2020/06/04 by Theo Johnson-Freyd, Johnson-Freyd, Theo
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Group Theory (math.GR) #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)
paper · pdf · doi:10.48550/arxiv.2006.02922
openalex publication_date 2020/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We explore the Atiyah-Hirzebruch spectral sequence for the tmf^\bullet[\frac12]-cohomology of the classifying space BM24 of the largest Mathieu group M24, twisted by a class ω∈ H4(BM24;Z[\frac12]) ≅ Z3. Our exploration includes detailed computations of the Z3-cohomology of M24 and of the first few differentials in the AHSS. We are specifically interested in the value of tmf^\bulletω(BM24)[\frac12] in cohomological degree -27. Our main computational result is that tmf-27ω(BM24)[\frac12] = 0 when ω≠ 0. For comparison, the restriction map tmf-3ω(BM24)[\frac12]→ tmf-3(pt)[\frac12] ≅ Z3 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 Co1-twisted-equivariant lifting [Vf\natural] of the class \24Δ\ ∈ TMF-24(pt), where Co1 denotes Conway's largest sporadic group. We conjecture that the product [Vf\natural] ν, where ν∈ TMF-3(pt) is the image of the generator of tmf-3(pt) ≅ Z24, does not vanish Co1-equivariantly, but that its restriction to M24-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 M24 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 "ℓ-complexes" and their relation to SU(2) Verlinde rings. The case ℓ=3 is used in our AHSS calculations.