The fundamental objects of interest in gauge theory are
connections on
vector bundles and
principal bundles. 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.
Principal bundles of the projection
π. \mathcal{F}(E) of the
Möbius strip E is a non-trivial principal \mathbb{Z}/2\mathbb{Z}-bundle over the circle. 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 fields, 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 G, or a
principal G-bundle, consists of a quintuple (P,X,\pi, G, \rho) where \pi: P \to X is a smooth
fibre bundle with fibre space isomorphic to a
Lie group G, and \rho represents a
free and
transitive right
group action of G on P which preserves the fibres, in the sense that for all p\in P, \pi(pg) = \pi(p) for all g\in G. Here P is the
total space, and X the
base space. Using the right group action for each x\in X and any choice of p\in P_x, the map g \mapsto pg defines a
diffeomorphism P_x \cong G between the fibre over x and the Lie group G as smooth manifolds. Note however there is no natural way of equipping the fibres of P with the structure of Lie groups, as there is no natural choice of element p\in P_x for every x\in X. The simplest examples of principal bundles are given when G=\operatorname{U}(1) is the
circle group. In this case the principal bundle has dimension \dim P = n + 1 where \dim X = n. Another natural example occurs when P=\mathcal{F}(TX) is the
frame bundle of the
tangent bundle of the manifold X, or more generally the frame bundle of a vector bundle over X. In this case the fibre of P is given by the
general linear group \operatorname{GL}(n, \mathbb{R}). Since a principal bundle is a fibre bundle, it locally has the structure of a product. That is, there exists an open covering \{U_{\alpha}\} of X and diffeomorphisms \varphi_\alpha: P_{U_\alpha} \to U_\alpha \times G commuting with the projections \pi and \operatorname{pr}_1, such that the
transition functions g_{\alpha\beta} :U_{\alpha}\cap U_{\beta} \to G defined by \varphi_\alpha \circ \varphi_{\beta}^{-1} (x,g) = (x, g_{\alpha\beta}(x) g) satisfy the
cocycle condition :g_{\alpha\beta}(x) g_{\beta\gamma}(x) = g_{\alpha\gamma}(x) on any triple overlap U_{\alpha}\cap U_{\beta}\cap U_\gamma. 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 U_\alpha\times G along the intersections U_\alpha\cap U_\beta using the transition functions. The cocycle condition ensures precisely that this defines an
equivalence relation on the disjoint union \bigsqcup_\alpha U_\alpha \times G and therefore that the
quotient space P=\bigsqcup_\alpha U_\alpha \times G/{\sim} is well-defined. This is known as the
fibre bundle construction theorem 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 s_\alpha: U_\alpha \to P_{U_\alpha} satisfying \pi \circ s_\alpha = \operatorname{Id} is an equivalent method of specifying a local trivialisation map. Namely, one can define \varphi_\alpha(p) = (\pi(p), \tilde s_\alpha(p)) where \tilde s_\alpha(p)\in G is the unique group element such that p\tilde s_\alpha(p)^{-1} =s_\alpha(\pi(p)).
Vector bundles A
vector bundle is a triple (E, X, \pi) where \pi: E\to X is a
fibre bundle with fibre given by a vector space \mathbb{K}^r where \mathbb{K}=\mathbb{R}, \mathbb{C} is a field. The number r 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 \{U_{\alpha}\} is such a cover, then under the isomorphism :\varphi_{\alpha}: E_{U_{\alpha}} \to U_{\alpha} \times \mathbb{K}^r one obtains r distinguished local
sections of E corresponding to the r coordinate basis vectors e_1,\dots,e_r of \mathbb{K}^r, denoted \boldsymbol{e}_1,\dots,\boldsymbol{e}_r. These are defined by the equation :\varphi_{\alpha} (\boldsymbol{e}_i (x)) = (x, e_i). To specify a trivialisation it is therefore equivalent to give a collection of r 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 g_{\alpha\beta}: U_{\alpha}\cap U_{\beta} \to \operatorname{GL}(r, \mathbb{K}) for a vector bundle, defined by :\varphi_{\alpha} \circ \varphi_{\beta}^{-1} (x, v) = (x, g_{\alpha\beta}(x) v). 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 \operatorname{GL}(r, \mathbb{K}), one obtains exactly the frame bundle of E, a principal \operatorname{GL}(r, \mathbb{K})-bundle.
Associated bundles Given a principal G-bundle P and a
representation \rho of G on a vector space V, one can construct an
associated vector bundle E=P\times_{\rho} V with fibre the vector space V. To define this vector bundle, one considers the right action on the product P\times V defined by (p,v)g = (pg, \rho(g^{-1})v) and defines P\times_{\rho} V = (P\times V)/G 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 P has transition functions g_{\alpha\beta} with respect to a local trivialisation \{U_{\alpha}\}, then one constructs the associated vector bundle using the transition functions \rho \circ g_{\alpha\beta}: U_{\alpha}\cap U_{\beta} \to \operatorname{GL}(V). The associated bundle construction can be performed for any fibre space F, not just a vector space, provided \rho: G\to \operatorname{Aut}(F) is a group homomorphism. One key example is the
capital A adjoint bundle \operatorname{Ad}(P) with fibre G, constructed using the group homomorphism \rho: G \to \operatorname{Aut}(G) defined by conjugation g \mapsto (h \mapsto g h g^{-1}). Note that despite having fibre G, the Adjoint bundle is neither a principal bundle, nor isomorphic as a fibre bundle to P itself. For example, if G is Abelian, then the conjugation action is trivial and \operatorname{Ad}(P) will be the trivial G-fibre bundle over X regardless of whether or not P is trivial as a fibre bundle. Another key example is the
lowercase a adjoint bundle \operatorname{ad}(P) constructed using the
adjoint representation \rho: G \to \operatorname{Aut}(\mathfrak{g}) where \mathfrak{g} is the
Lie algebra of G.
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 \varphi: P \to P commuting with the projection operator \pi and the right action \rho. For a vector bundle a gauge transformation is similarly defined by a diffeomorphism \varphi: E \to E commuting with the projection operator \pi which is a linear isomorphism of vector spaces on each fibre. The gauge transformations (of P or E) form a group under composition, called the
gauge group, typically denoted \mathcal{G}. This group can be characterised as the space of global
sections \mathcal{G} = \Gamma(\operatorname{Ad}(P)) of the adjoint bundle, or \mathcal{G} = \Gamma(\operatorname{Ad}(\mathcal{F} (E))) in the case of a vector bundle, where \mathcal{F}(E) denotes the frame bundle. One can also define a
local gauge transformation as a local bundle isomorphism over a trivialising open subset U_{\alpha}. This can be uniquely specified as a map g_{\alpha} : U_{\alpha} \to G (taking G=\operatorname{GL}(r, \mathbb{K}) in the case of vector bundles), where the induced bundle isomorphism is defined by :\varphi_{\alpha}(p) = pg_{\alpha}(\pi(p)) and similarly for vector bundles. Notice that given two local trivialisations of a principal bundle over the same open subset U_{\alpha}, the transition function is precisely a local gauge transformation g_{\alpha\alpha}: U_{\alpha} \to G. 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 s: X\to P 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 G itself, there is no canonical way of specifying which sections are constant. A choice of local trivialisation leads to one possible choice, where if P is trivial over a set U_{\alpha}, then a local section could be said to be horizontal if it is constant with respect to this trivialisation, in the sense that \varphi_{\alpha} (s(x)) = (x, g) for all x\in U_{\alpha} and one g\in G. In particular a trivial principal bundle P=X\times G comes equipped with a
trivial connection. In general a
connection is given by a choice of horizontal subspaces H_p \subset T_p P of the tangent spaces at every point p\in P, such that at every point one has T_p P = H_p \oplus V_p where V is the
vertical bundle defined by V=\ker d\pi. These horizontal subspaces must be compatible with the principal bundle structure by requiring that the horizontal
distribution H is invariant under the right group action: H_{pg} = d(R_g) (H_p) where R_g: P \to P denotes right multiplication by g. A section s is said to be
horizontal if T_p s \subset H_p where s is identified with its image inside P, which is a submanifold of P with tangent bundle Ts. Given a vector field v\in \Gamma(TX), there is a unique horizontal lift v^{\#}\in \Gamma(H). The
curvature of the connection H is given by the two-form with values in the adjoint bundle F\in \Omega^2(X, \operatorname{ad}(P)) defined by :F(v_1, v_2) = [v_1^{\#}, v_2^{\#}] - [v_1, v_2]^{\#} where [\cdot, \cdot] is the
Lie bracket of vector fields. Since the vertical bundle consists of the tangent spaces to the fibres of P and these fibres are isomorphic to the Lie group G whose tangent bundle is canonically identified with TG = G\times \mathfrak{g}, there is a unique
Lie algebra-valued two-form F\in \Omega^2(P, \mathfrak{g}) corresponding to the curvature. From the perspective of the
Frobenius integrability theorem, the curvature measures precisely the extent to which the horizontal distribution fails to be integrable, and therefore the extent to which H fails to embed inside P as a horizontal submanifold locally. The choice of horizontal subspaces may be equivalently expressed by a projection operator \nu: TP \to V which is equivariant in the correct sense, called the
connection one-form. For a horizontal distribution H, this is defined by \nu_H (h+v) = v where h+v denotes the decomposition of a tangent vector with respect to the direct sum decomposition TP = H\oplus V. Due to the equivariance, this projection one-form may be taken to be Lie algebra-valued, giving some \nu \in \Omega^1(P, \mathfrak{g}). A local trivialisation for P is equivalently given by a local section s_{\alpha}: U_{\alpha} \to P_{U_{\alpha}} and the connection one-form and curvature can be
pulled back along this smooth map. This gives the
local connection one-form A_{\alpha} = s_{\alpha}^* \nu\in \Omega^1(U_{\alpha}, \operatorname{ad}(P)) which takes values in the
adjoint bundle of P. Cartan's structure equation says that the curvature may be expressed in terms of the local one-form A_{\alpha} by the expression :F = dA_\alpha + \frac{1}{2} [A_\alpha, A_\alpha] where we use the Lie bracket on the Lie algebra bundle \operatorname{ad}(P) which is identified with U_\alpha \times \mathfrak{g} on the local trivialisation U_{\alpha}. Under a local gauge transformation g: U_\alpha \to G so that \tilde A_{\alpha} = (g\circ s)^* \nu, the local connection one-form transforms by the expression : \tilde A_\alpha = \operatorname{ad}(g)\circ A_\alpha + (g^{-1})^* \theta where \theta denotes the
Maurer–Cartan form of the Lie group G. In the case where G is a
matrix Lie group, one has the simpler expression \tilde A_{\alpha} = g A_\alpha g^{-1} - (dg)g^{-1}.
Connections on vector bundles A connection on a vector bundle may be specified similarly to the case for principal bundles above, known as an
Ehresmann connection. However vector bundle connections admit a more powerful description in terms of a differential operator. A
connection on a vector bundle is a choice of \mathbb{K}-linear differential operator :\nabla: \Gamma(E) \to \Gamma(T^*X \otimes E) = \Omega^1(E) such that :\nabla(fs) = df\otimes s + f \nabla s for all f\in C^{\infty}(X) and sections s\in \Gamma(E). The
covariant derivative of a section s in the direction of a vector field v is defined by :\nabla_v (s) = \nabla s (v) where on the right we use the natural pairing between \Omega^1(X) and TX. This is a new section of the vector bundle E, thought of as the derivative of s in the direction of v. The operator \nabla_v is the covariant derivative operator in the direction of v. The
curvature of \nabla is given by the operator F_{\nabla}\in \Omega^2(\operatorname{End}(E)) with values in the
endomorphism bundle, defined by :F_{\nabla}(v_1,v_2) = \nabla_{v_1} \nabla_{v_2} - \nabla_{v_2} \nabla_{v_1} - \nabla_{[v_1, v_2]}. In a local trivialisation the
exterior derivative d acts as a trivial connection (corresponding in the principal bundle picture to the trivial connection discussed above). Namely for a local frame \boldsymbol{e}_1, \dots, \boldsymbol{e}_r one defines :d (s^i \boldsymbol{e}_i) = ds^i \otimes \boldsymbol{e}_i where here we have used
Einstein notation for a local section s=s^i \boldsymbol{e}_i. Any two connections \nabla_1, \nabla_2 differ by an \operatorname{End}(E)-valued one-form A. To see this, observe that the difference of two connections is C^{\infty}(X)-linear: :(\nabla_1 - \nabla_2)(fs) = f(\nabla_1-\nabla_2)(s). In particular since every vector bundle admits a connection (using
partitions of unity and the local trivial connections), the set of connections on a vector bundle has the structure of an infinite-dimensional
affine space modelled on the vector space \Omega^1(\operatorname{End}(E)). This space is commonly denoted \mathcal{A}. Applying this observation locally, every connection over a trivialising subset U_{\alpha} differs from the trivial connection d by some local connection one-form A_{\alpha}\in \Omega^1(U_{\alpha}, \operatorname{End}(E)), with the property that \nabla = d + A_{\alpha} on U_{\alpha}. In terms of this local connection form, the curvature may be written as :F_A = dA_{\alpha} + A_{\alpha} \wedge A_{\alpha} where the wedge product occurs on the one-form component, and one composes endomorphisms on the endomorphism component. To link back to the theory of principal bundles, notice that A\wedge A = \frac{1}{2}[A, A] where on the right we now perform wedge of one-forms and commutator of endomorphisms. Under a gauge transformation u of the vector bundle E, a connection \nabla transforms into a connection u\cdot \nabla by the conjugation (u\cdot \nabla)_v(s) = u(\nabla_v(u^{-1}(s)). The difference u\cdot \nabla - \nabla = -(\nabla u)u^{-1} where here \nabla is acting on the endomorphisms of E. Under a
local gauge transformation g one obtains the same expression :\tilde A_{\alpha} = g A_{\alpha} g^{-1} - (dg)g^{-1} as in the case of principal bundles.
Induced connections A connection on a principal bundle induces connections on associated vector bundles. One way to see this is in terms of the local connection forms described above. Namely, if a principal bundle connection H has local connection forms A_{\alpha}\in \Omega^1(U_{\alpha}, \operatorname{ad}(P)), and \rho: G \to \operatorname{Aut}(V) is a representation of G defining an associated vector bundle E=P\times_{\rho} V, then the induced local connection one-forms are defined by :\rho_* A_{\alpha} \in \Omega^1(U_{\alpha}, \operatorname{End}(E)). Here \rho_* is the induced
Lie algebra homomorphism from \mathfrak{g} \to \operatorname{End}(V), and we use the fact that this map induces a homomorphism of vector bundles \operatorname{ad}(P) \to \operatorname{End}(E). The induced curvature can be simply defined by :\rho_* F_A \in \Omega^2(U_\alpha, \operatorname{End}(E)). Here one sees how the local expressions for curvature are related for principal bundles and vector bundles, as the Lie bracket on the Lie algebra \mathfrak{g} is sent to the commutator of endomorphisms of \operatorname{End}(V) under the Lie algebra homomorphism \rho_*.
Space of connections The central object of study in mathematical gauge theory is the space of connections on a vector bundle or principal bundle. This is an infinite-dimensional affine space \mathcal{A} modelled on the vector space \Omega^1(X, \operatorname{ad}(P)) (or \Omega^1(X, \operatorname{End}(E)) in the case of vector bundles). Two connections A, A'\in \mathcal{A} are said to be
gauge equivalent if there exists a gauge transformation u such that A' = u\cdot A. Gauge theory is concerned with gauge equivalence classes of connections. In some sense gauge theory is therefore concerned with the properties of the
quotient space \mathcal{A}/\mathcal{G}, which is in general neither a
Hausdorff space or a
smooth manifold. Many interesting properties of the base manifold X can be encoded in the geometry and topology of moduli spaces of connections on principal bundles and vector bundles over X. Invariants of X, such as
Donaldson invariants or
Seiberg–Witten invariants can be obtained by computing numeral quantities derived from moduli spaces of connections over X. The most famous application of this idea is
Donaldson's theorem, which uses the moduli space of Yang–Mills connections on a principal \operatorname{SU}(2)-bundle over a
simply connected four-manifold X to study its intersection form. For this work Donaldson was awarded a
Fields Medal. ==Notational conventions==