Quantization of Holomorphic Symplectic Manifolds: Analytic Continuation of Path Integrals and Coherent States
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910778450247680 |
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| author | Lackman, Joshua |
| author_facet | Lackman, Joshua |
| contents | 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^*$$\unicode{x2013}$algebra to the Poisson algebra of the associated hyperkähler manifold. Related to this, we show that the cotangent bundles of Grassmanians have commuting almost complex structures that are compatible with a holomorphic symplectic form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05428 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantization of Holomorphic Symplectic Manifolds: Analytic Continuation of Path Integrals and Coherent States Lackman, Joshua Symplectic Geometry High Energy Physics - Theory Mathematical Physics Algebraic Geometry Quantum Algebra We extend Berezin's quantization $q:M\to\mathbb{P}\mathcal{H}$ to holomorphic symplectic manifolds, which involves replacing the state space $\mathbb{P}\mathcal{H}$ with its complexification $\text{T}^*\mathbb{P}\mathcal{H}.$ We show that this is equivalent to replacing rank$\unicode{x2013}$1 Hermitian projections with all rank$\unicode{x2013}$1 projections. We furthermore allow the states to be points in the cotangent bundle of a Grassmanian. We also define a holomorphic path integral quantization as a certain idempotent in a convolution algebra and we prove that these two quantizations are equivalent. For each $n>0,$ we construct a faithful functor from the category of finite dimensional $C^*$$\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. |
| title | Quantization of Holomorphic Symplectic Manifolds: Analytic Continuation of Path Integrals and Coherent States |
| topic | Symplectic Geometry High Energy Physics - Theory Mathematical Physics Algebraic Geometry Quantum Algebra |
| url | https://arxiv.org/abs/2501.05428 |