Quantization of Holomorphic Symplectic Manifolds: Analytic Continuation of Path Integrals and Coherent States
Joshua Lackman
TL;DR
The paper addresses quantization on holomorphic symplectic manifolds by extending Berezin’s framework to allow the target to be $T^*\mathbb{P}\mathcal{H}$ and to Grassmannians, and by formulating a holomorphic path integral quantization that yields an idempotent in a convolution algebra. It develops the geometry of $T^*\mathbf{G}_n\mathcal{H}$ with commuting integrable almost complex structures $I,J$ and a holomorphic symplectic form $\Omega$, establishing a Hermitian, polarization-rich setting that supports both pointwise and path-integral quantizations. The work shows an equivalence between holomorphic Berezin quantization and holomorphic path-integral quantization, and constructs a functor from finite-dimensional $C^*$-algebras to hyperkähler manifolds, linking quantum commutators to Poisson brackets on these geometric spaces. It also provides explicit quantization constructions on $T^*\mathbf{G}_n\mathcal{H}$ and discusses the role of LS-submanifolds in defining integration cycles, along with concrete examples tying the formalism to well-known quantization schemes such as Toeplitz quantization. The results collectively broaden the bridge between operator-algebraic quantization and complex- and hyperkähler-geometry, with potential implications for A-model perspectives and noncommutative geometry.
Abstract
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^*$$\unicode{x2013}$algebras to to the category of hyperkähler manifolds and we show that our quantization recovers the original $C^*$$\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 hyperkähler 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.