Topological quantum field theory in quaternionic geometry
The paper is devoted to the generalization of Witten's topological field theory to \(4k\) dimensions (where \(k\geq 1\)), to the case when manifolds admit an almost quaternionic structure and hyper-Kähler structure. Donaldson-like smooth invariants of \(4k\)-dimensional manifolds admitting an almost quaternionic structure are constructed by means of quantum field theory techniques. This structure plays the same role as a conformal structure does in conventional theories of Witten and Donaldson in four dimensions. It is an auxiliary tool for introducing the notion of instanton and expressing the resulting smooth invariants in terms of the associated moduli space. Final results do not depend on any particular choice of it. For \(k=1\) the proposed theory reproduces the Witten interpretation of four-dimensional Donaldson invariants. For the topological field theory on a manifold admitting a hyper-Kähler structure the authors show that in addition to the Witten-type supersymmetry there is present a two-parameter family of new supersymmetries associated with a two-parameter family of complex structures. The analogue of the four-dimensional Donaldson map provides a number of candidates for new invariants of \(4k\)-dimensional hypercomplex structures. The paper consists of four sections: 1. Introduction, 2. A comment on Witten's topological field theory, 3. Topological field theory on almost quaternionic manifolds, 4. Topological field theory on a hyper-Kähler manifold.
scientific article; zbMATH DE number 663779
scientific article