From the Hopf Fibration to Instantons: Geometry in Gauge Theory

Author

• Emily D. Wessman, Utah State University

Date of Award

5-2026

Degree Type

Thesis

Degree Name

Departmental Honors

Department

Mathematics and Statistics

Abstract

This paper explores the relationship between topology, differential geometry, and gauge theory through the study of Yang-Mills theory and its solutions, known as instantons. Beginning with the Hopf fibration, we show how principal fiber bundles encode topological information and appear in physical contexts such as electromagnetism. In particular, we consider how the fibration of S³ over CP¹ ≅ S² represents the Dirac magnetic monopole, and how the Chern number associated with the bundle is exactly the winding number for the monopole.

We then develop the framework of gauge theory, focusing on connections on principal SU(2) bundles over 4-manifolds. We introduce the Yang-Mills equations as a set of partial differential equations for a connection A on a principal bundle over a smooth 4-manifold. The solutions to these equations are instantons, and we examine the quaternionic Hopf fibration SU(2) → S⁷ → S⁴ as a canonical example of such an instanton in reference to the famous BPST solution over R⁴.

We next reduce the anti-self-dual condition on curvature of a Kähler surface to the Hermitian Yang-Mills equations. The equivalence highlights how we can use complex geometry to simplify gauge-theoretic structures, where curvature decomposes into its types and holomorphicity conditions naturally arise. Extending this, we consider product manifolds and use the process of dimensional reduction through the ASD equations to derive the Hitchin system. With this system, components of the curvature corresponding to invariance on one of the factors become Higgs fields, providing a lower-dimensional structure of the original gauge theory to work with.

Finally, we discuss other approaches to instanton theory, including the Nahm transform and ADHM construction. We show that through the ADHM construction we can construct certain matrix-valued operators and maps, then assemble a matrix equation that corresponds with instanton solutions on R⁴ . We describe the Nahm transform from the perspective of elliptic curves and the Fourier-Mukai transform. Throughout the paper, we emphasize how geometric structures encode physical phenomena, illustrating the important role that modern geometry plays in theoretical physics.

Recommended Citation

Wessman, Emily D., "From the Hopf Fibration to Instantons: Geometry in Gauge Theory" (2026). Undergraduate Honors Capstone Projects. 1086.
https://digitalcommons.usu.edu/honors/1086