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arXiv · 2605.10695

A Simplicial Approach to Higher Geometric Quantization

Qian Zhang
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Abstract

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.

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Qian Zhang. 2026-05-11. A Simplicial Approach to Higher Geometric Quantization. https://arxiv.org/abs/2605.10695

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