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N=2 Supergravity and N=2 Super Yang-Mills Theory on General Scalar Manifolds: Symplectic Covariance, Gaugings and the Momentum Map

L. Andrianopoli, M. Bertolini, A. Ceresole, R. D'Auria, S. Ferrara, P. Fre', T. Magri

TL;DR

This work provides a coordinate-free, symplectic-covariant formulation of N=2 supergravity with arbitrary vector and hypermultiplets and general gaugings of isometries of the scalar manifolds. It derives the complete Lagrangian and SUSY variations, expresses the scalar potential in terms of Killing vectors and momentum maps for Special Kähler and quaternionic geometries, and presents the master Gaillard–Zumino formula for the period matrix ${\cal N}$ under symplectic transformations. It develops symplectic embeddings of homogeneous spaces such as ${\cal ST}[m,n]$ and details holomorphic and triholomorphic momentum maps that govern gauging of composite connections. It then takes the rigid limit to obtain general N=2 matter-coupled Yang–Mills theory from the supergravity construction, clarifying Planck-scale rescalings and how rigid special geometry and HyperKähler/quaternionic geometry emerge. The framework provides a unified geometric toolkit for moduli, dualities and string-compactification contexts beyond special coordinates.

Abstract

The general form of N=2 supergravity coupled to an arbitrary number of vector multiplets and hypermultiplets, with a generic gauging of the scalar manifold isometries is given. This extends the results already available in the literature in that we use a coordinate independent and manifestly symplectic covariant formalism which allows to cover theories difficult to formulate within superspace or tensor calculus approach. We provide the complete lagrangian and supersymmetry variations with all fermionic terms, and the form of the scalar potential for arbitrary quaternionic manifolds and special geometry, not necessarily in special coordinates. Lagrangians for rigid theories are also written in this general setting and the connection with local theories elucidated. The derivation of these results using geometrical techniques is briefly summarized.

N=2 Supergravity and N=2 Super Yang-Mills Theory on General Scalar Manifolds: Symplectic Covariance, Gaugings and the Momentum Map

TL;DR

This work provides a coordinate-free, symplectic-covariant formulation of N=2 supergravity with arbitrary vector and hypermultiplets and general gaugings of isometries of the scalar manifolds. It derives the complete Lagrangian and SUSY variations, expresses the scalar potential in terms of Killing vectors and momentum maps for Special Kähler and quaternionic geometries, and presents the master Gaillard–Zumino formula for the period matrix under symplectic transformations. It develops symplectic embeddings of homogeneous spaces such as and details holomorphic and triholomorphic momentum maps that govern gauging of composite connections. It then takes the rigid limit to obtain general N=2 matter-coupled Yang–Mills theory from the supergravity construction, clarifying Planck-scale rescalings and how rigid special geometry and HyperKähler/quaternionic geometry emerge. The framework provides a unified geometric toolkit for moduli, dualities and string-compactification contexts beyond special coordinates.

Abstract

The general form of N=2 supergravity coupled to an arbitrary number of vector multiplets and hypermultiplets, with a generic gauging of the scalar manifold isometries is given. This extends the results already available in the literature in that we use a coordinate independent and manifestly symplectic covariant formalism which allows to cover theories difficult to formulate within superspace or tensor calculus approach. We provide the complete lagrangian and supersymmetry variations with all fermionic terms, and the form of the scalar potential for arbitrary quaternionic manifolds and special geometry, not necessarily in special coordinates. Lagrangians for rigid theories are also written in this general setting and the connection with local theories elucidated. The derivation of these results using geometrical techniques is briefly summarized.

Paper Structure

This paper contains 25 sections, 1 theorem, 404 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Duality Rotations and Symplectic Covariance
  3. Symplectic embeddings of homogenous spaces
  4. Symplectic embedding of the ${\cal ST}\left [ m,n \right ]$ homogeneous manifolds
  5. Special Kähler Geometry
  6. Hodge--Kähler manifolds
  7. Special Kähler Manifolds: general discussion
  8. Special Kähler manifolds: the local case
  9. Special Kähler manifolds: the rigid case
  10. Special Kähler manifolds: the issue of special coordinates
  11. Hypergeometry
  12. Quaternionic, versus HyperKähler manifolds
  13. The Gauging
  14. The Momentum Map
  15. Holomorphic momentum map on Kähler manifolds
  16. ...and 10 more sections

Key Result

Lemma 7.1

The following identity is true: where $C_{\Lambda \Sigma}$ is a constant fulfilling the cocycle condition

Theorems & Definitions (6)

  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Definition 5.1
  • Definition 5.2
  • Lemma 7.1