Gauge theory and Higgs mechanism based on differential geometry on discrete space M4 * ZN

Abstract

Weinberg-Salam theory and $SU(5)$ grand unified theory are reconstructed using the generalized differential calculus extended on the discrete space $M_4\times Z_{{}_N}$. Our starting point is the generalized gauge field expressed by $A(x,n)={\sum }_i{a}_i^{†}(x,n)\mathbf{d}{a}_i(x,n),(n=1,2,\cdots N)$, where $a_i(x,n)$ is the square matrix valued function defined on $M_4\times Z_{{}_N}$ and $\mathbf{d}=d+{\sum }_{m=1}^{{}_N}{d}_{{\chi }_m}$ is generalized exterior derivative. We can construct the consistent algebra of $d_{{\chi }_m}$ which is exterior derivative with respect to $Z_{{}_N}$ and the spontaneous breakdown of gauge symmetry is coded in $d_{{\chi }_m}$. The unified picture of the gauge field and Higgs field as the generalized connection in non-commutative geometry is realized. Not only Yang-Mills-Higgs lagrangian but also Dirac lagrangian, invariant against the gauge transformation, are reproduced through the inner product between the differential forms. Three sheets ($Z_3)$ are necessary for Weinberg-Salam theory including strong interaction and $SU(5)$ Gut. Our formalism is applicable to more realistic model like $SO(10)$ unification model.

32 pages, CHU9401

References in corpus (3)

• Grand Unification in Non-Commutative Geometry
• Parameter restrictions in a non-commutative geometry model do not survive standard quantum corrections
• Higgs Mass and Noncommutative Geometry

Cited by in corpus (6)

• The Higgs Mass, Superconnections and the TeV-scale Left-Right Symmetric Model
• BRST invariant Lagrangian of spontaneously broken gauge theories in noncommutative geometry
• SU(5) orbifold GUT in noncommutative geometry
• BRST invariant formulation of spontaneously broken gauge theory in generalized differential geometry
• Generalized Chern-Simons Form and Descent Equation
• Reconstruction of the standard model with classical conformal invariance in noncommutative geometry