Abstract
Attention is given basically to the construction of auxiliary bundles and the development of a right-invariant formalism: the Lie algebra of right-invariant vertical vector fields cal P instead of the algebra of fundamental fields fund P, the corresponding 1-forms as means for interpreting Faddeev-Popov “ghosts.”
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Literature cited
V. Abramov, “Generalized p-forms and Faddeev-Popov field,” Uch. Zap. Tartusk. Univ., No. 665, 3–13 (1984).
V. Abramov, “Supermanifolds with connection and symmetries of quantum effective action,” [in Russian], Preprint FI-28 (1985). Section of Physicomathematical and Technical Sciences, Academy of Sciences of the EstSSR, Tartu (1985).
N. Bourbaki, Algebra. Algebraic Structures. Linear and Multilinear Algebra [Russian translation], Fizmatgiz, Moscow (1962).
N. Bourbaki, Differentiable and Analytic Manifolds. Summary of Results [Russian translation], Mir, Moscow (1975).
A. M. Vasil'ev, “Differential algebra. Covariant analytic methods in differential geometry,” J. Sov. Math.,14, No. 3 (1980).
L. E. Evtushik, Yu. G. Lumiste, N. M. Ostianu, and A. P. Shirokov, “Differential-geometric structures on manifolds,” J. Sov. Math.,14, No. 6 (1980).
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry [Russian translation], Vol. 1, Nauka, Moscow (1981).
N. P. Konopleva and V. N. Popov, Gauge Fields [in Russian], 2nd edn., Moscow (1980).
G. F. Laptev, “Structural equations of a principal fibered manifold,” in: Proceedings of the Geometry Seminar [in Russian], Vol. 2, In-t. Nauchn. Inf. Akad. Nauk SSSR (1969), pp. 161–178.
G. F. Laptev, “Manifolds immersed in generalized spaces,” in: Proceedings of the 4th All-Union Mathematical Congress, 1961 [in Russian], Vol. 2, Nauka, Leningrad (1964), pp. 226–233.
Yu. G. Lumiste, “Theory of connections in fibered space,” in: Algebra. Topology. Geometry. 1969 [in Russian], Itogi Nauki i Tekhniki, VINITI AN SSSR, Moscow (1971), pp. 123–168.
Yu. G. Lumiste, “Connections in the geometric interpretation of Yang-Mills and Faddeev-Popov fields, “Izv. Vyssh. Uchebn. Zaved., Mat., No. 1, 46–54 (1983).
Yu. I. Manin, “Gauge fields and holomorphic geometry,” J. Sov. Math.,21, No. 4 (1983).
Yu. I. Manin, Gauge Fields and Complex Geometry [in Russian], Nauka, Moscow (1984).
A. S. Mishchenko, Vector Bundles and Their Applications [in Russian], Nauka, Moscow (1984).
A. G. Reiman, M. A. Semenov-Tyan-Shanskii, and L. D. Faddeev, “Quntum anomalies and cocycles on gauge groups,” Funkts. Anal. Prilozhen.,18, No. 4, 64–72 (1984).
A. A. Slavnov and L. D. Faddeev, Introduction to the Quantum Theory of Gauge Fields [in Russian], Nauka, Moscow (1978).
S. Sternberg, Lectures on Differential Geometry [Russian translation], Mir, Moscow (1970).
N. Steenrod, Topology of Fiber Bundles [Russian translation], IL, Moscow (1953).
M. Hall, Theory of Groups [Russian translation], IL, Moscow (1962).
M. F. Atiyah, N. J. Hitchin, and I. M. Singer, “Self-duality in four-dimensional Riemannian geometry,” Proc. R. Soc. London,A362, No. 1711, 425–461 (1978).
M. F. Atiyah and J. D. S. Jones, “Topological aspects of Yang-Mills theory,” Commun. Math. Phys.,61, No. 2, 97–118 (1978).
C. Becchi, A. Rouet, and R. Stora, “Renormalization of the Abelian Higgs-Kibble model,” Commun. Math. Phys.,42, No. 2, 127–162 (1975).
J. P. Bourguignon and H. B. Lawson, Jr., “Yang-Mills theory: Its physical origins and differential geometric aspects,” Ann. Math. Stud., No. 102, 395–421 (1982).
M. L. Bruschi, “About Singer theorem,” Lett. Nuovo Cimento,38, No. 14, 485–490 (1983).
M. Daniel and C. M. Viallet, “The geometrical setting of the gauge theories of Yang-Mills type,” Rev. Mod. Phys.,52, 175–197 (1980).
Th. Friedrich and L. Habermann, “The Yang-Mills equation over the two-dimensional sphere,” Prepr. Humboldt-Univ. Berlin Sekt. Math., No. 78 (1984).
J. M. Leinaas and K. Olaussen, “Ghosts and geometry,” Phys. Lett.,B108, No. 3, 199–202 (1982).
E. Lubkin, “Geometric definition of gauge invariance,” Ann. Phys. (NY),23, No. 2, 233–283 (1963).
P. K. Mitter and C. M. Viallet, “On the bundle of connections and the gauge orbit manifold in Yang-Mills theory,” Commun. Math. Phys.,79, No. 4, 457–472 (1981).
Y. Ne'eman, “Proposed geometrizations for unitarity-imposing superalgebras of the field/ ghost systems,” in: Coll. on Group Theor. Methods in Physics, TAUP, Trieste (1983), pp. 151–183.
I. M. Singer, “Some remarks on Gribov's ambiguity,” Commun. Math. Phys.,60, No. 1, 7–12 (1978).
S. Sternberg, “On the role of field theories in our physical conception of geometry,” Lect. Notes Math.,676, 1–80 (1978).
R. Utiyama, “Invariant theory of interaction,” Phys. Rev.,101, 1597–1607 (1956).
S. Vergeles, “Some properties of orbit space in Yang-Mills theory,” Lett. Math. Phys.,7, No. 5, 399–406 (1983).
T. Wu and C. N. Yang, “Concept of nonintegrable phase factors and global formulation of gauge fields,” Phys. Rev.,D12, No. 12, 3845–3857 (1975).
C. N. Yang, “Fiber bundles and the physics of the magnetic monopole,” in: Chern Symp., 1979, Proc. Int. Symp. Differ. Geometry in Honor of S. S. Chern, Berkeley, Calif., June, 1979, NY e.a. (1980), pp. 247–253.
C. N. Yang and R. L. Mills, “Conservation of isotopic spin and isotopic gauge invariance,” Phys. Rev.,96, 191–195 (1954).
Additional information
The content of the plenary report of the author “Differential-Geometric Methods in Gauge Theories” to the Seventh All-Union Geometry Conference (Odessa, September 18–19, 1984) is recounted in extended and supplemented form.
Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 17, pp. 153–171, 1985.
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Lumiste, Y.G. Differential-geometric structures and Gauge theories. J Math Sci 37, 1254–1268 (1987). https://doi.org/10.1007/BF01091862
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DOI: https://doi.org/10.1007/BF01091862