Chirality, a new key for the definition of the connection and curvature of a Lie-Kac superalgebra

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 [Formula: see text] which defines the supertrace of the superalgebra: [Formula: see text] , we construct a covariant differential: [Formula: see text] , where A is the standard even Lie-subalgebra connection 1-form and [Formula: see text] a scalar field valued in the odd module. Despite the fact that [Formula: see text] is a scalar, [Formula: see text] anticomtes with ([Formula: see text]) because [Formula: see text] anticommutes with the odd generators hidden in [Formula: see text]. 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.