There are crucially two different incarnations of supersymmetry, whose conflation is the source of much confusion in public discussion.
On the one hand there is "low-energy" supersymmetry. This has traditionally been motivated by appeal to naturalness of the standard model, gauge-coupling unification, and dark matter candidates. It is this version of supersymmetry which is being put under high pressure from current experimental data, leading people to recognize that arguments via naturalness etc. are not as solid as they may have felt.
On the other hand there is "high-energy" supersymmetry, namely supergravity. By nature of high energies, this is much less accessible to experiment, if at all, but it is this version of supersymmetry that actually has solid theoretical motivation.
For instance what string theory actually predicts as a mathematical result is that combining the assumption of strings with the existence of fermions, hence considering what was originally called the "spinning string", miraculously implies local spacetime supersymmetry, i.e. supergravity. On the other hand, the reduction of this solid prediction of high energy supersymmetry to low energy supersymmetry needs to assume by hand that the spacetime vacuum is a KK-compactification on Calabi-Yau fibers, or similar. This is not something that the theory predicts, this is something imposed by hand. What the theory predicts is high energy supersymmetry. (This is well known, and yet it is frequently forgotten in public discussion.)
But even without the assumption of strings, there are good formal arguments for high energy supersymmetry. The most popular among these is the Coleman-Mandula theorem combined with the Haag–Lopuszanski–Sohnius theorem. This says that if you believe that fundamentally (i.e. at high energy, small distances) spacetime symmetry should be "unified with internal symmetry" then supersymmetry is one possibility to achieve this.
Popular as this argument is, it has its weaknesses. First of all you might not buy into the assumption that spacetime symmetry needs to be unified with internal symmetry. Even if you do, the theorem only says that supersymmetry is one way to achieve this.
But actually there is a little-known theorem by a very famous mathematician which has considerably stronger implications as to the motivation of high energy supersymmetry. This is Deligne's theorem on tensor categories.
Paraphrased to the realm of particle physics, the theorem says the following, informally: Given any reasonable choice of elementary particle species, then 1) there always is a spacetime symmetry group G such such that these particles arise in the Wigner classification of elementary particles transforming under G, and 2) the class of spacetime groups arising this way are precisely the (algebraic) super-groups. (As opposed to, say, more non-commutative groups etc.)
More exposition of this is in the recent PhysicsForums Insights article
Now, Deligne's theorem is from 2002, hence it is not really new anymore. But it seems essentially unknown among physicists. I had known about if for a while, and appreciated its implication for high energy supersymmetry. But it was only due to the recent increase of confused public discussion on the motivations for supersymmetry that I began to think that this theorem might deserve more public exposition among physicists.
But I am wondering: Did anyone else, in the physics community, point to Deligne's theorem on tensor categories before, as a motivation for fundamental (high energy) supersymmetry? What would be references?
