nLab special geometry





Given a field theory with scalar fields, a frequent question is whether it can be reinterpreted as a sigma-model, where the scalars are the coordinates of some target space. For d=4 N=2 supergravity and d=5 N=2 supergravity, such target spaces correspond to smooth manifolds with structures collectively known as special geometry.


The term special geometry first appears in (Strominger (1990)) in the context of d=4 N=2 theories. The structure described therein eventually became known as special Kähler, allowing to relate structures other than Kähler to the “special” structure, which essentially refers to the existence of some special affine coordinates. These variants lead to different supersymmetry multiplets.


The starting point of special geometries is an Hermitian manifold (see (Lopes Cardoso & Mohaupt (2020))). One then introduces additional compatible structure to get the different variants of special geometries.

Affine special real geometry


An affine special real manifold is a Hessian manifold (M,g,)(M,g,\nabla) such that the Hessian potential is cubic polynomial in \nabla-affine (i.e. special) coordinates.

This structure appears in the context of five-dimensional theories with vector multiplets (CM20).

Special Kähler geometry

There are at least two definitions of special Kähler geometry on a smooth manifold MM. One is local in nature, phrased in terms of the existence of coordinates satisfying certain conditions, first appearing in (de Wit, Lauwers & van Proeyen (1985)). The other definition is global, which we recall below. These two definitions are shown to be equivalent in Strominger (1990).


Let MM be an nn-dimensional Kähler manifold such that its Kähler form JJ represents in de Rham cohomology the first Chern class of a complex line bundle LL. Let HH be a holomorphic Sp(2n+2,)\text{Sp}(2n+2,\mathbb{R}) (the symplectic group) vector bundle over MM and (,)(-,-) a compatible Hermitian metric on HH. Then MM is a special Kähler manifold, meaning it admits special geometry, if there exists a holomorphic section Ω\Omega of HLH\otimes L such that

J=i2π¯ln(Ω,Ω¯). J \,=\, - \frac{\mathrm{i}}{2\pi} \partial\bar{\partial} \text{ln} (\Omega,\bar{\Omega}) \,.

A more general notion is special complex geometry defined in ACD02: a complex manifold (M,J)(M,J) with a flat torsionfree connection \nabla such that J\nabla J is symmetric.



We show equivalences of some definitions and give examples which show that earlier definitions are not equivalent, and are not sufficient to restrict the Kähler metric to one that occurs in N = 2 supersymmetry.

  • Pietro Fré, Lectures on special Kähler geometry and electric-magnetic duality rotations, Nucl. Phys. B Proc. Suppl. 45:2-3, pp. 59–114 (1996) (hep-th/9512043 doi)

Freed’s approach to special Kähler geometry is generalized to special complex geometry in

  • D. V. Alekseevsky, V. Cortés, C. Devchand, Special complex manifolds, Journal of Geometry and Physics 42:1-2 (2002) 85–105 doi

Last revised on November 4, 2023 at 19:53:59. See the history of this page for a list of all contributions to it.