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Yang-Mills connections over compact strongly pseudoconvex CR manifolds. (English) Zbl 0815.32008

The author extends the (Hermitian-Einstein) Yang-Mills theory (known for holomorphic vector bundles over compact Kähler manifolds) to the real odd-dimensional case. He bases on the ideas of Tanaka, who gave an affine connection and defined the notions of a holomorphic vector bundle and its canonical connection over a strongly pseudoconvex CR manifold. The author shows similarities between Tanaka’s connection in the CR case and the Hermitian-Einstein connection in the Kähler case. A suitably chosen connection is called the Yang-Mills-Tanaka connection by the author, and he asks if every holomorphic vector bundle over a CR manifold admits a unique Yang-Mills-Tanaka connection. To make the first step to a solution, he considers the moduli space of Yang-Mills-Tanaka connections \(D\) of a complex vector bundle \(E\) on a compact normal CR manifold. He shows that the tangent space of the moduli space at \([D]\) is isomorphic to some cohomology group. As an application he shows a unique existence theorem for the Yang-Mills equation for some vector bundle over a compact CR manifold being a \(U(1)\)-bundle over a compact Kähler manifold. He also shows that the moduli space of anti self-dual connections of \(P^ 2(\mathbb{C})\) induces the moduli space of Yang-Mills-Tanaka connections of the sphere \(S^ 5\). Finally he examines the holomorphic tangent bundle of a compact normal CR manifold and he gives conditions for the Tanaka’s connection to be a Yang-Mills-Tanaka connection.

MSC:

32V99 CR manifolds
32T99 Pseudoconvex domains
32Q20 Kähler-Einstein manifolds
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
32G13 Complex-analytic moduli problems
32J27 Compact Kähler manifolds: generalizations, classification
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References:

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