Implicit representations of codimension-2 submanifolds and their prequantum structure

TL;DR

This paper introduces a novel implicit representation for codimension-2 submanifolds using complex functions, revealing a prequantum bundle structure over their symplectic space, with implications for geometric quantization.

Contribution

It demonstrates that the space of implicit submanifold representations admits a prequantum bundle structure, linking symplectic geometry with complex function representations.

Findings

The space of implicit representations has a prequantum bundle structure.

The Marsden-Weinstein symplectic form is interpreted as the curvature of a connection.

The geometric interpretation involves the phase level sets of complex functions.

Abstract

This paper explores the geometry of the space of codimension-2 submanifolds. We implicitly represent these submanifolds by a class of complex-valued functions. We show that the space of all these implicit representations admits a prequantum bundle structure over the space of submanifolds, equipped with the well-known Marsden-Weinstein symplectic structure. This bundle allows a new geometric interpretation of the Marsden-Weinstein structure as the curvature of a connection form, which measures the average of volumes swept by the deformation of the S^1-family of hypersurfaces, defined as the phase level sets of the complex function implicitly representing a submanifold.