Instantons and geometric invariant theory (Q1068143)

By the theorem of Atiyah-Ward (anti-)self-dual Yang-Mills potentials on \(S^ 4\) correspond to certain holomorphic vector bundles on \({\mathbb{P}}_ 3({\mathbb{C}})\) via the twistor fibering \({\mathbb{P}}_ 3({\mathbb{C}})\to S^ 4\). In this paper a new correspondence is established by considering instead holomorphic vector bundles on \({\mathbb{P}}_ 2({\mathbb{C}})\) with a trivialization on a fixed line. More precisely let G be one of the groups SU(r), SO(r), Sp(r) and let M(G,k) be the set of isomorphism classes of pairs (A,\(\alpha)\), where A is a (anti-)self-dual G-connection on a G- principal bundle over \(S^ 4={\mathbb{R}}^ 4\cup \{\infty \}\) with Pontrjagin index k, and \(\alpha\) is a trivialization \(P_{\infty}\simeq G\). On the other hand let \(VB(G_{{\mathbb{C}}},k)\) be the set of isomorphism classes of holomorphic \(G_{{\mathbb{C}}}\)-vector bundles E on \({\mathbb{P}}_ 2({\mathbb{C}})\) with Chern classes \(c_ 1=0\), \(c_ 2=k\), where \(G_{{\mathbb{C}}}\) is the complexification of G, together with a trivialization \(E| \ell_{\infty}\simeq \ell_{\infty}\times {\mathbb{C}}^ 2\) on the line at infinity. It is proved that there is a natural bijection M(G,k)\(\to_{\approx}VB(G_{{\mathbb{C}}},k)\). The proof is given for \(G=SU(r)\) by analyzing the matrices in monad descriptions of the bundles on \({\mathbb{P}}_ 2({\mathbb{C}})\) and of the bundles on \({\mathbb{P}}_ 3({\mathbb{C}})\) which correspond to the connections by the Atiyah-Ward correspondence. The point is that: (1) the structure of the matrices is simplified considerably by fixing a trivalization of \(E| \ell_{\infty}\), and that (2) the reality condition for instanton bundles on \({\mathbb{P}}_ 3({\mathbb{C}})\) is exactly that for the vanishing of the moment map \(\mu\) associated to the natural group action of GL(k,\({\mathbb{C}})\) on the space \(W\subset {\mathbb{C}}^ N\) of the monad matrices for \(VB(G_{{\mathbb{C}}},k)\). Then by the result of Kempf-Ness on closest points on orbits of stable points it follows that \(VB(G_{{\mathbb{C}}},k)=W/GL(k,{\mathbb{C}})\) is in bijection with \((\mu^{- 1}(0)\cap W)/U(k)=M(G,k)\). As a corollary M(G,k) inherits a complex algebraic structure. It would be interesting to know whether \(VB(G_{{\mathbb{C}}},k)\) is smooth. Since the above bijection should at least be topological it follows from the connectivity of \(VB(G_{{\mathbb{C}}},k)\), which is inherited from that of the usual moduli spaces M(0,k) of stable bundles on \({\mathbb{P}}_ 2({\mathbb{C}})\), that also M(G,k) is connected.