SU(2/1) superchiral self-duality: a new quantum, algebraic and geometric paradigm to describe the electroweak interactions
Jean Thierry-Mieg, Peter Jarvis
Abstract
A bstract 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 $$ \tilde{A} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> </mml:math> mixing exterior-forms of all degrees and satisfying the chiral self-duality condition $$ \tilde{A} =^{\ast }{\tilde{A}}_{\chi } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:msup> <mml:mspace/> <mml:mo>∗</mml:mo> </mml:msup> <mml:msub> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mi>χ</mml:mi> </mml:msub> </mml:math> , 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.