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In a 2017 article More on supersymmetric and 2d analogs of the SYK model by Murugan, Stanford and Witten, the authors take a model called the SYK model (named after Sachdev, Ye and Kitaev) and study supersymmetric versions in dimensions one and two.

The physics motivation for the original SYK model in condensed matter physics can be found here and in related works (it originally models the spin-$S$ quantum Heisenberg magnet with Gaussian-random, infinite-range exchange interactions). At the end of the article, it is mentioned that constructing a supersymmetric SYK model in dimension three is still an open problem.

Such a model would be more realistic and would not suffer from some of the problems which one encounters with global symmetries in dimensions one and two, so I was curious if this had been looked into in the meantime. Would there be some peculiar difficulties involved in doing this, or has no-one studied it because it would necessitate very involved calculations and not many new ideas which are not already in the paper I cite above?

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The disordered SYK model in three dimensions with supersymmetry was studied by Fedor Popov in Supersymmetric tensor model at large $N$ and small $\varepsilon$.

The complications are discussed in A 3d disordered superconformal fixed point:

There are two problems with the generalization of SYK-like models with disorder to dimensions higher than two. Firstly, pure fermionic models do not generically have relevant operators (a four-Fermi interaction being marginally irrelevant in two dimensions). Secondly, including bosonic degrees of freedom is problematic, since disordered Hamiltonians fail to be generically positive definite.

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