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 χ which defines the supertrace of the superalgebra: S T r ( … ) = T r ( 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 anticomtes with ( χ A ) 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.