Quantization of Holomorphic Symplectic Manifolds: Analytic Continuation of Path Integrals and Coherent States

We extend Berezin's quantization $q:M\to\mathbb{P}\mathcal{H}$ to holomorphic symplectic manifolds, which involves replacing the state space $\mathbb{P}\mathcal{H}$ with its complexification $\text{T}^*\mathbb{P}\mathcal{H}.$ We show that this is equivalent to replacing rank$\unicode{x2013}$1 Hermitian projections with all rank$\unicode{x2013}$1 projections. We furthermore allow the states to be points in the cotangent bundle of a Grassmanian. We also define a holomorphic path integral quantization as a certain idempotent in a convolution algebra and we prove that these two quantizations are equivalent. For each $n>0,$ we construct a faithful functor from the category of finite dimensional $C^*$$\unicode{x2013}$algebras to to the category of hyperk\"{a}hler manifolds and we show that our quantization recovers the original $C^*$$\unicode{x2013}$algebra. In particular, this functor comes with a homomorphism from the commutator algebra of the $C^*$$\unicode{x2013}$algebra to the Poisson algebra of the associated hyperk\"{a}hler manifold. Related to this, we show that the cotangent bundles of Grassmanians have commuting almost complex structures that are compatible with a holomorphic symplectic form.