Topological Mathieu Moonshine

Abstract

We explore the Atiyah-Hirzebruch spectral sequence for the tmf • [ 1 2 ]-cohomology of the classifying space BM 24 of the largest Mathieu group M 24 , twisted by a class ω ∈ H 4 (BM 24 ; Z[ 1 2 ]) ∼ = Z 3 . Our exploration includes detailed computations of the F 3 -cohomology of M 24 and of the first few differentials in the AHSS. We are specifically interested in the value of tmfis nonzero 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 canonicallydefined Co 1 -twisted-equivariant lifting [V f ♮ ] of the class {24∆} ∈ TMF −24 (pt), for a specific value ω of the twisting, where Co 1 denotes Conway's largest sporadic group. We conjecture that the product [V f ♮ ]ν, where ν ∈ TMF −3 (pt) is the image of the generator of tmf −3 (pt) ∼ = Z 24 , does not vanish Co 1 -equivariantly, but that its restriction to M 24 -twisted-equivariant TMF does vanish. We explain why 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 "ℓ-complexes," in which the differential D satisfies D ℓ = 0 rather than D 2 = 0, and their relation to SU(2) Verlinde rings. The case ℓ = 3 is used in our AHSS calculations.

Topological modular forms and the absence of all heterotic global anomalies

“…This point was already noted in [GPPV18,JF20], but the comparison to [ABG10] was not made in these earlier references.…”

“…Theorem 1.4 and Conjecture 1.3 of Segal, Stolz and Teichner posit that KO(X) and TMF(X) classify 1-dimensional unitary N =1 supersymmetric quantum field theories with pin− structure and 2-dimensional unitary N =(0, 1) supersymmetric quantum field theories systems with spin structure up to continuous deformations, respectively. From this perspective, it is natural to identify the twists of KO(X) and TMF(X) with the anomalies of respective systems parameterized over X, as was already mentioned in [GPPV18,JF20]. According to Physics Assumption 1.1, they are respectively given by (4.4) (I Z Ω pin− ) 3 (X) and (I Z Ω spin ) 4 (X).…”

Snowmass White Paper: Moonshine

“…We content ourselves with mentioning just a few new directions. One recent proposal [110] for understanding Mathieu moonshine takes place in the arena of algebraic topology, employing topological modular forms 5 , and building on recent work which defines a new mock modular "secondary elliptic genus" as a natural invariant of 2d supersymmetric QFTs [86]. Non-invertible symmetries and topological defect lines have been studied in the Monster VOA [124] and naturally capture its self-dualities.…”

Heterotic global anomalies and torsion Witten index

“…It was conjectured [13,14] that the "space of N = (0, 1) SQFTs" with the worldsheet pure gravitational anomaly ν ∈ Z is homotopy equivalent to the (−ν)-th space TMF −ν of a generalized cohomology theory known as topological modular forms, TMF. See [15][16][17][18][19][20][21][22] for the physics literature discussing this conjecture. If the conjecture is correct, it implies that there is new obstruction to spontaneous supersymmetry breaking beyond the Witten index or the elliptic genus [23], no matter what continuous deformation we perform.…”