Symplectic double groupoids and the generalized K\"ahler potential

A description of the fundamental degrees of freedom underlying a generalized K\"ahler manifold, which separates its holomorphic moduli from the space of compatible metrics 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, which involves the new concept of a holomorphic symplectic Morita 2-equivalence between double symplectic groupoids, equipped with a Lagrangian bisection of its real symplectic core. Essentially, any generalized K\"ahler manifold has an associated holomorphic symplectic manifold of quadruple dimension and equipped with an anti-holomorphic involution; the metric is determined by a Lagrangian submanifold of its fixed point locus. This finally 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 2-equivalence 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 decorated boundary, through a quasi-Hamiltonian reduction.