# Disk-sphere field duality theorem

## TL;DR

This paper introduces a new duality theorem linking fields on disks and spheres, supported by classical topological theorems, to analyze boundary-related properties of symmetry fields.

## Contribution

It formulates the disk-sphere field duality theorem, connecting boundary behavior with topological properties using Poincaré-Hopf and boundary number theorems.

## Key findings

- Disk-sphere duality relates boundary behavior to topological field properties.
- The theorem provides a new perspective for analyzing symmetry fields on disks and spheres.
- Boundary effects are crucial in topological field analysis.

## Abstract

This paper presents a new reformulated theorem for fields embedded on a sphere or a disk. We focus in particular on the associated sphere of a disk when closing its only one boundary. We call this the disk-sphere duality theorem for the study of fields topological properties. For that purpose, we use the Poincar{\'e}-Hopf theorem and the boundary number theorem to firmly support our developments. In this context, the state of a $n$-symmetry direction field will be analyzed to show that disk-sphere field duality is closely related to the behavior near the disk's boundary.