Citation: Johnson-Freyd, Theo, and David Treumann. "Third homology of some sporadic finite groups." SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 15 (2019): 059.
Web: https://arxiv.org/abs/1810.00463
Tags: Mathematical, Conformal-field-theory
In this lovely paper, Theo Johnson-Freyd and David Treumann do exactly what they say they'll do in the title. Namely, they compute the (third) homology groups of some sporadic finite groups. This is not the first time that Theo has done this. He has an earlier paper which is also devoted to cohomological computations of this sort:
> Johnson-Freyd, Theo, and David Treumann. "H4 (Co0; Z)= Z/24." International Mathematics Research Notices 2020.21 (2020): 7873-7907.
I find these papers great for several reasons. First, it feels like they're making real progress in mathematics. There are some cool sporadic groups. We want to understand them more. Let's compute their (co)homology. Another reason to like these papers is their relation to physics.
The monster group is related to conformal field theory. Namely, the monster group has a canonical action on a certain conformal field theory - its "moonshine representation". It is a wonderful theorem that this moonshine representation, in fact, is anomalous. This nonzero anomaly lives in the third cohomology group of the monster group with U(1) coefficients. Thus, the third cohomology group of the monster is interesting.
> Johnson-Freyd, Theo. "The moonshine anomaly." arXiv preprint arxiv:1707.08388 (2018)
The computations in these papers are quite nice. At times, they use deep theory. Namely, to prove that there is a copy of Z24 in the cohomology of the monster group they use this whole deep relationship with conformal field theory. At times, the work is completely computational - they run their data from some computer library to get answers. The sporadic groups are too big to be handled entirely computationally, and two compilcated to be studied entirely using theory. One needs a mastery of both the computational and theoretical tools related to sporadic groups and group cohomology, and in these papers Theo Johnson-Freyd and David Treumann are showing that they are masters of that craft.