From Supergravity to ball bearings. Cohomology for physicists and engineers
Supersymmetry was formulated in 1974 by Wess and Zumino. Roughly speaking it states: "WE LIVE ON A SUPERMANIFOLD". Physicists have several models of this supermanifold, called N-extended Minkowski superspace, where N runs 1 to 8 (or ad infinitum in some unconventional models due to M. Vasiliev). For N=1 several analogs of Einstein Equations, supergravity equations, or SUGRA, were written during 1974-76. For N=2 it took ten more years and the answer (by V. Ogievetsky group) was unexpected: one had to add an extra dimension to the underlying Minkowski space-time. The amount of computations made the further advancement problematic.
It turns out that the difficulty lies not in "superization" of Minkowski space: recipes for calculating Curvature tensor (the left hand side of Einstein equations) from textbooks on differential geometry are easy to superize but the result is not SUGRA! The problem is that Minkowski SUPERspace is not just a manifold (or supermanifold) with a so-called G-structure, but a nonholomomic one. The term NONHOLONOMIC was coined by H. Hertz and means the phase space of any dynamical system with nonintegrable constraints on velocities. Examples are numerous: bicycles and cars, ball bearings; persueing rockets, waves in plasma, etc.
In the second half of the talk (for experts) I will define analogs of curvature tensors for manifolds with nonholonomic structure. The answer is given in terms of Lie algebra cohomology. (This is a very simple notion of linear algebra to be defined.) In particular, I will reproduce the results obtained jointly with Grozman: describe analogs of the Riemann tensor for N-extended SUGRA for any N.
I will also speculate on why time travel is unlikely if we assume that we live on any of Minkowski superspaces.
Since the curvature tensor governs stability of the dynamical systems, the possibility to compute it is essential for engineering problems. Unfortunately, at the moment the amount of computations for any problem of practical interest is beyond reach even for brainiest of the computers. So we had a lucky break with SUGRA.
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View article "Orthogonal polynomials of a discrete variable and Lie algebras of complex size matrices" by Leites and Sergeev in latex, dvi, ps and pdf .