Yang-Mills equations in 4-dimensional conformally connected manifolds.

*(English. Russian original)*Zbl 1188.58006
Russ. Math. 53, No. 3, 56-60 (2009); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 3, 67-72 (2009).

After a short general description on real quadrics in a 5-dimensional
projective space, the author presents the notion of conformal
connectivity
on 4-dimensional manifolds. The further search of the necessary invariant
is related with the Hodge operator. Once the first variation of the Yang-Mills
functional and the Yang Mills equation are sketched, the paper ends with
three examples of different type of conformal manifolds.

On these examples, it is shown that the only invariant, which is quadratic with respect to the curvature \(\Phi\) of the connectivity is the Yang-Mills functional \(\int|{\mathrm{tr}} (*\Phi\wedge\Phi)|\).

Reviewer’s remark: The first sections presented are not new.

On these examples, it is shown that the only invariant, which is quadratic with respect to the curvature \(\Phi\) of the connectivity is the Yang-Mills functional \(\int|{\mathrm{tr}} (*\Phi\wedge\Phi)|\).

Reviewer’s remark: The first sections presented are not new.

Reviewer: Marian Ioan Munteanu (Iaşi)

##### MSC:

58E15 | Variational problems concerning extremal problems in several variables; Yang-Mills functionals |

53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

##### Keywords:

Bianchi identity; compact 4-dimensional manifold; conformal connectivity; curvature of connectivity; Hodge operator; quadric signature; real quadrics; Yang-Mills functional
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\textit{A. V. Luk'yanov}, Russ. Math. 53, No. 3, 56--60 (2009; Zbl 1188.58006); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 3, 67--72 (2009)

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##### References:

[1] | A. L. Besse, Einstein Manifolds (Springer, Berlin, 1987; Mir, Moscow, 1990), Vol. I. |

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