Morita equivalence and the generalized K\"ahler potential

We solve the problem of determining the fundamental degrees of freedom underlying a generalized K\"ahler structure of symplectic type. For a usual K\"ahler structure, it is well-known that the geometry is determined by a complex structure, a K\"ahler class, and the choice of a positive $(1,1)$-form in this class, which depends locally on only a single real-valued function: the K\"ahler potential. Such a description for generalized K\"ahler geometry has been sought since it was discovered in 1984. We show that a generalized K\"ahler structure of symplectic type is determined by a pair of holomorphic Poisson manifolds, a holomorphic symplectic Morita equivalence between them, and the choice of a positive Lagrangian brane bisection, which depends locally on only a single real-valued function, which we call the generalized K\"ahler potential. Our solution draws upon, and specializes to, the many results in the physics literature which solve the problem under the assumption (which we do not make) that the Poisson structures involved have constant rank. To solve the problem we make use of, and generalize, two main tools: the first is the notion of symplectic Morita equivalence, developed by Weinstein and Xu to study Poisson manifolds; the second is Donaldson's interpretation of a K\"ahler metric as a real Lagrangian submanifold in a deformation of the holomorphic cotangent bundle.