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

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

10 pages, 24 references. This is the revised version published in JHEP. Our new definition of the superconnection is unchanged relative to version 1 of this preprint, but thanks to the careful help of the referee, the discussion of the literature is streamlined and much improved

References in corpus (3)

• Noncommutative Geometry and the standard model with neutrino mixing
• Supersymmetric Wilson Loops via Integral Forms
• Scalar anomaly cancellation reveals the hidden superalgebraic structure of the quantum chiral SU(2/1) model of leptons and quarks

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• Indecomposable doubling for representations of the type I Lie superalgebras sl(m/n) and osp(2/2n)