to obtain a parameterization of the section X in terms of coordinates σ, φ of the manifold SO(1, 1) × SO(1, n)/SO(2) × SO(n). This achieves the desired proof that the group Giso has a transitive action on the special manifold and consequently that the cubic norm (8.6.37), (8.6.38) is admissible as a definition of a very special manifold.

8.6.3 Quaternionic Geometry

Next we turn our attention to the hypermultiplet sector of an N = 2 supergravity. For these multiplets no distinction arises between the D = 4 and D = 5. Each hypermultiplet contains 4 real scalar fields and, at least locally, they can be regarded as the four components of a quaternion. The locality caveat is, in this case, very substantial because global quaternionic coordinates can be constructed only occasionally even on those manifolds that are denominated quaternionic in the mathematical literature [6, 17]. Anyhow, what is important is that, in the hypermultiplet sector, the scalar manifold QM has dimension multiple of four:

and, in some appropriate sense, it has a quaternionic structure.

We name Hypergeometry that pertaining to the hypermultiplet sector, irrespectively whether we deal with global or local N = 2 theories. Yet there are two kinds of hypergeometries. Supersymmetry requires the existence of a principal SU(2)- bundle

The bundle S U is flat in the rigid supersymmetry case while its curvature is proportional to the Kähler forms in the local case.

These two versions of hypergeometry were already known in mathematics prior to their use [14, 18–20, 22, 23] in the context of N = 2 supersymmetry and are identified as:

rigid hypergeometry ≡ HyperKähler geometry

(8.6.50)

local hypergeometry ≡ quaternionic geometry

8.6.4 Quaternionic, Versus HyperKähler Manifolds

Both a quaternionic or a HyperKähler manifold QM is a 4m-dimensional real manifold endowed with a metric h:

and three complex structures

Equation (8.6.67) is the explicit statement that the Levi Civita connection associated with the metric h has a holonomy group contained in SU(2) Sp(2m). Consider now (8.6.53), (8.6.55) and (8.6.59). We easily deduce the following relation:

that holds true both in the HyperKähler and in the quaternionic case. In the latter case, using (8.6.59), (8.6.69) can be rewritten as follows:

Equation (8.6.70) implies that the intrinsic components of the curvature 2-form Ωx yield a representation of the quaternion algebra. In the HyperKähler case such a representation is provided only by the HyperKähler form. In the quaternionic case we can write:

The quaternionic manifolds are not requested to be homogeneous spaces, however there exists a subclass of quaternionic homogeneous spaces that are displayed in Table 8.3.