Graded Riemannian geometry and graded fibre bundles, a context for local super-gauge theories
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Starting from Kostant's definition of a graded manifold , we present graded versions of tensor algebra, Riemannian metrics, Lie groups and actions (these are also defined by Kostant but we present a directly geometrical definition which is more convenient for our purposes), vector bundles, and principal bundles.
With these notions in place, we can define a graded G-structure on a graded manifold In the simplest non-trivial case, this leads immediately to a local version of the 14-dimensional graded Lie algebra discovered by Volkov and Akulov, and described by Ne'eman in .
Let M be a graded manifold (in the sense of Kostant) of even dimension 4 and odd dimension 4. We suppose that T(M), the total graded tangent bundle of M, has a distinguished sub-bundle of even dimension 0 and odd dimension 4 . We suppose further that the pairing S ⊗ S → T(M)/S induced by the Lie-bracket induces a conformal class of Lorentz metrics on T/S by singling out the “squares” as null vectors. We observe that there is a complex structure on S which makes this Lie-bracket pairing the real part of a Hermitian pairing from S ⊗ S → T/S ⊗ C . Using this structure, it appears that one can give graded analogues of the Yang-Mills Equation and of the conformally invariant part of Einstein's equations.
KeywordsGauge Transformation Gauge Field Invariance Condition Principal Bundle Geometric Quantization
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- B. Kostant, Graded manifolds, graded Lie theory and prequantization, Differential Geometrical Methods in Mathematical Physics, Springer Lecture Notes in Mathematics #570 pp. 177–306.Google Scholar
- Y. Ne'eman, The application of graded Lie algebras to invariance considerations in particle physics, ibid pp. 109–144.Google Scholar