Topological Mathieu Moonshine

Abstract

We explore the Atiyah-Hirzebruch spectral sequence for the $tm{f}^{\bullet }[\frac{1}{2}]$-cohomology of the classifying space $B{M}_{24}$ of the largest Mathieu group $M_{24}$, twisted by a class $\omega \in H^4(B{M}_{24};Z[\frac{1}{2}])\cong Z_3$. 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 $tm{f}_{\omega }^{\bullet }(B{M}_{24})[\frac{1}{2}]$ in cohomological degree $-27$. Our main computational result is that $tm{f}_{\omega }^{-27}(B{M}_{24})[\frac{1}{2}]=0$ when $\omega \ne 0$. For comparison, the restriction map $tm{f}_{\omega }^{-3}(B{M}_{24})[\frac{1}{2}]\to tm{f}^{-3}(pt)[\frac{1}{2}]\cong Z_3$ is surjective for one of the two nonzero values of $\omega$. Our motivation comes from Mathieu Moonshine. Assuming a well-studied conjectural relationship between $TMF$ and supersymmetric quantum field theory, there is a canonically-defined $C{o}_1$-twisted-equivariant lifting $[{\bar{V}}^{f♮}]$ of the class $\{24\Delta \}\in TM{F}^{-24}(pt)$, where $C{o}_1$ denotes Conway's largest sporadic group. We conjecture that the product $[{\bar{V}}^{f♮}]\nu$, where $\nu \in TM{F}^{-3}(pt)$ is the image of the generator of $tm{f}^{-3}(pt)\cong Z_{24}$, does not vanish $C{o}_1$-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.

arXiv comment1+57+7 pages. v2 contains small edits

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