nLab
field strength

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Differential cohomology

Contents

Idea

In gauge theory cocycles in differential cohomology model gauge fields.

By definition, every differential cohomology theory Γ ()\Gamma^\bullet(-) comes with a characteristic curvature form morphism

F:Γ¯ (X)Ω (X)π *Γ, F : \bar \Gamma^\bullet(X) \to \Omega^\bullet(X)\otimes \pi_*\Gamma \otimes \mathbb{R} \,,

the (generalized) Chern character.

For cΓ (X)c \in \Gamma^\bullet(X) a cocycle representing a gauge field in gauge theory, its image F(c)Ω (X)F(c) \in \Omega^\bullet(X) is the field strength of the gauge field. If we think of this cocycle as being (a generalization of) a connection on a bundle, this is essentially the curvature of that connection.

Often gauge fields are named after their field strength. For instance the field strength of the electromagnetic field is the 22-form FΩ 2(X)F \in \Omega^2(X) whose components are the electric and the magnetic fields.

Examples

gauge field: models and components

physicsdifferential geometrydifferential cohomology
gauge fieldconnection on a bundlecocycle in differential cohomology
instanton/charge sectorprincipal bundlecocycle in underlying cohomology
gauge potentiallocal connection differential formlocal connection differential form
field strengthcurvatureunderlying cocycle in de Rham cohomology
gauge transformationequivalencecoboundary
minimal couplingcovariant derivativetwisted cohomology
BRST complexLie algebroid of moduli stackLie algebroid of moduli stack
extended Lagrangianuniversal Chern-Simons n-bundleuniversal characteristic map

Revised on January 7, 2013 21:53:42 by Urs Schreiber (89.204.154.29)