nLab
gauge transformation

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Equality and Equivalence

\infty-Chern-Weil theory

Contents

Idea

Local gauge transformations

In physics the term local gauge transformation or gauge equivalence means essentially isomorphism or rather equivalence in an (infinity,1)-category: the configuration space of a physical theory is typically a groupoid (an orbifold) and a gauge transformation between configurations is a morphism in this groupoid.

More specifically, in physical theories called gauge theory – such as describing electromagnetism and Yang-Mills fields – the configuration space is a space of GG-principal bundles (over spacetime XX) with connection over a Lie group GG called the gauge group. A gauge transformation in the strict and original sense of the word is a morphism in the groupoid GBund (X)G Bund_\nabla(X) of these bundles with connection.

In this context the connection \nabla itself – usually thought of in terms of its local Lie-algebra valued 1-forms (A iΩ 1(U iX,𝔤))(A_i \in \Omega^1(U_i \subset X, \mathfrak{g})) – is the gauge field . The equation

A i=h i 1A ih i+h 1dh i;h iC (U i,G) A'_i = h_i^{-1} A_i h_i + h^{-1} d h_i \;\;\;\; ; \;\; h_i \in C^\infty(U_i,G)

that characterizes morphisms A igA iA_i \stackrel{g}{\to} A'_i in the groupoid of Lie-algebra valued forms is historically the hallmark of the “gauge principle” and is often what is meant specifically by gauge transformation .

But from there on the terminology generalizes to almost all physical theories. For one this is because the configuration space of many theories may be thought of as spaces of bundles with connections, notably also for gravity.

Moreover, many formal physical theories such as Chern-Simons theory, supergravity, etc. are described by higher categorical generalizations of bundles with connections: principal ∞-bundles with connection on a principal ∞-bundle such as the Chern-Simons circle 3-bundle. Their configuration spaces form not just groupoids but ∞-groupoids. The higher morphisms in these are called higher gauge transformations.

For instance the configuration space of the Kalb-Ramond field is the 2-groupoid of circle 2-bundles with connection over spacetime. An object in there is locally given by a 2-form B iΩ 2(U iX)B_i \in \Omega^2(U_i \subset X). A 1-morphism in there is a first order gauge transformation B iλB iB_i \stackrel{\lambda}{\to} B'_i characterized by the equation B i=B i+d dRλ iB_i' = B_i + d_{dR} \lambda_i. A 2-morphism in there is a second order gauge transformation λ ig iλ i\lambda_i \stackrel{g_i}{\Rightarrow} \lambda'_i characterized by λ i=λ i+d dRg i\lambda'_i = \lambda_i + d_{dR} g_i.

The Lie algebroid of the groupoid of configurations and gauge transformations is known in physics in terms of its dual Chevalley-Eilenberg algebra called the BRST-complex. The degree 1 generators in this dg-algebra are hence the functions on infinitesimal gauge transformations. (A discussion of such infinitesimal transformations is here.) These graded functions on infintesimal gauge transformations are called ghost fields or ghosts for short, in the physics literature.

If the space of configurations is not just a groupoid in ordinary spaces but a groupoid in derived spaces such as derived smooth manifolds, then the CE-algebra of the corresponding derived ∞-Lie algebroid is called the BV-BRST complex.

Global gauge transformations

See global gauge group.

Details

For local connection forms

The formulas for the local gauge transformations and higher gauge transformations for connections on bundles, connections on 2-bundles and connections on 3-bundles are discussed, respectively, at

The fully general description for connections on ∞-bundles is at

Examples

In electromagnetism

(…)

In Yang-Mills theory

(…)

In gravity and topological field theories

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 August 25, 2014 01:05:11 by Urs Schreiber (82.113.98.69)