Editing Wu-Yang Dictionary (section)

== Mathematical aspects ==

At its core, the Wu–Yang dictionary equates physical concepts from gauge theory with geometric ones from fiber bundle theory. Fiber bundles are mathematical structures that describe how a "total space" (like a global manifold) is built from local "fibers" attached to a base space, similar to how gauge fields describe local symmetries in physics.

A simplified mapping includes:

{| class="wikitable"
|-
! Gauge Theory (Physics) Term
! Fiber Bundle Theory (Math) Equivalent
|-
| Gauge (or global gauge)
| Principal coordinate bundle
|-
| Gauge type
| Principal fiber bundle
|-
| Gauge potential \( b_{\mu}^{k} \)
| Connection on principal fiber bundle
|-
| Field strength \( f_{\mu\nu}^{k} \)
| Curvature
|-
| Phase factor \( \Phi_{QP} \)
| Parallel displacement
|-
| Gauge transformation
| Change of bundle coordinates
|-
| Gauge group
| Structure group
|-
| Electromagnetism
| Connection on a U(1) bundle
|-
| Isotopic spin gauge field
| Connection on an SU(2) bundle
|-
| Dirac's monopole quantization
| Classification of U(1) bundles by first Chern class
|-
| Electromagnetism without monopole
| Connection on a trivial U(1) bundle
|-
| Electromagnetism with monopole
| Connection on a nontrivial U(1) bundle
|}

In the original 1975 paper, some entries (like the mathematical equivalent of an electric current source) were left blank, as physicists understood them intuitively but mathematicians needed further development—this gap spurred later work, such as applying the Chern–Weil theorem to gauge fields. The dictionary highlights how gauge potentials correspond to connections (rules for parallel transport), and field strengths to curvatures (measuring how much the connection "twists").