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Super D-Differentiation for R  ∞ -Supermanifolds

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Abstract

The concept of ‘D-Differentiation’, which, in the context of smooth manifolds, generalises Lie and covariant differentiation, is extended to R  ∞ -supermanifolds under the name of ‘Super D-Differentiation’. This is done by defining new (non-linear) mappings, called ‘μ-mappings’ and by relating their non-linearity to the Leibniz rule that a derivation must satisfy when it acts on a tensor product. The resulting axiomatics, which is basis-independent and coordinate-free, is then expressed in a general basis (not necessarily holonomic). Super Lie and Super covariant differentiation are, amongst others, special cases of Super D-Differentiation. In particular, the transformation rules for the connection coefficients and the commutation coefficients of non-holonomic bases are obtained. These special cases are found to be in agreement with the DeWitt Super covariant and Super Lie derivatives.

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Authors and Affiliations

  1. Mathematics Department, National University of Ireland, Cork, Ireland

    D. Hurley

  2. Physics Department, National University of Ireland, Cork, Ireland

    M. Vandyck

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  1. D. Hurley
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  2. M. Vandyck
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Correspondence to D. Hurley.

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Hurley, D., Vandyck, M. Super D-Differentiation for R  ∞ -Supermanifolds. Math Phys Anal Geom 10, 123–134 (2007). https://doi.org/10.1007/s11040-007-9024-5

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  • DOI: https://doi.org/10.1007/s11040-007-9024-5

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