A Simplicial Approach to Higher Geometric Quantization
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917480914485248 |
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| author | Zhang, Qian |
| author_facet | Zhang, Qian |
| contents | This paper develops a unified framework for observables in n-plectic geometry, extending the L_infty-algebra of Hamiltonian (n-1)-forms to Hamiltonian forms of all degrees via a degree-shifting Grassmann variable u that encodes submanifold codimension. Interpreting k-form observables as k-dimensional topological defects yields a recursive gluing construction that assembles into a semi-simplicial set sOb_bullet(M), which we prove satisfies the Kan filling property, thereby providing an n-groupoid model for observables. From this semi-simplicial perspective we extract cohomological invariants and construct a recursive inner product leading to a categorified pre-n-Hilbert space. The hierarchical structure of polarizations yields a natural quantization scheme matching the 1-polarization classification of multisymplectic geometry. The resulting framework bridges higher algebraic structures with higher categorical geometry and establishes a systematic foundation for the geometric quantization of extended objects. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10695 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Simplicial Approach to Higher Geometric Quantization Zhang, Qian Mathematical Physics Category Theory Differential Geometry This paper develops a unified framework for observables in n-plectic geometry, extending the L_infty-algebra of Hamiltonian (n-1)-forms to Hamiltonian forms of all degrees via a degree-shifting Grassmann variable u that encodes submanifold codimension. Interpreting k-form observables as k-dimensional topological defects yields a recursive gluing construction that assembles into a semi-simplicial set sOb_bullet(M), which we prove satisfies the Kan filling property, thereby providing an n-groupoid model for observables. From this semi-simplicial perspective we extract cohomological invariants and construct a recursive inner product leading to a categorified pre-n-Hilbert space. The hierarchical structure of polarizations yields a natural quantization scheme matching the 1-polarization classification of multisymplectic geometry. The resulting framework bridges higher algebraic structures with higher categorical geometry and establishes a systematic foundation for the geometric quantization of extended objects. |
| title | A Simplicial Approach to Higher Geometric Quantization |
| topic | Mathematical Physics Category Theory Differential Geometry |
| url | https://arxiv.org/abs/2605.10695 |