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Symplectic double groupoids and the generalized Kähler potential

Daniel Álvarez, Marco Gualtieri, Yucong Jiang

Abstract

A description of the fundamental degrees of freedom underlying generalized Kähler geometry, which separates its holomorphic moduli from its compatible Riemannian metric in a similar way to the Kähler case, has been sought since its discovery in 1984. In this paper, we describe a full solution to this problem for arbitrary generalized Kähler manifolds. We discover that the holomorphic structure underlying a generalized Kähler manifold is a holomorphic symplectic Morita double bimodule between double symplectic groupoids, and that each compatible Riemannian metric is given by a Lagrangian submanifold forming a bisection of the real symplectic core of this double bimodule. In other words, a generalized Kähler manifold has an associated holomorphic symplectic manifold of quadruple dimension and equipped with an anti-holomorphic involution; the generalized Kähler metric is then determined by the choice of a Lagrangian submanifold of the fixed point locus of this involution. This resolves affirmatively a long-standing conjecture by physicists concerning the existence of a generalized Kähler potential. We demonstrate the theory by constructing explicitly the above Morita double bimodule and Lagrangian bisection for the well-known generalized Kähler structures on compact even-dimensional semisimple Lie groups, which have until now escaped such analysis. We construct the required holomorphic symplectic manifolds by expressing them as moduli spaces of flat connections on surfaces with Lagrangian boundary conditions, through a quasi-Hamiltonian reduction.

Symplectic double groupoids and the generalized Kähler potential

Abstract

A description of the fundamental degrees of freedom underlying generalized Kähler geometry, which separates its holomorphic moduli from its compatible Riemannian metric in a similar way to the Kähler case, has been sought since its discovery in 1984. In this paper, we describe a full solution to this problem for arbitrary generalized Kähler manifolds. We discover that the holomorphic structure underlying a generalized Kähler manifold is a holomorphic symplectic Morita double bimodule between double symplectic groupoids, and that each compatible Riemannian metric is given by a Lagrangian submanifold forming a bisection of the real symplectic core of this double bimodule. In other words, a generalized Kähler manifold has an associated holomorphic symplectic manifold of quadruple dimension and equipped with an anti-holomorphic involution; the generalized Kähler metric is then determined by the choice of a Lagrangian submanifold of the fixed point locus of this involution. This resolves affirmatively a long-standing conjecture by physicists concerning the existence of a generalized Kähler potential. We demonstrate the theory by constructing explicitly the above Morita double bimodule and Lagrangian bisection for the well-known generalized Kähler structures on compact even-dimensional semisimple Lie groups, which have until now escaped such analysis. We construct the required holomorphic symplectic manifolds by expressing them as moduli spaces of flat connections on surfaces with Lagrangian boundary conditions, through a quasi-Hamiltonian reduction.
Paper Structure (36 sections, 30 theorems, 197 equations, 12 figures)

This paper contains 36 sections, 30 theorems, 197 equations, 12 figures.

Table of Contents

  1. Summary of the main Theorem
  2. Consequences of the main Theorem
  3. Generalized Kähler structures on Lie groups
  4. Outline of the paper
  5. Generalized Kähler geometry and holomorphic Manin triples
  6. Holomorphic Courant algebroids and matched pairs
  7. Manin triples from generalized Kähler structures
  8. Gauge equivalence of Manin triples
  9. Generalized Kähler structures from Manin triples
  10. Symplectic double groupoids and higher Morita equivalences
  11. Quasi-symplectic groupoids
  12. Poisson groupoids
  13. Symplectic double groupoids
  14. Double Lie groupoids and their generalized morphisms
  15. Symplectic Morita equivalences and multiplicative bisections
  16. ...and 21 more sections

Key Result

Proposition 1.3

Let $\mathcal{E}$ be a holomorphic exact Courant algebroid. An involutive isotropic splitting of its imaginary part $\mathcal{E}_I$ induces an isomorphism between its real part $\mathcal{E}_R$ and $(\mathbb{T} M, H)$, where $H=d^c\omega$, for a unique pluriclosed real $(1,1)$-form $\omega$.

Figures (12)

  • Figure 1: A holomorphic symplectic Morita 2-equivalence
  • Figure 2: Lagrangian bisections $\mathcal{L}$ of $\mathcal{S}$ correspond to generalized Kähler metrics
  • Figure 3: Matched pair functor from holomorphic to smooth Courant algebroids
  • Figure 4: Gauge equivalences between matched pairs of $\mathcal{E}_\pm$
  • Figure 5: An element $d\in D$
  • ...and 7 more figures

Theorems & Definitions (99)

  • Definition 1.1
  • Example 1.2
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • Lemma 1.5
  • proof
  • Theorem 1.6
  • proof
  • Theorem 1.7
  • ...and 89 more