Distribution of the Four Fundamental Forces in the Fiber‑Bundle Model

Author: Zhang Suhang, Luoyang, Henan

Abstract

Based on the established correspondence between gauge fields and fiber‑bundle theory, this paper presents a hierarchical classification for the geometric carriers of the four fundamental interactions. The proposition is as follows: the topology of gravity is rooted in the base manifold, while the topologies of electromagnetism, weak and strong interactions are rooted in fiber bundles. Gravity occupies the base, and gauge forces reside in the fibers. This classification is an inductive collation of the geometric structures of the four forces, rather than a dynamical unification programme; it does not require a unified field equation. Within this framework, each of the four forces possesses its own structural conserved quantity. Calculations are performed separately, while unification is achieved at the structural level.

Keywords: fiber bundle; gauge field; gravity; structural conservation; base manifold; topological invariant

1 Introduction

The correspondence between gauge fields and fiber‑bundle theory was established by Wu Tai‑Tsun and Yang Chen‑Ning in 1975. This work uncovered the mathematical skeleton of gauge fields: a gauge field is a connection on a principal fiber bundle, and the field strength is the curvature of this connection.

After this correspondence was established, a natural geometric question arises:
Where do the four fundamental interactions reside within the language of fiber‑bundle theory?

This paper proposes a classification:

- Gravity: rooted in the base manifold;
- Electromagnetism, weak interaction, strong interaction: rooted in fiber bundles.

This classification is not a dynamical unification programme, and does not aim to construct a unified field equation. It is an inductive collation at the level of geometric structure.

2 Gauge Fields and Fiber Bundles

A principal fiber bundle is written as:
P \to M
where M is the base manifold and P is the principal fiber bundle, with fibers modelled by the gauge group G.

The gauge field A_\mu is a connection on P. The field strength F_{\mu\nu} is the curvature of the connection.

- Electromagnetic force: G=\mathrm{U}(1)
- Weak nuclear force: G=\mathrm{SU}(2)
- Strong nuclear force: G=\mathrm{SU}(3)

Gauge transformations act on the fiber degrees of freedom, while the base manifold M remains unchanged.

Geometric location of gauge forces: connections on fiber bundles.

3 Gravity and the Base Manifold

The connection of general relativity is the Levi‑Civita connection.
It is defined on the spacetime manifold itself, not on an extra fiber.

The gravitational field is described by the metric g_{\mu\nu}. The metric‑compatible Levi‑Civita connection is an intrinsic connection on the spacetime base manifold.

Diffeomorphism transformations move points on the base manifold directly.

Geometric location of gravity: intrinsic geometry of the base manifold itself.

4 Essential Differences Between the Two Types of Connections

Gauge‑field connections and gravitational connections bear formal similarities, yet differ in their carrier.

Gauge field Gravity
Connection defined on Fiber bundle Base manifold
Transformation acts on Fiber degrees of freedom Points of the base manifold
Base manifold Fixed background Object being transformed
Gauge group   Diffeomorphism group
Curvature Field strength   Riemann tensor  

Yang Chen‑Ning observed in 1969 that the algebraic structures of the Riemann tensor and the Yang‑Mills field‑strength tensor are analogous.

However, similarity does not imply equivalence:

- The Yang‑Mills connection is defined on an additional internal fiber;
- The Levi‑Civita connection is defined on the spacetime manifold itself.

This distinction forms the basis of the classification proposed in this paper.

5 Geometric Distribution of the Four Forces

Within the above framework, the geometric carrier hierarchy for the four forces is given below.

Force Geometric location Structural conserved quantity
Gravity Base manifold Spacetime topological invariants
Electromagnetism Fiber bundle   Chern class
Weak interaction Fiber bundle   instanton number
Strong interaction Fiber bundle   Chern‑Simons number

Gravity resides on the base, and gauge forces reside in the fibers.
This is not a mechanical separation, but a distinction of geometric carrier hierarchy.

6 Structural Conservation and Division of Computation

The four forces are not unified at the metric level, but can be unified at the structural level.

Unification at the structural level: all four forces are physical realizations of structural conservation.

- Gravity = structural conservation of the base manifold
- Gauge forces = structural conservation of fiber bundles

Division at the computational level:

- Gravity is calculated via general relativity;
- Electromagnetism, weak and strong interactions are calculated via Yang‑Mills equations.

The two computational frameworks each govern their own domain and are valid within their respective scopes.
Unification is achieved at the structural level, while division holds at the computational level.

7 Discussion: Relation to the Wu‑Yang Work

The 1975 work by Wu Tai‑Tsun and Yang Chen‑Ning established the correspondence between gauge fields and principal fiber‑bundle connections.

At that time, the mainstream research direction was to incorporate gravity into the gauge‑field framework, treating the Lorentz group or diffeomorphism group as a gauge group. The thinking was oriented toward merging, rather than classification.

Yang Chen‑Ning was aware that the gravitational connection is the intrinsic Riemannian connection of the spacetime base manifold. Nevertheless, constrained by the research objectives of the era, he did not distill the proposition that “the topology of gravity is rooted in the base manifold, whereas the topologies of electromagnetism, weak and strong interactions are rooted in fiber bundles” as a geometric classification for the four fundamental interactions.

The classification presented in this paper is an inductive sorting of the geometric carrier hierarchy of the four interactions, built upon the Wu‑Yang correspondence.
It addresses a question of geometric structural position, rather than dynamical problems.

8 Conclusions

1. The topology of gravity is rooted in the base manifold; the topologies of electromagnetism, weak and strong interactions are rooted in fiber bundles.
2. Gravity resides on the base, and gauge forces reside in the fibers. This classification is an inductive collation built upon the Wu‑Yang correspondence.
3. The four forces are unified at the structural level: they are all physical realizations of structural conservation.
4. The four forces are divided at the computational level: gravity is described by general relativity, and gauge forces by Yang‑Mills equations.
5. This paper does not attempt to derive a unified field equation; it merely provides a classification of geometric carrier hierarchy.

References

Omitted