Handbook of Pseudo-Riemannian Geometry and Supersymmetry
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Special geometry for arbitrary signatures
María A. Lledó
, Óscar Maciá
, Antoine Van Proeyen
and Veeravalli S. Varadarajan
(1) Universidad de Valencia, Burjassot (Valencia), Spain
(2) Universidad de Valencia, Burjassot (Valencia), Spain
(3) Katholieke Universiteit Leuven, Belgium
(4) UCLA, Los Angeles, USA
In this paper we generalize special geometry to arbitrary signatures
in target space. We formulate the definitions in a precise mathematical
setting and give a translation to the coordinate formalism used in
physics. For the projective case, we first discuss in detail projective
Kähler manifolds, appearing in N = 1 supergravity. We develop a
new point of view based on the intrinsic construction of the line bundle.
The topological properties are then derived and the Levi-Civita
connection in the projective manifold is obtained as a particular projection
of a Levi-Civita connection in a ‘mother’ manifold with one
extra complex dimension. The origin of this approach is in the superconformal
formalism of physics, which is also explained in detail.
Finally, we specialize these results to projective special Kähler manifolds
and provide explicit examples with different choices of signature.
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