nLab special geometry

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Idea

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.

History

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.

Definition

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

Definition

An affine special real manifold is a Hessian manifold $(M,g,\nabla)$ 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 $M$. 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).

Definition

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

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

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

Examples

• The moduli spaces of Calabi-Yau threefolds and of $c=9$ $(2,2)$ SCFTs both admit special geometries.

• The base of any algebraically completely integrable system has canonically defined special geometry and conversely, any special Kähler manifolds is a base of some integrable system (Freed 1999).

Related concepts

• supergravity

• Hessian manifold

References

• Bernard de Wit, P.G. Lauwers, Antoine Van Proeyen. Lagrangians of N = 2 supergravity-matter systems, Nuclear Physics B 255 (1985) 569–608 doi

• Andrew Strominger, Special geometry, Comm. Math. Phys. 133:1 (1990), 163–180 [euclid:cmp/1104201320]

• Daniel Freed, Special Kähler manifolds, Comm. Math. Phys. 203 (1999) 31–52 [doi:10.1007/s002200050604]

• Gabriel Lopes Cardoso, Thomas Mohaupt. Special geometry, Hessian structures and applications. Physics Reports 855 (2020) 1–141. (doi)

• B. Craps, F. Roose, W. Troost, Antoine Van Proeyen, What is special Kähler geometry? , Nuclear Physics B 503:3 (1997) 565–613 (doi arXiv:hep-th/9703082)

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