Topological quantum field theory in quaternionic geometry
type: Item
instance of: scholarly article
title: Topological quantum field theory in quaternionic geometry
zbMATH Open document ID: 0818.53044
DOI: 10.1016/0393-0440(94)90004-3
published in: Journal of Geometry and Physics
publication date: 1995-08-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 81T70
Mathematics Subject Classification ID: 11R52
zbMATH DE Number: 663779
zbMATH Keywords: topological quantum field theory
zbMATH Keywords: almost quaternionic structure
zbMATH Keywords: hyper- Kähler structure
zbMATH Keywords: instanton
zbMATH Keywords: supersymmetry
author: Sergei Merkulov
author: Henrik Pedersen
author: Andrew Swann
reviewed by: Andrzej Frydryszak
MaRDI profile type: Publication
full work available at URL: 0393-0440(94)90004-3
OpenAlex ID: W2072278063
cites work: Vector fields on spheres
cites work: Self-duality in four-dimensional Riemannian geometry
cites work: Topological Lagrangians and cohomology
cites work: Complex Paraconformal Manifolds – their Differential Geometry and Twistor Theory
cites work: Moduli space of 1-instantons on a quaternionic projective space \({\mathbb{H}}{\mathbb{P}}^ n\)
cites work: Q3475945
cites work: Q3998602
cites work: Q5185713
cites work: Holonomy groups, complex structures and \(D=4\) topological Yang-Mills theory
cites work: Sphere transitive structures and the triality automorphism
cites work: Yang-Mills fields on quaternionic spaces
cites work: Q3697706
cites work: Q3657003
cites work: Superconnections, Thom classes, and equivariant differential forms
cites work: Vector bundles over quaternionic Kähler manifolds
cites work: Moduli spaces of Yang-Mills connections over quaternionic Kähler manifolds
cites work: Q3317724
cites work: Differential geometry of quaternionic manifolds
cites work: HyperKähler and quaternionic Kähler geometry
cites work: Topological quantum field theory
description: scientific article; zbMATH DE number 663779