Gauge & Field-Theoretic Structures

← Back to Domain-Specific Families

Abstractions about geometric structures underlying gauge and field theories — connections and curvature such as the Bismut connection and the Yang–Mills equations, dualities between coupling regimes like Montonen–Olive duality, and models unifying or classifying fields such as the non-linear sigma model and unified field theory.

8 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Bismut Connection — The unique connection on a Hermitian manifold that preserves both the metric and complex structure while having totally skew-symmetric torsion, reducing to Levi–Civita in the Kähler case.
• Classification of Electromagnetic Fields — A pointwise relativistic taxonomy that uses the two Lorentz scalar invariants of the electromagnetic field tensor to distinguish null and non-null fields and determine which electric–magnetic simplifications some observer can realize.
• Etherington's Reciprocity Theorem — A metric-null-geodesic light-bundle reciprocity between source and observer area distances that yields luminosity–angular-distance duality when photons are conserved.
• Montonen–Olive Duality — A strong–weak equivalence conjecture in supersymmetric gauge theory that exchanges electric gauge particles with magnetic monopoles, coupling with its reciprocal, and Noether with topological charge.
• Non-Linear Sigma Model — Represent fields as maps from spacetime or another base into a curved target manifold, with dynamics governed by the pullback of the target metric and any explicitly added potential or topological terms.
• Stable Yang–Mills–Higgs Pair — A critical gauge–Higgs configuration whose specified Yang–Mills–Higgs functional has no negative second-variation direction in the admissible perturbation space after gauge redundancy is removed.
• Unified field theory — Seek a single field-theoretic structure whose fields, symmetries, or geometric components recover interactions that otherwise appear as separate fundamental forces.
• Yang–Mills Equations — The Yang–Mills equations require a connection's curvature to have zero gauge-covariant divergence, making the connection stationary for the Yang–Mills action.