NONCOMMUTATIVE GEOMETRY AND GRADED ALGEBRAS IN ELECTROWEAK INTERACTIONS

Abstract

The Standard Model of Electroweak Interactions can be described by a generalized Yang-Mills field incorporating both the usual gauge bosons and the Higgs fields. The graded derivative by means of which the Yang-Mills field strength is constructed involves both a differential acting on space-time and a differential acting on an associative graded algebra of matrices. The square of the curvature for the corresponding covariant derivative yields the bosonic Lagrangian of the Standard Model. We show how to recover the whole fermionic part of the Standard Model in this framework. Quarks and leptons fit naturally into the smallest typical and nontypical irreducible representations of the graded algebra Lie SU(2|1) associated with the above associative ℤ2-graded algebra. The existence of reducible indecomposable representations leads naturally to flavor mixing in the quark sector, possibility of existence for a right neutrino and possible mixing in the leptonic sector. We therefore bridge the gap between noncommutative geometry and graded Lie algebras. The Z2 grading refers to left and right chiralities in the fermionic sector and to even and odd forms in the bosonic sector. Supergauge transformations could only be defined in an extension of the theory incorporating tensor fields of higher rank. The Standard Model contains only one-forms and zero-forms in the bosonic sector, therefore only the even part of the above graded Lie algebra — i.e. Lie[SU(2)×U(1)] — acts.