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

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 $\chi$ which defines the supertrace of the superalgebra: $STr(…) = Tr (\chi …)$,weconstructacovariantdifferential:$D = \chi (d + A) + \Phi$, where A is the standard even Lie-subalgebra connection 1-form and $\mathrm{\Phi }$ a scalar field valued in the odd module. Despite the fact that $\mathrm{\Phi }$ is a scalar, $\mathrm{\Phi }$ anticommutes with $(\chi A)$ because $\chi$ anticommutes with the odd generators hidden in $\mathrm{\Phi }$. 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.