Like a parent cautiously returning to a firework which failed to launch at the first attempt, scientists at CERN are about to switch the Large Hadron Collider (LHC) on again.
Anil Ananthaswamy duly provides a decent summary in New Scientist of the prospects for the LHC finding evidence of supersymmetric particles as well as the Higgs boson.
The basic idea of supersymmetry is that the two types of elementary particles with which we are familiar, bosons and fermions, are actually just different states of single particle types. In this respect, it is postulated that each type of boson has a fermionic partner, and each type of fermion has a bosonic partner. Supersymmetry therefore predicts the existence of numerous particles which have not hitherto been detected. For example, the photon, (a boson) has a hypothetical supersymmetric partner called the photino (a fermion).
The particle ontology of supersymmetry is then twinned with a cosmological explanation of why bosons and fermions are observed as distinct particles in terrestrial laboratories. It is proposed that the symmetry between bosons and fermions was respected at the higher energy levels found in the early universe, but as the universe expanded and energy levels dropped, supersymmetric symmetry breaking took place, with the consequence that the bosons and fermions with which we are familiar interact very rarely with their supersymmetric partners. These weakly-interacting supersymmetric particles then provide a nice candidate to explain the existence of dark matter in astronomy and cosmology.
Mathematically, supersymmetry also entails an interesting modification to the definition of what an elementary particle is. The latter is intimately related to the local space-time symmetry group, the group of symmetries possessed by every small patch of space-time, irrespective of how those patches are sewn into a global space-time. Without supersymmetry, the local space-time symmetry group of our universe appears to be a subgroup of the Poincare group. Wigner established that each type of elementary particle corresponds to an irreducible unitary Hilbert space representation of this subgroup of the Poincare group (with a few technical qualifications concerning so-called covering groups).
However, if it transpires that our universe is a supersymmetric universe, then the definition of an elementary particle has a straightforward generalisation. The local space-time symmetry group becomes (a subgroup of) the super-Poincare group, and the set of possible supersymmetric elementary particles is then defined by the irreducible unitary Hilbert space representations of (a subgroup of) the super-Poincare group. Each such representation decomposes into a direct sum of unitary irreducible representations of the Poincare group, and the members of such a supersymmetric 'multiplet' are said to be super-partners of each other.
Mathematicians interested in symmetry, and philosophers interested in the mereological concept of elementarity, should therefore share a common interest in the data physicists are set to harvest from the LHC's detectors.