##
**Graded gauge theory.**
*(English)*
Zbl 0536.53069

The author deals with two things: first he defines a class of graded principal fibre bundles, secondly he introduces there some differential operators and studies a Lagrangian of the supergauge field theory. For the purpose of his graded bundles he presents a realization of a super Lie group in the case when a linear representation of its algebra is given. The total space of the supergroup is provided with a semigroup structure. Then the author constructs a graded fibre bundle. The resulting bundle contains both Lorentz spinors (at the base) and G- spinors (at the fibre). Note that there exist also non-trivial bundles of this type; an example is outlined by the reviewer in [Prepr., Inst. Math., Pol. Acad. Sci., Warsz. 277, 88 p. (1983; Zbl 0515.58013)].

In the second part the author tries to describe gauge fields in the above bundle and their Lagrangians. Commutation relations in some sectors of values of the fields are discussed. Physical phenomena like conformal symmetry breaking and mass in related fields are studied. Three coupling scale constants of length-dimension are pointed out. The number of variables is big as the author is unable to give an explicit form for the Lagrangian. On the other hand some physical fields cannot be described in this model. Note that the whole machinery of the first part is not used in the second one. For needs of the Lagrangian local coordinates of the bundle are quite sufficient. Formula 1.18 is not proved. 4.6 seems to be not true. Several notational mistakes can be noticed.

In the second part the author tries to describe gauge fields in the above bundle and their Lagrangians. Commutation relations in some sectors of values of the fields are discussed. Physical phenomena like conformal symmetry breaking and mass in related fields are studied. Three coupling scale constants of length-dimension are pointed out. The number of variables is big as the author is unable to give an explicit form for the Lagrangian. On the other hand some physical fields cannot be described in this model. Note that the whole machinery of the first part is not used in the second one. For needs of the Lagrangian local coordinates of the bundle are quite sufficient. Formula 1.18 is not proved. 4.6 seems to be not true. Several notational mistakes can be noticed.

Reviewer: J.Czyz

### MSC:

53C80 | Applications of global differential geometry to the sciences |

58H99 | Pseudogroups, differentiable groupoids and general structures on manifolds |

55R99 | Fiber spaces and bundles in algebraic topology |

81T08 | Constructive quantum field theory |

17B70 | Graded Lie (super)algebras |

### Keywords:

graded principal fibre bundles; Lagrangian; supergauge field theory; supergroup; Lorentz spinors; G-spinors### Citations:

Zbl 0515.58013
Full Text:
DOI

### References:

[1] | Kostant, B.: In: Differential geometrical methods in theoretical physics. Math. Ser. Vol. 570. Bleuler, K., Reetz, A. (eds.). Berlin, Heidelberg, New York: Springer 1975 · Zbl 0372.22009 |

[2] | Rogers, A.: Super Lie groups: Global topology and local structure. J. Math. Phys.22, 939 (1981) · Zbl 0473.22013 |

[3] | Balantekin, A., Baha, Bars, I.: Dimension and character formulas for Lie supergroups. J. Math. Phys.22, 1149 (1981) · Zbl 0469.22017 |

[4] | Kerner, R., da Silva Maia, E.M.: Graded gauge theories over supersymmetric space. J. Math. Phys.24, 361 (1983) · Zbl 0511.58019 |

[5] | Kerner, R.: Nuovo Cimento A73, 309 (1983) |

[6] | Freedman, D.Z., Nieuwenhuizen, P., Van, Ferrara, S.: Progress toward a theory of supergravity. Phys. Rev. D13, 3214 (1976) |

[7] | Yates, R.G.: Fibre bundles and supersymmetries. Commun. Math. Phys.76, 255 (1980) · Zbl 0447.53027 |

[8] | Bonora, L., Pasti, P., Tonon, M.: Ann. Phys.144, 15 (1982) |

[9] | D’Auria, R., Fré, P., Regge, T.: Graded-Lie-algebra cohomology and supergravity. Riv. Nuovo Cimento3, 12 (1980) |

[10] | See also: Gawedzki, K.: Supersymmetries-mathematics of supergeometry. Ann. Inst. Henri Poincaré27, 335 (1977) · Zbl 0369.53061 |

[11] | Haag, R., Lopuszanski, J., Sohnius, M.: All possible generators of supersymmetries of theS-matrix. Nucl. Phys. B88, 257 (1975) |

[12] | Corwin, L., Ne’eman, Y., Sternberg, S.: Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry). Rev. Mod. Phys.47, 573 (1975) · Zbl 0557.17004 |

[13] | Witten, E.: Dynamical breaking of supersymmetry. Nucl. Phys. B188, 513 (1981); Search for a realistic Klein-Kaluza theory.186, 412 (1981) · Zbl 1258.81046 |

[14] | Kerner, R.: Geometrical background for the unified field theories: the Einstein-Cartan theory over a principal fibre bundle. Ann. Inst. Poincaré34, 437 (1981) · Zbl 0474.53060 |

[15] | Domokos, G., Kövesi-Domokos, S.: Internal symmetries and ghost symmetries. Phys. Rev. D16, 3060 (1977) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.