {24} $的$ Z_3 $ - 元素学的详细计算以及AHSS中的前几个差异。我们对$ tmf^\bullet_Ω的值特别感兴趣（bm_ {24}）[\ frac12] $在共同体学度中$ -27 $。我们的主要计算结果是$ tmf^{ - 27}_Ω（bm_ {24}）[\ frac12] = 0 $当$ω\ neq 0 $。为了进行比较，限制映射$ tmf^{ - 3}_Ω（bm_ {24}）[\ frac12] \ to tmf^{ - 3}（pt）[\ frac12] \ frac12] \ cong z_3 $是两个非零值$ $ω$中的两个非零值之一。 我们的动机来自Mathieu Moonshine。假设$ tmf $与超对称量子场理论之间存在良好的猜想关系，则有一个规范定义的$ co_1 $ -twisted-equivariant lift $ [\ bar {v}^{v}^{f \ antural} $ class $康威最大的零星群体。我们认为产品$ [\ bar {v}^{f \ natural}]ν$，其中$ν\ in tmf^{ - 3}（-3}（pt）$是$ tmf^{ - 3}（-3}（pt）（pt）（pt）（pt）\ cong z_ {24} $的生成器的图像$ m_ {24} $ - 扭曲 - equivariant $ tmf $ do nish。这个猜想回答了Mathieu Moonshine中的一些问题：这意味着使用$ M_ {24} $对称性的微小超对称量子场理论存在，其扭曲和缠绕的分区功能具有与Mathieu Moonshine中相同的模拟模块性。我们的AHSS计算在奇数上“扰动地”建立了这种猜想。 附录主要用于娱乐目的，讨论了“ $ \ ell $ -complexes”及其与$ \ Mathrm {su}（2）$ Verlinde戒指的关系。我们的AHSS计算中使用了$ \ ell = 3 $。
We explore the Atiyah-Hirzebruch spectral sequence for the $tmf^\bullet[\frac12]$-cohomology of the classifying space $BM_{24}$ of the largest Mathieu group $M_{24}$, twisted by a class $ω\in H^4(BM_{24};Z[\frac12]) \cong Z_3$. Our exploration includes detailed computations of the $Z_3$-cohomology of $M_{24}$ and of the first few differentials in the AHSS. We are specifically interested in the value of $tmf^\bullet_ω(BM_{24})[\frac12]$ in cohomological degree $-27$. Our main computational result is that $tmf^{-27}_ω(BM_{24})[\frac12] = 0$ when $ω\neq 0$. For comparison, the restriction map $tmf^{-3}_ω(BM_{24})[\frac12]\to tmf^{-3}(pt)[\frac12] \cong Z_3$ is surjective for one of the two nonzero values of $ω$. Our motivation comes from Mathieu Moonshine. Assuming a well-studied conjectural relationship between $TMF$ and supersymmetric quantum field theory, there is a canonically-defined $Co_1$-twisted-equivariant lifting $[\bar{V}^{f\natural}]$ of the class $\{24Δ\} \in TMF^{-24}(pt)$, where $Co_1$ denotes Conway's largest sporadic group. We conjecture that the product $[\bar{V}^{f\natural}] ν$, where $ν\in TMF^{-3}(pt)$ is the image of the generator of $tmf^{-3}(pt) \cong Z_{24}$, does not vanish $Co_1$-equivariantly, but that its restriction to $M_{24}$-twisted-equivariant $TMF$ does vanish. This conjecture answers some of the questions in Mathieu Moonshine: it implies the existence of a minimally supersymmetric quantum field theory with $M_{24}$ symmetry, whose twisted-and-twined partition functions have the same mock modularity as in Mathieu Moonshine. Our AHSS calculation establishes this conjecture "perturbatively" at odd primes. An appendix included mostly for entertainment purposes discusses "$\ell$-complexes" and their relation to $\mathrm{SU}(2)$ Verlinde rings. The case $\ell=3$ is used in our AHSS calculations.
