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What is Special Kähler Geometry ?

B. Craps, F. Roose, W. Troost, A. Van Proeyen

TL;DR

This paper clarifies the notion of special Kähler geometry in N=2 theories by presenting multiple, equivalent definitions for rigid and local cases and by proving their equivalence under symplectic duality. It develops the symplectic framework for dualities, introduces the vector-kinetic matrix ${\cal N}$ and the prepotential, and analyzes when a prepotential exists, including how symplectic rotations can reveal a prepotential even when it is not manifest. The authors connect these geometric structures to moduli spaces of Riemann surfaces and Calabi–Yau threefolds, illustrating the general framework with matrix formulations and examples. The results clarify the interplay between field theory, geometry, and string-inspired moduli, and provide a robust foundation for constructing N=2 supergravity actions and understanding CY moduli spaces through special geometry. Overall, any special Kähler manifold can be obtained from a prepotential via a symplectic transformation, tying together dualities, kinetic terms, and the underlying holomorphic data that govern vector multiplet couplings.

Abstract

The scalars in vector multiplets of N=2 supersymmetric theories in 4 dimensions exhibit `special Kaehler geometry', related to duality symmetries, due to their coupling to the vectors. In the literature there is some confusion on the definition of special geometry. We show equivalences of some definitions and give examples which show that earlier definitions are not equivalent, and are not sufficient to restrict the Kaehler metric to one that occurs in N=2 supersymmetry. We treat the rigid as well as the local supersymmetry case. The connection is made to moduli spaces of Riemann surfaces and Calabi-Yau 3-folds. The conditions for the existence of a prepotential translate to a condition on the choice of canonical basis of cycles.

What is Special Kähler Geometry ?

TL;DR

This paper clarifies the notion of special Kähler geometry in N=2 theories by presenting multiple, equivalent definitions for rigid and local cases and by proving their equivalence under symplectic duality. It develops the symplectic framework for dualities, introduces the vector-kinetic matrix and the prepotential, and analyzes when a prepotential exists, including how symplectic rotations can reveal a prepotential even when it is not manifest. The authors connect these geometric structures to moduli spaces of Riemann surfaces and Calabi–Yau threefolds, illustrating the general framework with matrix formulations and examples. The results clarify the interplay between field theory, geometry, and string-inspired moduli, and provide a robust foundation for constructing N=2 supergravity actions and understanding CY moduli spaces through special geometry. Overall, any special Kähler manifold can be obtained from a prepotential via a symplectic transformation, tying together dualities, kinetic terms, and the underlying holomorphic data that govern vector multiplet couplings.

Abstract

The scalars in vector multiplets of N=2 supersymmetric theories in 4 dimensions exhibit `special Kaehler geometry', related to duality symmetries, due to their coupling to the vectors. In the literature there is some confusion on the definition of special geometry. We show equivalences of some definitions and give examples which show that earlier definitions are not equivalent, and are not sufficient to restrict the Kaehler metric to one that occurs in N=2 supersymmetry. We treat the rigid as well as the local supersymmetry case. The connection is made to moduli spaces of Riemann surfaces and Calabi-Yau 3-folds. The conditions for the existence of a prepotential translate to a condition on the choice of canonical basis of cycles.

Paper Structure

This paper contains 34 sections, 10 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Symplectic transformations
  3. Rigid special geometry
  4. The vector superfield
  5. Actions for $N=2$ vector multiplets
  6. Duality transformations in superspace
  7. Definitions of rigid special Kähler manifolds
  8. Definition 1
  9. Definition 2
  10. Definition 3
  11. Matrix formulation.
  12. Riemann surfaces moduli spaces
  13. Useful concepts of Riemann surfaces
  14. Special moduli spaces of Riemann surfaces
  15. Examples
  16. ...and 19 more sections

Key Result

Lemma A.1

If the $2m\times m$ matrix $V=\pmatrix {X^I\cr Y_I}$, where $X^I$ and $Y_I$ are $m$-vectors, is of rank $m$, then there exists a symplectic rotation ${\cal S}\in \hbox{$Sp\left( 2m,{\hbox{\rm$\hbox{I}\!\hbox{R}$}} \right)$}$ such that $\tilde{V}={\cal S}V=\pmatrix {\tilde{X}\cr \tilde{Y}}$ has the p

Theorems & Definitions (10)

  • Lemma A.1
  • Lemma A.2
  • Theorem A.3
  • Corollary A.4
  • Lemma B.1
  • Lemma B.2
  • Lemma B.3
  • Lemma B.4
  • Lemma B.5
  • Corollary B.6