A Simplicial Approach to Higher Geometric Quantization

TL;DR

This paper introduces a unified, higher categorical framework for observables in n-plectic geometry, extending algebraic structures and providing a systematic approach to geometric quantization of extended objects.

Contribution

It develops a semi-simplicial n-groupoid model for observables, extracts cohomological invariants, and constructs a categorified pre-n-Hilbert space, advancing higher geometric quantization.

Findings

Proves the semi-simplicial set satisfies the Kan filling property.

Constructs a recursive inner product for categorified pre-n-Hilbert space.

Establishes a quantization scheme aligned with polarization classifications.

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.