Symplectic double groupoids and the generalized K\"ahler potential

A description of the fundamental degrees of freedom underlying generalized K\"ahler geometry, which separates its holomorphic moduli from its compatible Riemannian metric in a similar way to the K\"ahler case, has been sought since its discovery in 1984. In this paper, we describe a full solution to this problem for arbitrary generalized K\"ahler manifolds. We discover that the holomorphic structure underlying a generalized K\"ahler 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 K\"ahler manifold has an associated holomorphic symplectic manifold of quadruple dimension and equipped with an anti-holomorphic involution; the generalized K\"ahler 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 K\"ahler potential. We demonstrate the theory by constructing explicitly the above Morita double bimodule and Lagrangian bisection for the well-known generalized K\"ahler 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.