A quaternionic kähler manifold is a special type of Riemannian manifold equipped with a metric that is not only Kähler but also allows for the existence of a hypercomplex structure. This means it has a holonomy group that is a subgroup of the unitary group, specifically related to quaternionic structures. These manifolds play a significant role in the study of special holonomy, which is important in various areas of mathematics and theoretical physics.
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Quaternionic kähler manifolds can be seen as an extension of Kähler manifolds to higher dimensions, specifically when considering real dimensions that are multiples of four.
These manifolds have a holonomy group that is contained within the group Sp(n) × U(1), which reflects their richer geometric structure compared to ordinary Kähler manifolds.
Every quaternionic kähler manifold admits a unique metric connection, known as the quaternionic kähler connection, which preserves both the metric and the hypercomplex structure.
Examples of quaternionic kähler manifolds include flag manifolds and certain symmetric spaces, which showcase their interesting properties in various contexts.
In physics, quaternionic kähler manifolds are relevant to theories like supergravity and string theory, where they provide geometric frameworks for describing supersymmetry.
Review Questions
How do quaternionic kähler manifolds relate to Kähler manifolds, and what additional structure do they possess?
Quaternionic kähler manifolds extend the concept of Kähler manifolds by incorporating a hypercomplex structure alongside the Kähler metric. While Kähler manifolds are defined solely by their complex structures and compatible metrics, quaternionic kähler manifolds require that these structures satisfy certain compatibility conditions with respect to quaternion multiplication. This additional layer allows quaternionic kähler manifolds to exhibit richer geometric properties and behaviors.
Discuss the significance of the holonomy group for quaternionic kähler manifolds and how it differs from ordinary Kähler manifolds.
The holonomy group of a quaternionic kähler manifold is more complex than that of an ordinary Kähler manifold because it includes components related to both hypercomplex structures and quaternionic multiplication. Specifically, for quaternionic kähler manifolds, the holonomy group lies within Sp(n) × U(1), reflecting how these manifolds incorporate both quaternionic and complex structures. This distinction in holonomy groups indicates that quaternionic kähler manifolds possess additional symmetries and geometric properties not found in standard Kähler manifolds.
Evaluate the implications of quaternionic kähler geometry in theoretical physics, particularly in relation to supergravity or string theory.
Quaternionic kähler geometry has significant implications in theoretical physics, especially in supergravity and string theory. These geometries provide suitable frameworks for constructing models that incorporate supersymmetry, as they allow for consistent formulations of scalar fields and interactions governed by the underlying geometric structures. The rich topological properties and special holonomy associated with quaternionic kähler manifolds also facilitate the study of compactification scenarios in string theory, revealing deeper insights into the connections between geometry and physical phenomena.
A Kähler manifold is a complex manifold that has a Riemannian metric compatible with its complex structure, allowing for a symplectic form derived from the Kähler metric.
Holonomy group: The holonomy group of a Riemannian manifold captures how much the parallel transport of vectors around closed loops can change those vectors, giving insight into the curvature and topology of the manifold.
Hypercomplex structure: A hypercomplex structure is a generalization of complex structures, where there are multiple compatible complex structures related through quaternion multiplication.