nLab gauge transformation

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 $G$-principal bundles (over spacetime $X$) with connection over a Lie group $G$ called the gauge group. A gauge transformation in the strict and original sense of the word is a morphism in the groupoid $G Bund_\nabla(X)$ 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 $(A_i \in \Omega^1(U_i \subset X, \mathfrak{g}))$ – is the gauge field . The equation

$$A'_i = h_i^{-1} A_i h_i + h_i^{-1} d h_i \;\;\;\; ; \;\; h_i \in C^\infty(U_i,G)$$

that characterizes morphisms $A_i \stackrel{g}{\to} A'_i$ 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 $B_i \in \Omega^2(U_i \subset X)$. A 1-morphism in there is a first order gauge transformation $B_i \stackrel{\lambda}{\to} B'_i$ characterized by the equation $B_i' = B_i + d_{dR} \lambda_i$. A 2-morphism in there is a second order gauge transformation $\lambda_i \stackrel{g_i}{\Rightarrow} \lambda'_i$ characterized by $\lambda'_i = \lambda_i + d_{dR} g_i$.

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 infinitesimal 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

• groupoid of Lie-algebra valued 1-forms

• 2-groupoid of Lie 2-algebra valued forms

• 3-groupoid of Lie 3-algebra valued forms.

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

• ∞-groupoid of ∞-Lie-algebra valued forms.

Examples

In electromagnetism

(…)

In Yang-Mills theory

(…)

In gravity and topological field theories

• general covariance

Related concepts

• gauge

• gauge parameter

• gauge group, gauge fixing

• large gauge transformation

• higher gauge transformation

• gauge invariance

• group averaging

• Haag–Łopuszański–Sohnius theorem on symmetries of an S-matrix

• spontaneously broken symmetry

• principle of equivalence

• general covariance

• generalized global symmetry

gauge field: models and components

physics	differential geometry	differential cohomology
gauge field	connection on a bundle	cocycle in differential cohomology
instanton/charge sector	principal bundle	cocycle in underlying cohomology
gauge potential	local connection differential form	local connection differential form
field strength	curvature	underlying cocycle in de Rham cohomology
gauge transformation	equivalence	coboundary
minimal coupling	covariant derivative	twisted cohomology
BRST complex	Lie algebroid of moduli stack	Lie algebroid of moduli stack
extended Lagrangian	universal Chern-Simons n-bundle	universal characteristic map