\(k\)-instantons on \(G_2(\mathbb{C}^{n+2})\) and stable vector bundles
Twistor spaces are very important for understanding Yang-Mills fields on a four dimensional manifold \(M\). As established by \textit{M. F. Atiyah, N. J. Hitchin} and \textit{I. M. Singer} [Proc. R. Soc. Lond., Ser. A 362, 425-461 (1978; Zbl 0389.53011)] there is a very nice correspondence between Yang-Mills fields on \(M\) and holomorphic vector bundles on the twistor space \(F\) of \(M\). This was used to classify instantons on \(S^4\) and on \({\mathbf P}^2\). For quaternionic-Kähler manifolds \(M\) twistor spaces \(F\) were studied by \textit{S. Salamon} [Invent. Math. 67, 143-171 (1982; Zbl 0486.53048)] and latter on self-dual and anti-self-dual connections on \(M\) by Mamone Capria and Salamon, and Nitta. In this paper the authors consider complex Grassmannians \(M=G_2({\mathbb{C}}^{n+2})\) (as quaternionic-Kähler manifolds) and classify a special class of anti-self-dual \(Sp(r)\)-bundles, \(r\leq 1\). Classification is expressed as the corresponding bundle \(\tilde E\) on the twistor space \(F=SU(n+1)/S(U(1)\otimes U(n)\otimes U(1))\) (flag manifold) of \(M\) is the cohomology bundle of some monad. In the proof Riemann-Roch formula and the appropriate vanishing theorems are used.
scientific article; zbMATH DE number 1389852