Yang-Mills theory over compact symplectic manifolds
Over the past thirty years, Yang-Mills theory has undergone tremendous developments in both mathematics and theoretical physics, especially with regard to compact Riemannian four-manifolds and compact Kähler manifolds as underlying spaces. In particular, the moduli spaces of the respective Yang-Mills connections have been studied by many authors, largely in connection with the moduli theory for stable holomorphic vector bundles. Recently, some attempts have been undertaken to extend the ``classical'' Yang-Mills theory to the class of compact symplectic manifolds, and the paper under review represents another original effort in this direction. More precisely, the author considers an arbitrary vector bundle \(E\) with inner product \(h\) over an arbitrary compact symplectic manifold \((M,\omega)\), together with an arbitrary almost complex structure \(J\) compatible with the symplectic form \(\omega\). In this setting, he describes the solutions of the Yang-Mills equation in terms of the data \(\omega\) and \(J\), exhibits the moduli space of the corresponding Yang-Mills connections, and shows the finite dimensionality of this moduli space. In addition, the author's construction of the moduli space of Yang-Mills connections over a compact symplectic manifold also yields an explicit characterization of the absolute minima of the Yang-Mills functional, which is then used to prove the finite dimensionality of the moduli space of these minimizers as well. Finally, the author's fairly general, purely symplectic-geometric methods and results shed some new light on earlier results of A. Lichnerowicz on the absolute minima of the energy functional on the space of smooth mappings between two compact symplectic manifolds [\textit{A. Lichnerowicz}, Applications harmoniques et variétés kähleriennes, Sympos. Math., Roma 3, 341--402 (1970; Zbl 0193.50101)]. Alltogether, the present paper provides a highly interesting approach toward a general Yang-Mills theory on compact symplectic manifolds, with a lot of inspirations for further research in this direction. For instance, it would be very desirable to understand the geometry of the occuring moduli spaces more completely. Anyway, the paper contains many new and important results, and it is written in pleasantly detailed, thorough and comprehensive style.
scientific article; zbMATH DE number 2093687