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 point *x* of the space *X* we associate (or "attach") a vector space *V*(*x*) in such a way that these vector spaces fit together to form another space of the same kind as *X* (e.g. a topological space, manifold, or algebraic variety), which is then called a **vector bundle over X**.

- Definition and first consequences
- Transition functions
- Subbundles
- Vector bundle morphisms
- Sections and locally free sheaves
- Operations on vector bundles
- Additional structures and generalizations
- Smooth vector bundles
- K-theory
- See also
- General notions
- Topology and differential geometry
- Algebraic and analytic geometry
- Notes
- Sources
- External links

The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space *V* such that *V*(*x*) = *V* for all *x* in *X*: in this case there is a copy of *V* for each *x* in *X* and these copies fit together to form the vector bundle *X* × *V* over *X*. Such vector bundles are said to be *trivial*. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial.

Vector bundles are almost always required to be *locally trivial*, however, which means they are examples of fiber bundles. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the category of topological spaces.

A **real vector bundle** consists of:

- topological spaces
*X*(*base space*) and*E*(*total space*) - a continuous surjection π:
*E*→*X*(*bundle projection*) - for every
*x*in*X*, the structure of a finite-dimensional real vector space on the fiber π^{−1}({*x*})

where the following compatibility condition is satisfied: for every point *p* in *X*, there is an open neighborhood *U* ⊆ *X* of *p*, a natural number *k*, and a homeomorphism

such that for all *x* ∈ *U*,

- for all vectors
*v*in**R**^{k}, and - the map is a linear isomorphism between the vector spaces
**R**^{k}and π^{−1}({*x*}).

The open neighborhood *U* together with the homeomorphism is called a **local trivialization** of the vector bundle. The local trivialization shows that *locally* the map π "looks like" the projection of *U* × **R**^{k} on *U*.

Every fiber π^{−1}({*x*}) is a finite-dimensional real vector space and hence has a dimension *k*_{x}. The local trivializations show that the function *x*↦*k _{x}* is locally constant, and is therefore constant on each connected component of

The Cartesian product *X* × **R**^{k}, equipped with the projection *X* × **R**^{k} → *X*, is called the **trivial bundle** of rank *k* over *X*.

Given a vector bundle *E* → *X* of rank *k*, and a pair of neighborhoods *U* and *V* over which the bundle trivializes via

the composite function

is well-defined on the overlap, and satisfies

for some GL(*k*)-valued function

These are called the **transition functions** (or the **coordinate transformations**) of the vector bundle.

The set of transition functions forms a Čech cocycle in the sense that

for all *U*, *V*, *W* over which the bundle trivializes satisfying . Thus the data (*E*, *X*, π, **R**^{k}) defines a fiber bundle; the additional data of the *g*_{UV} specifies a GL(*k*) structure group in which the action on the fiber is the standard action of GL(*k*).

Conversely, given a fiber bundle (*E*, *X*, π, **R**^{k}) with a GL(*k*) cocycle acting in the standard way on the fiber **R**^{k}, there is associated a vector bundle. This is an example of the fibre bundle construction theorem for vector bundles, and can be taken as an alternative definition of a vector bundle.

One simple method of constructing vector bundles is by taking subbundles of other vector bundles. Given a vector bundle over a topological space, a subbundle is simply a subspace for which the restriction of to gives the structure of a vector bundle also. In this case the fibre is a vector subspace for every .

A subbundle of a trivial bundle need not be trivial, and indeed every real vector bundle can be viewed as a subbundle of a trivial bundle of sufficiently high rank. For example the Möbius band, a non-trivial line bundle over the circle, can be seen as a subbundle of the trivial rank 2 bundle over the circle.

A ** morphism ** from the vector bundle π_{1}: *E*_{1} → *X*_{1} to the vector bundle π_{2}: *E*_{2} → *X*_{2} is given by a pair of continuous maps *f*: *E*_{1} → *E*_{2} and *g*: *X*_{1} → *X*_{2} such that

*g*∘ π_{1}= π_{2}∘*f*- for every
*x*in*X*_{1}, the map π_{1}^{−1}({*x*}) → π_{2}^{−1}({*g*(*x*)}) induced by*f*is a linear map between vector spaces.

Note that *g* is determined by *f* (because π_{1} is surjective), and *f* is then said to **cover g**.

The class of all vector bundles together with bundle morphisms forms a category. Restricting to vector bundles for which the spaces are manifolds (and the bundle projections are smooth maps) and smooth bundle morphisms we obtain the category of smooth vector bundles. Vector bundle morphisms are a special case of the notion of a bundle map between fiber bundles, and are also often called **(vector) bundle homomorphisms**.

A bundle homomorphism from *E*_{1} to *E*_{2} with an inverse which is also a bundle homomorphism (from *E*_{2} to *E*_{1}) is called a **(vector) bundle isomorphism**, and then *E*_{1} and *E*_{2} are said to be **isomorphic** vector bundles. An isomorphism of a (rank *k*) vector bundle *E* over *X* with the trivial bundle (of rank *k* over *X*) is called a **trivialization** of *E*, and *E* is then said to be **trivial** (or **trivializable**). The definition of a vector bundle shows that any vector bundle is **locally trivial**.

We can also consider the category of all vector bundles over a fixed base space *X*. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the identity map on *X*. That is, bundle morphisms for which the following diagram commutes:

(Note that this category is *not* abelian; the kernel of a morphism of vector bundles is in general not a vector bundle in any natural way.)

A vector bundle morphism between vector bundles π_{1}: *E*_{1} → *X*_{1} and π_{2}: *E*_{2} → *X*_{2} covering a map *g* from *X*_{1} to *X*_{2} can also be viewed as a vector bundle morphism over *X*_{1} from *E*_{1} to the pullback bundle *g***E*_{2}.

Given a vector bundle π: *E* → *X* and an open subset *U* of *X*, we can consider **sections** of π on *U*, i.e. continuous functions *s*: *U* → *E* where the composite π∘*s* is such that (π∘*s*)(*u*) = *u* for all *u* in *U*. Essentially, a section assigns to every point of *U* a vector from the attached vector space, in a continuous manner. As an example, sections of the tangent bundle of a differential manifold are nothing but vector fields on that manifold.

Let *F*(*U*) be the set of all sections on *U*. *F*(*U*) always contains at least one element, namely the **zero section**: the function *s* that maps every element *x* of *U* to the zero element of the vector space π^{−1}({*x*}). With the pointwise addition and scalar multiplication of sections, *F*(*U*) becomes itself a real vector space. The collection of these vector spaces is a sheaf of vector spaces on *X*.

If *s* is an element of *F*(*U*) and α: *U* → **R** is a continuous map, then α*s* (pointwise scalar multiplication) is in *F*(*U*). We see that *F*(*U*) is a module over the ring of continuous real-valued functions on *U*. Furthermore, if O_{X} denotes the structure sheaf of continuous real-valued functions on *X*, then *F* becomes a sheaf of O_{X}-modules.

Not every sheaf of O_{X}-modules arises in this fashion from a vector bundle: only the locally free ones do. (The reason: locally we are looking for sections of a projection *U* × **R**^{k} → *U*; these are precisely the continuous functions *U* → **R**^{k}, and such a function is a *k*-tuple of continuous functions *U* → **R**.)

Even more: the category of real vector bundles on *X* is equivalent to the category of locally free and finitely generated sheaves of O_{X}-modules.

So we can think of the category of real vector bundles on *X* as sitting inside the category of sheaves of O_{X}-modules; this latter category is abelian, so this is where we can compute kernels and cokernels of morphisms of vector bundles.

A rank *n* vector bundle is trivial if and only if it has *n* linearly independent global sections.

Most operations on vector spaces can be extended to vector bundles by performing the vector space operation *fiberwise*.

For example, if *E* is a vector bundle over *X*, then there is a bundle *E** over *X*, called the ** dual bundle **, whose fiber at *x* ∈ *X* is the dual vector space (*E _{x}*)*. Formally

There are many functorial operations which can be performed on pairs of vector spaces (over the same field), and these extend straightforwardly to pairs of vector bundles *E*, *F* on *X* (over the given field). A few examples follow.

- The
**Whitney sum**(named for Hassler Whitney) or**direct sum bundle**of*E*and*F*is a vector bundle*E*⊕*F*over*X*whose fiber over*x*is the direct sum*E*⊕_{x}*F*of the vector spaces_{x}*E*and_{x}*F*._{x} - The
**tensor product bundle***E*⊗*F*is defined in a similar way, using fiberwise tensor product of vector spaces. - The
**Hom-bundle**Hom(*E*,*F*) is a vector bundle whose fiber at*x*is the space of linear maps from*E*to_{x}*F*(which is often denoted Hom(_{x}*E*_{x},*F*) or_{x}*L*(*E*_{x},*F*_{x})). The Hom-bundle is so-called (and useful) because there is a bijection between vector bundle homomorphisms from*E*to*F*over*X*and sections of Hom(*E*,*F*) over*X*. - Building on the previous example, given a section
*s*of an endomorphism bundle Hom(*E*,*E*) and a function*f*:*X*→**R**, one can construct an**eigenbundle**by taking the fiber over a point*x*∈*X*to be the*f*(*x*)-eigenspace of the linear map*s*(*x*):*E*_{x}→*E*_{x}. Though this construction is natural, unless care is taken, the resulting object will not have local trivializations. Consider the case of*s*being the zero section and*f*having isolated zeroes. The fiber over these zeroes in the resulting "eigenbundle" will be isomorphic to the fiber over them in*E*, while everywhere else the fiber is the trivial 0-dimensional vector space. - The dual vector bundle
*E**is the Hom bundle Hom(*E*,**R**×*X*) of bundle homomorphisms of*E*and the trivial bundle**R**×*X*. There is a canonical vector bundle isomorphism Hom(*E*,*F*) =*E**⊗*F*.

Each of these operations is a particular example of a general feature of bundles: that many operations that can be performed on the category of vector spaces can also be performed on the category of vector bundles in a functorial manner. This is made precise in the language of smooth functors. An operation of a different nature is the ** pullback bundle ** construction. Given a vector bundle *E* → *Y* and a continuous map *f*: *X* → *Y* one can "pull back" *E* to a vector bundle *f*E* over *X*. The fiber over a point *x* ∈ *X* is essentially just the fiber over *f*(*x*) ∈ *Y*. Hence, Whitney summing *E* ⊕ *F* can be defined as the pullback bundle of the diagonal map from *X* to *X* × *X* where the bundle over *X* × *X* is *E* × *F*.

**Remark**: Let *X* be a compact space. Any vector bundle *E* over *X* is a direct summand of a trivial bundle; i.e., there exists a bundle *E*' such that *E* ⊕ *E*' is trivial. This fails if *X* is not compact: for example, the tautological line bundle over the infinite real projective space does not have this property.^{ [1] }

Vector bundles are often given more structure. For instance, vector bundles may be equipped with a vector bundle metric. Usually this metric is required to be positive definite, in which case each fibre of *E* becomes a Euclidean space. A vector bundle with a complex structure corresponds to a complex vector bundle, which may also be obtained by replacing real vector spaces in the definition with complex ones and requiring that all mappings be complex-linear in the fibers. More generally, one can typically understand the additional structure imposed on a vector bundle in terms of the resulting reduction of the structure group of a bundle. Vector bundles over more general topological fields may also be used.

If instead of a finite-dimensional vector space, if the fiber *F* is taken to be a Banach space then a ** Banach bundle ** is obtained.^{ [2] } Specifically, one must require that the local trivializations are Banach space isomorphisms (rather than just linear isomorphisms) on each of the fibers and that, furthermore, the transitions

are continuous mappings of Banach manifolds. In the corresponding theory for C^{p} bundles, all mappings are required to be C^{p}.

Vector bundles are special fiber bundles, those whose fibers are vector spaces and whose cocycle respects the vector space structure. More general fiber bundles can be constructed in which the fiber may have other structures; for example sphere bundles are fibered by spheres.

A vector bundle (*E*, *p*, *M*) is **smooth**, if *E* and *M* are smooth manifolds, p: *E* → *M* is a smooth map, and the local trivializations are diffeomorphisms. Depending on the required degree of smoothness, there are different corresponding notions of *C ^{p}* bundles, infinitely differentiable

A smooth vector bundle can be characterized by the fact that it admits transition functions as described above which are *smooth* functions on overlaps of trivializing charts *U* and *V*. That is, a vector bundle *E* is smooth if it admits a covering by trivializing open sets such that for any two such sets *U* and *V*, the transition function

is a smooth function into the matrix group GL(k,**R**), which is a Lie group.

Similarly, if the transition functions are:

*C*then the vector bundle is a^{r},*C*vector bundle^{r}*real analytic*then the vector bundle is a**real analytic vector bundle**(this requires the matrix group to have a real analytic structure),*holomorphic*then the vector bundle is a**holomorphic vector bundle**(this requires the matrix group to be a complex Lie group),*algebraic functions*then the vector bundle is an**algebraic vector bundle**(this requires the matrix group to be an algebraic group).

The *C*^{∞}-vector bundles (*E*, *p*, *M*) have a very important property not shared by more general *C*^{∞}-fibre bundles. Namely, the tangent space *T _{v}*(

The vertical lift can also be seen as a natural *C*^{∞}-vector bundle isomorphism *p*E* → *VE*, where (*p*E*, *p*p*, *E*) is the pull-back bundle of (*E*, *p*, *M*) over *E* through *p*: *E* → *M*, and *VE* := Ker(*p*_{*}) ⊂ *TE* is the *vertical tangent bundle*, a natural vector subbundle of the tangent bundle (*TE*, π_{TE}, *E*) of the total space *E*.

The total space *E* of any smooth vector bundle carries a natural vector field *V*_{v} := vl_{v}*v*, known as the *canonical vector field*. More formally, *V* is a smooth section of (*TE*, π_{TE}, *E*), and it can also be defined as the infinitesimal generator of the Lie-group action (*t*, *v*) ↦ *e*^{t}*v* given by the fibrewise scalar multiplication. The canonical vector field *V* characterizes completely the smooth vector bundle structure in the following manner. As a preparation, note that when *X* is a smooth vector field on a smooth manifold *M* and *x* ∈ *M* such that *X*_{x} = 0, the linear mapping

does not depend on the choice of the linear covariant derivative ∇ on *M*. The canonical vector field *V* on *E* satisfies the axioms

- The flow (
*t*,*v*) → Φ^{t}_{V}(*v*) of*V*is globally defined. - For each
*v*∈*V*there is a unique lim_{t→∞}Φ^{t}_{V}(*v*) ∈*V*. *C*_{v}(*V*)∘*C*_{v}(*V*) =*C*_{v}(*V*) whenever*V*_{v}= 0.- The zero set of
*V*is a smooth submanifold of*E*whose codimension is equal to the rank of*C*_{v}(*V*).

Conversely, if *E* is any smooth manifold and *V* is a smooth vector field on *E* satisfying 1–4, then there is a unique vector bundle structure on *E* whose canonical vector field is *V*.

For any smooth vector bundle (*E*, *p*, *M*) the total space *TE* of its tangent bundle (*TE*, π_{TE}, *E*) has a natural secondary vector bundle structure (*TE*, *p*_{*}, *TM*), where *p*_{*} is the push-forward of the canonical projection *p*: *E* → *M*. The vector bundle operations in this secondary vector bundle structure are the push-forwards +_{*}: *T*(*E* × *E*) → *TE* and λ_{*}: *TE* → *TE* of the original addition +: *E* × *E* → *E* and scalar multiplication λ: *E* → *E*.

The K-theory group, *K*(*X*), of a compact Hausdorff topological space is defined as the abelian group generated by isomorphism classes [*E*] of complex vector bundles modulo the relation that whenever we have an exact sequence

then

in topological K-theory. KO-theory is a version of this construction which considers real vector bundles. K-theory with compact supports can also be defined, as well as higher K-theory groups.

The famous periodicity theorem of Raoul Bott asserts that the K-theory of any space *X* is isomorphic to that of the *S*^{2}*X*, the double suspension of *X*.

In algebraic geometry, one considers the K-theory groups consisting of coherent sheaves on a scheme *X*, as well as the K-theory groups of vector bundles on the scheme with the above equivalence relation. The two constructs are the same provided that the underlying scheme is smooth.

- Grassmannian: classifying spaces for vector bundle, among which projective spaces for line bundles
- Characteristic class
- Splitting principle
- Stable bundle

- Gauge theory: the general study of connections on vector bundles and principal bundles and their relations to physics.
- Connection: the notion needed to differentiate sections of vector bundles.

- ↑ Hatcher 2003, Example 3.6.
- ↑ Lang 1995.

- Abraham, Ralph H.; Marsden, Jerrold E. (1978),
*Foundations of mechanics*, London: Benjamin-Cummings, see section 1.5, ISBN 978-0-8053-0102-1 . - Hatcher, Allen (2003),
*Vector Bundles & K-Theory*(2.0 ed.). - Jost, Jürgen (2002),
*Riemannian Geometry and Geometric Analysis*(3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-42627-1 , see section 1.5. - Lang, Serge (1995),
*Differential and Riemannian manifolds*, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94338-1 . - Lee, Jeffrey M. (2009),
*Manifolds and Differential Geometry*, Graduate Studies in Mathematics, Vol. 107, Providence: American Mathematical Society, ISBN 978-0-8218-4815-9`|volume=`

has extra text (help). - Lee, John M. (2003),
*Introduction to Smooth Manifolds*, New York: Springer, ISBN 0-387-95448-1 see Ch.5 - Rubei, Elena (2014),
*Algebraic Geometry, a concise dictionary*, Berlin/Boston: Walter De Gruyter, ISBN 978-3-11-031622-3 .

In mathematics, the **tangent space** of a manifold generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold.

In differential geometry, the **tangent bundle** of a differentiable manifold is a manifold which assembles all the tangent vectors in . As a set, it is given by the disjoint union of the tangent spaces of . That is,

In mathematics, and particularly topology, a **fiber bundle** is a space that is *locally* a product space, but *globally* may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, , that in small regions of behaves just like a projection from corresponding regions of to . The map , called the **projection** or **submersion** of the bundle, is regarded as part of the structure of the bundle. The space is known as the **total space** of the fiber bundle, as the **base space**, and the **fiber**.

In mathematics, a **principal bundle** is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian product, a principal bundle is equipped with

- An action of on , analogous to for a product space.
- A projection onto . For a product space, this is just the projection onto the first factor, .

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 naturally on F(*E *) via a change of basis, giving the frame bundle the structure of a principal GL(*k*, **R**)-bundle.

In the mathematical field of topology, a **section** of a fiber bundle is a continuous right inverse of the projection function . In other words, if is a fiber bundle over a base space, :

In mathematics, a **foliation** is an equivalence relation on an *n*-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension *p*, modeled on the decomposition of the real coordinate space **R**^{n} into the cosets *x* + **R**^{p} of the standardly embedded subspace **R**^{p}. The equivalence classes are called the **leaves** of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable, or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class *C ^{r}* it is usually understood that

In mathematics, the theory of fiber bundles with a structure group allows an operation of creating an **associated bundle**, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . For a fiber bundle *F* with structure group *G*, the transition functions of the fiber in an overlap of two coordinate systems *U*_{α} and *U*_{β} are given as a *G*-valued function *g*_{αβ} on *U*_{α}∩*U*_{β}. One may then construct a fiber bundle *F*′ as a new fiber bundle having the same transition functions, but possibly a different fiber.

Suppose that *φ* : *M* → *N* is a smooth map between smooth manifolds *M* and *N*. Then there is an associated linear map from the space of 1-forms on *N* to the space of 1-forms on *M*. This linear map is known as the **pullback**, and is frequently denoted by *φ*^{∗}. More generally, any covariant tensor field – in particular any differential form – on *N* may be pulled back to *M* using *φ*.

This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:

In differential geometry, a **Poisson structure** on a smooth manifold is a Lie bracket on the algebra of smooth functions on , subject to the Leibniz rule

In differential geometry, **pushforward** is a linear approximation of smooth maps on tangent spaces. Suppose that *φ* : *M* → *N* is a smooth map between smooth manifolds; then the **differential** of *φ, ,* at a point *x* is, in some sense, the best linear approximation of *φ* near *x*. It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of *M* at *x* to the tangent space of *N* at *φ*(*x*), . Hence it can be used to *push* tangent vectors on *M**forward* to tangent vectors on *N*. The differential of a map *φ* is also called, by various authors, the **derivative** or **total derivative** of *φ*.

In mathematics, a **pullback bundle** or **induced bundle** is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle *π* : *E* → *B* and a continuous map *f* : *B*′ → *B* one can define a "pullback" of *E* by *f* as a bundle *f*^{*}*E* over *B*′. The fiber of *f*^{*}*E* over a point *b*′ in *B*′ is just the fiber of *E* over *f*(*b*′). Thus *f*^{*}*E* is the disjoint union of all these fibers equipped with a suitable topology.

In mathematics, a **differentiable manifold** is a type of manifold that is locally similar enough to a vector space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In differential geometry, in the category of differentiable manifolds, a **fibered manifold** is a surjective submersion

In mathematics, a **vector-valued differential form** on a manifold *M* is a differential form on *M* with values in a vector space *V*. More generally, it is a differential form with values in some vector bundle *E* over *M*. Ordinary differential forms can be viewed as **R**-valued differential forms.

In mathematics, a **holomorphic vector bundle** is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : *E* → *X* is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A **holomorphic line bundle** is a rank one holomorphic vector bundle.

In mathematics, a **bundle map** is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely which category of fiber bundles is under consideration. In the first three sections, we will consider general fiber bundles in the category of topological spaces. Then in the fourth section, some other examples will be given.

In mathematics, an **affine bundle** is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.

In mathematics, and especially differential geometry and mathematical physics, **gauge theory** is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics *theory* means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a physical model of some natural phenomenon.

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