Abstract
A natural generalization of a Lie algebra connection, or Yang-Mills field, to the case of a Lie-Kac superalgebra, for example SU(m/n), just in terms of ordinary complex functions and differentials, is proposed. Using the chirality χ which defines the supertrace of the superalgebra: STr(…) = Tr(χ…), we construct a covariant differential: D = χ(d + A) + Φ, where A is the standard even Lie-subalgebra connection 1-form and Φ a scalar field valued in the odd module. Despite the fact that Φ is a scalar, Φ anticommutes with (χA) because χ anticommutes with the odd generators hidden in Φ. Hence the curvature F = DD is a superalgebra-valued linear map which respects the Bianchi identity and correctly defines a chiral parallel transport compatible with a generic Lie superalgebra structure.
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Thierry-Mieg, J. Chirality, a new key for the definition of the connection and curvature of a Lie-Kac superalgebra. J. High Energ. Phys. 2021, 111 (2021). https://doi.org/10.1007/JHEP01(2021)111
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DOI: https://doi.org/10.1007/JHEP01(2021)111