Abstract
We propose an extension of the Yang-Mills paradigm from Lie algebras to internal chiral superalgebras. We replace the Lie algebra-valued connection one-form A, by a superalgebra-valued polyform tilde{A} mixing exterior-forms of all degrees and satisfying the chiral self-duality condition tilde{A} =^{ast }{tilde{A}}_{chi } , where χ denotes the superalgebra grading operator. This superconnection contains Yang-Mills vectors valued in the even Lie subalgebra, together with scalars and self-dual tensors valued in the odd module, all coupling only to the charge parity CP-positive Fermions. The Fermion quantum loops then induce the usual Yang-Mills-scalar Lagrangian, the self-dual Avdeev-Chizhov propagator of the tensors, plus a new vector-scalar-tensor vertex and several quartic terms which match the geometric definition of the supercurvature. Applied to the SU(2/1) Lie-Kac simple superalgebra, which naturally classifies all the elementary particles, the resulting quantum field theory is anomaly-free and the interactions are governed by the super-Killing metric and by the structure constants of the superalgebra.
Highlights
We propose an extension of the Yang-Mills paradigm from Lie algebras to internal chiral superalgebras
In Quillen [11, 12], the covariant differential is defined as D = d + A + L, where L = Liλi is as for us a mixed exterior-form of even degree valued in the odd module of the superalgebra
Because L must be odd relative to the differential calculus to ensure that the curvature F = DD defines a linear map, Quillen assumes that the components Li of L are valued in another graded algebra which anticommute with the exterior-forms
Summary
This can as well be expressed in terms of the dual 1-form ∗c (C.6): Applying this transformation to the 2, 3 and 4 forms (b, c, e), the Dirac operator associated to the superconnection A acting on the left Fermions can be rewritten as. This term plays a crucial role in the self consistency of the theory Given these algebraic and geometrical definitions, let us study how the Dirac action of the superconnection on the chiral Fermions gets promoted in the quantum field theory into the definition of the propagators and interactions of its components, the complex scalar field Φ, the vector A, and the complex self-dual anti-self-dual antisymmetric tensor BB all correctly satisfying the spin-statistics relation
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