In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and especially
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
and
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
, gauge theory is the general study of
connections on
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
s,
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
s, and
fibre bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E an ...
s. Gauge theory in mathematics should not be confused with the closely related concept of a
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie grou ...
in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, which is a
field theory which admits
gauge symmetry
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
. In mathematics ''theory'' means a
mathematical theory
A mathematical theory is a mathematical model of a branch of mathematics that is based on a set of axioms
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reason ...
, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
of some natural phenomenon.
Gauge theory in mathematics is typically concerned with the study of gauge-theoretic equations. These are
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s involving connections on vector bundles or principal bundles, or involving sections of vector bundles, and so there are strong links between gauge theory and
geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of ...
. These equations are often physically meaningful, corresponding to important concepts in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
or
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
, but also have important mathematical significance. For example, the
Yang–Mills equations
In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the E ...
are a system of
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
for a connection on a principal bundle, and in physics solutions to these equations correspond to
vacuum solutions to the equations of motion for a
classical field theory
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
, particles known as
instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
s.
Gauge theory has found uses in constructing new
invariants of
smooth manifolds, the construction of exotic geometric structures such as
hyperkähler manifolds, as well as giving alternative descriptions of important structures in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
such as
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
s of vector bundles and
coherent sheaves.
History
Gauge theory has its origins as far back as the formulation of
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
describing classical electromagnetism, which may be phrased as a gauge theory with structure group the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
. Work of
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
on
magnetic monopole
In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...
s and relativistic
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
encouraged the idea that bundles and connections were the correct way of phrasing many problems in quantum mechanics. Gauge theory in mathematical physics arose as a significant field of study with the seminal work of
Robert Mills and
Chen-Ning Yang
Yang Chen-Ning or Chen-Ning Yang (; born 1 October 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese theoretical physicist who made significant contributions to statistical mechanics, integrable systems, gauge t ...
on so-called Yang–Mills gauge theory, which is now the fundamental model that underpins the
standard model of particle physics.
The mathematical investigation of gauge theory has its origins in the work of
Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...
,
Isadore Singer, and
Nigel Hitchin
Nigel James Hitchin FRS (born 2 August 1946) is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University ...
on the self-duality equations on a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
in four dimensions. In this work the moduli space of self-dual connections (instantons) on Euclidean space was studied, and shown to be of dimension
where
is a positive integer parameter. This linked up with the discovery by physicists of
BPST instantons, vacuum solutions to the Yang–Mills equations in four dimensions with
. Such instantons are defined by a choice of 5 parameters, the center
and scale
, corresponding to the
-dimensional moduli space. A BPST instanton is depicted to the right.
Around the same time Atiyah and
Richard Ward discovered links between solutions to the self-duality equations and algebraic bundles over the
complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
. Another significant early discovery was the development of the
ADHM construction
In mathematical physics and gauge theory, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Cons ...
by Atiyah,
Vladimir Drinfeld, Hitchin, and
Yuri Manin. This construction allowed for the solution to the anti-self-duality equations on Euclidean space
from purely linear algebraic data.
Significant breakthroughs encouraging the development of mathematical gauge theory occurred in the early 1980s. At this time the important work of Atiyah and
Raoul Bott about the Yang–Mills equations over Riemann surfaces showed that gauge theoretic problems could give rise to interesting geometric structures, spurring the development of infinite-dimensional
moment map In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the act ...
s, equivariant
Morse theory, and relations between gauge theory and algebraic geometry.
[Atiyah, M.F. and Bott, R., 1983. The yang-mills equations over riemann surfaces. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 308(1505), pp. 523–615.] Important analytical tools in
geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of ...
were developed at this time by
Karen Uhlenbeck
Karen Keskulla Uhlenbeck (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W. Richard ...
, who studied the analytical properties of connections and curvature proving important compactness results. The most significant advancements in the field occurred due to the work of
Simon Donaldson
Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. H ...
and
Edward Witten
Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...
.
Donaldson used a combination of algebraic geometry and geometric analysis techniques to construct new
invariants of
four manifolds, now known as
Donaldson invariant
In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricting ...
s.
[Donaldson, S.K., 1983. An application of gauge theory to four-dimensional topology. Journal of Differential Geometry, 18(2), pp. 279–315.][Donaldson, S.K., 1990. Polynomial invariants for smooth four-manifolds. Topology, 29(3), pp. 257–315.] With these invariants, novel results such as the existence of topological manifolds admitting no smooth structures, or the existence of many distinct smooth structures on the Euclidean space
could be proved. For this work Donaldson was awarded the
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...
in 1986.
Witten similarly observed the power of gauge theory to describe topological invariants, by relating quantities arising from
Chern–Simons theory
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and Jam ...
in three dimensions to the
Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomi ...
, an invariant of
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ...
s.
[Witten, E., 1989. Quantum field theory and the Jones polynomial. Communications in Mathematical Physics, 121(3), pp. 351–399.] This work and the discovery of Donaldson invariants, as well as novel work of
Andreas Floer on
Floer homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer in ...
, inspired the study of
topological quantum field theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathe ...
.
After the discovery of the power of gauge theory to define invariants of manifolds, the field of mathematical gauge theory expanded in popularity. Further invariants were discovered, such as
Seiberg–Witten invariants and
Vafa–Witten invariants.
[Witten, Edward (1994), "Monopoles and four-manifolds.", Mathematical Research Letters, 1 (6): 769–796, arXiv:hep-th/9411102, Bibcode:1994MRLet...1..769W, doi:10.4310/MRL.1994.v1.n6.a13, MR 1306021, archived from the original on 2013-06-29] Strong links to algebraic geometry were realised by the work of Donaldson, Uhlenbeck, and
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
on the
Kobayashi–Hitchin correspondence relating Yang–Mills connections to
stable vector bundle In mathematics, a stable vector bundle is a ( holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable ...
s.
[Simon K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proceedings of the London Mathematical Society (3) 50 (1985), 1-26.][Karen Uhlenbeck and Shing-Tung Yau, On the existence of Hermitian–Yang-Mills connections in stable vector bundles.Frontiers of the mathematical sciences: 1985 (New York, 1985). Communications on Pure and Applied] Work of Nigel Hitchin and
Carlos Simpson on
Higgs bundle
In mathematics, a Higgs bundle is a pair (E,\varphi) consisting of a holomorphic vector bundle ''E'' and a Higgs field \varphi, a holomorphic 1-form taking values in the bundle of endomorphisms of ''E'' such that \varphi \wedge \varphi=0. Such pai ...
s demonstrated that moduli spaces arising out of gauge theory could have exotic geometric structures such as that of
hyperkähler manifolds, as well as links to
integrable systems
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
through the
Hitchin system
In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the ...
.
[Hitchin, N.J., 1987. The self-duality equations on a Riemann surface. Proceedings of the London Mathematical Society, 3(1), pp. 59–126.] Links to
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
and
mirror symmetry were realised, where gauge theory is essential to phrasing the
homological mirror symmetry conjecture and the
AdS/CFT correspondence
In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter ...
.
Fundamental objects of interest
The fundamental objects of interest in gauge theory are
connections on
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
s and
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
s. In this section we briefly recall these constructions, and refer to the main articles on them for details. The structures described here are standard within the differential geometry literature, and an introduction to the topic from a gauge-theoretic perspective can be found in the book of Donaldson and
Peter Kronheimer
Peter Benedict Kronheimer (born 1963) is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is William Caspar Graustein Professor of Mathematics at Harvard University and for ...
.
[Donaldson, S.K., Donaldson, S.K. and Kronheimer, P.B., 1990. The geometry of four-manifolds. Oxford University Press.]
Principal bundles
The central objects of study in gauge theory are principal bundles and vector bundles. The choice of which to study is essentially arbitrary, as one may pass between them, but principal bundles are the natural objects from the physical perspective to describe
gauge field
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie grou ...
s, and mathematically they more elegantly encode the corresponding theory of connections and curvature for vector bundles associated to them.
A principal bundle with structure group
, or a principal
-bundle, consists of a quintuple
where
is a smooth
fibre bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E an ...
with fibre space isomorphic to a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
, and
represents a
free and
transitive right
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of
on
which preserves the fibres, in the sense that for all
,
for all
. Here
is the ''total space'', and
the ''base space''. Using the right group action for each
and any choice of
, the map
defines a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two ...
between the fibre over
and the Lie group
as smooth manifolds. Note however there is no natural way of equipping the fibres of
with the structure of Lie groups, as there is no natural choice of element
for every
.
The simplest examples of principal bundles are given when
is the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
. In this case the principal bundle has dimension
where
. Another natural example occurs when
is the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
of the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of the manifold
, or more generally the frame bundle of a vector bundle over
. In this case the fibre of
is given by the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
.
Since a principal bundle is a fibre bundle, it locally has the structure of a product. That is, there exists an open covering
of
and diffeomorphisms
commuting with the projections
and
, such that the ''transition functions''
defined by
satisfy the
cocycle condition
In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in group cohomology. In autonomous ...
:
on any triple overlap
. In order to define a principal bundle it is enough to specify such a choice of transition functions, The bundle is then defined by gluing trivial bundles
along the intersections
using the transition functions. The cocycle condition ensures precisely that this defines an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
on the disjoint union
and therefore that the
quotient space is well-defined. This is known as the
fibre bundle construction theorem
In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space, fiber and a suitable set of transition functions. The theorem also gives conditions under which two such bundles are isomor ...
and the same process works for any fibre bundle described by transition functions, not just principal bundles or vector bundles.
Notice that a choice of ''local section''
satisfying
is an equivalent method of specifying a local trivialisation map. Namely, one can define
where
is the unique group element such that
.
Vector bundles
A vector bundle is a triple
where
is a
fibre bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E an ...
with fibre given by a vector space
where
is a field. The number
is the ''rank'' of the vector bundle. Again one has a local description of a vector bundle in terms of a trivialising open cover. If
is such a cover, then under the isomorphism
:
one obtains
distinguished local sections of
corresponding to the
coordinate basis vectors
of
, denoted
. These are defined by the equation
:
To specify a trivialisation it is therefore equivalent to give a collection of
local sections which are everywhere linearly independent, and use this expression to define the corresponding isomorphism. Such a collection of local sections is called a ''frame''.
Similarly to principal bundles, one obtains transition functions
for a vector bundle, defined by
:
If one takes these transition functions and uses them to construct the local trivialisation for a principal bundle with fibre equal to the structure group
, one obtains exactly the frame bundle of
, a principal
-bundle.
Associated bundles
Given a principal
-bundle
and a
representation of
on a vector space
, one can construct an associated vector bundle
with fibre the vector space
. To define this vector bundle, one considers the right action on the product
defined by
and defines
as the
quotient space with respect to this action.
In terms of transition functions the associated bundle can be understood more simply. If the principal bundle
has transition functions
with respect to a local trivialisation
, then one constructs the associated vector bundle using the transition functions
.
The associated bundle construction can be performed for any fibre space
, not just a vector space, provided
is a group homomorphism. One key example is the ''capital A adjoint bundle''
with fibre
, constructed using the group homomorphism
defined by conjugation
. Note that despite having fibre
, the Adjoint bundle is neither a principal bundle, or isomorphic as a fibre bundle to
itself. For example, if
is Abelian, then the conjugation action is trivial and
will be the trivial
-fibre bundle over
regardless of whether or not
is trivial as a fibre bundle. Another key example is the ''lowercase a
adjoint bundle In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nonassociative) algebra bundle. Adjoint bundles ...
''
constructed using the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is ...
where
is the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of
.
Gauge transformations
A gauge transformation of a vector bundle or principal bundle is an automorphism of this object. For a principal bundle, a gauge transformation consists of a diffeomorphism
commuting with the projection operator
and the right action
. For a vector bundle a gauge transformation is similarly defined by a diffeomorphism
commuting with the projection operator
which is a linear isomorphism of vector spaces on each fibre.
The gauge transformations (of
or
) form a group under composition, called the gauge group, typically denoted
. This group can be characterised as the space of global sections
of the adjoint bundle, or
in the case of a vector bundle, where
denotes the frame bundle.
One can also define a local gauge transformation as a local bundle isomorphism over a trivialising open subset
. This can be uniquely specified as a map
(taking
in the case of vector bundles), where the induced bundle isomorphism is defined by
:
and similarly for vector bundles.
Notice that given two local trivialisations of a principal bundle over the same open subset
, the transition function is precisely a local gauge transformation
. That is, ''local gauge transformations are changes of local trivialisation'' for principal bundles or vector bundles.
Connections on principal bundles
A connection on a principal bundle is a method of connecting nearby fibres so as to capture the notion of a section
being ''constant'' or ''horizontal''. Since the fibres of an abstract principal bundle are not naturally identified with each other, or indeed with the fibre space
itself, there is no canonical way of specifying which sections are constant. A choice of local trivialisation leads to one possible choice, where if
is trivial over a set
, then a local section could be said to be horizontal if it is constant with respect to this trivialisation, in the sense that
for all
and one
. In particular a trivial principal bundle
comes equipped with a trivial connection.
In general a connection is given by a choice of horizontal subspaces
of the tangent spaces at every point
, such that at every point one has
where
is the
vertical bundle defined by
. These horizontal subspaces must be compatible with the principal bundle structure by requiring that the horizontal
distribution Distribution may refer to:
Mathematics
* Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
*Probability distribution, the probability of a particular value or value range of a vari ...
is invariant under the right group action:
where
denotes right multiplication by
. A section
is said to be horizontal if
where
is identified with its image inside
, which is a submanifold of
with tangent bundle
. Given a vector field
, there is a unique horizontal lift
. The curvature of the connection
is given by the two-form with values in the adjoint bundle
defined by
:
where