Conjugate connections and SU(3)-instanton invariants. (English) Zbl 1019.53009
The author applies a notion of conjugate connection introduced by S. Kobayashi and E. Shinozaki to study the geometry of 4-manifolds dealing exclusively with the simplest compact simple Lie group \(SU(3)\). In this case the outer automorphism group is isomorphic to the cyclic group \(\mathbb{Z}_2\). One of the aims of this paper is to prove a fixed point theorem under the action of \(\mathbb{Z}_2\) on the irreducible part \({\mathcal B}^{*}(P)\) of the quotient space \({\mathcal B}(P)\) of gauge-equivalent connections on a reducible principal \(SU(3)\)-bundle \(P\) along the irreducible part \({\mathcal B}^{*}(Q)\) of the quotient spaces \({\mathcal B}(Q)\) of \(SO(3)\)-subbundles \(Q\) of \(P\). Following a line suggested by Donaldson, the author defines simple \(SU(3)\)-instanton invariants \(q_k(X)\) for certain smooth 4-manifolds.
Reviewer: Viktor Abramov (Tartu)
MSC:
53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
58E15 | Variational problems concerning extremal problems in several variables; Yang-Mills functionals |
70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
81T13 | Yang-Mills and other gauge theories in quantum field theory |
14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |