“Despite the success of the Large Hadron Collider, evidence for the follow-up theory – supersymmetry – has proved elusive” #
“All would be perfect except that no one has detected any of the many expected supersymmetric particles. “ #
Thus, it seems that supersymmetries, described by generalization of Lie algebras to Lie superalgebras, are illusive. This is also about supergravity based on a super extension of a Poincare Lie algebra.
At the same time, we observe particles both of the even Grassmann parity (photons) and the odd one (fermions). Moreover, gauge symmetries are parameterized by odd ghosts, and BRST theory at present is the generally accepted technique of gauge field quantization. These facts motivate us to develop Grassmann-graded Lagrangian theory of even and odd fields, in general.
G. Sardanashvily, Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory (2013)
G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced Classical Field Theory (2009)