# Relationship between different $Z_{16}$ classifications

I find that there exist two classifications which have a $Z_{16}$ group structure:

1. The sixteen fold way of classifying Majorana fermions, vortex systems appearing in Kitaev's paper on his honeycomb model: https://arxiv.org/abs/cond-mat/0506438.

2. In the classification of surface states of topological superconductors belonging to class DIII in the presence of interactions: https://arxiv.org/abs/1406.3032

Are the two related in any way?

## 1 Answer

Besides the obvious relationship that the representation theory of Majorana zero modes is the leverage point for each, the two are related in an intricate network of bosonization and dimensional reduction.

First, this paper of Gaiotto and Kapustin related $\Omega^3_{spin}(BG)$, which classifies $G$-SPTs in 2+1D, with Kitaev's 16 topological gauge theories through gauging a certain fermionic higher symmetry. You can combine this with the dimensional reduction procedure we worked out here to relate this to $\Omega^4_{pin^+}$, which classifies topological superconductors in 3+1D with $T^2 = (-1)^F$. We later worked out a direct relationship for unitary symmetries here, but using an extension of Kitaev's theories to a family of $\mathbb{Z}_{16}$ different 3+1D $\mathbb{Z}_2$ gauge theories. Working that out for anti-unitary symmetries would be interesting, though I think very difficult unless you come up with a clever way to do things.

The point is that certain singularities of the gauge field can carry majorana zero modes, fermionic charges, or even more fun things in higher dimensions.

On an even deeper level perhaps, it's all related to the structure of quadratic forms, and the magic of the Arf-Brown-Kervaire invariant, which connects various $\mathbb{Z}_8$'s in dimensions $4k+2$ to the 8 of Bott periodicity. Any extra factors of two are because the number of components of a spinor grows like $2^{d/2}$.

Here are some more interesting papers: