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 that 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 mathematical model of some natural phenomenon. Gauge theory in mathematics is typically concerned with the study of gauge-theoretic equations. These are differential equations 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. These equations are often physically meaningful, corresponding to important concepts in quantum field theory or string theory, but also have important mathematical significance. For example, the Yang–Mills equations are a system of partial differential equations 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, particles known as instantons. 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 such as moduli spaces of vector bundles and coherent sheaves.
History
Gauge theory has its origins as far back as the formulation of Maxwell's equations describing classical electromagnetism, which may be phrased as a gauge theory with structure group the circle group. Work of Paul Dirac on magnetic monopoles and relativistic quantum mechanics 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 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, Isadore Singer, and Nigel Hitchin on the self-duality equations on a Riemannian manifold 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 8 k − 3 {\displaystyle 8k-3} where k {\displaystyle k} 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 k = 1 {\displaystyle k=1} . Such instantons are defined by a choice of 5 parameters, the center z ∈ R 4 {\displaystyle z\in \mathbb {R} ^{4}} and scale ρ ∈ R > 0 {\displaystyle \rho \in \mathbb {R} _{>0}} , corresponding to the 8 − 3 = 5 {\displaystyle 8-3=5} -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 C P 3 {\displaystyle \mathbb {CP} ^{3}} . Another significant early discovery was the development of the ADHM construction by Atiyah, Vladimir Drinfeld, Hitchin, and Yuri Manin. This construction allowed for the solution to the anti-self-duality equations on Euclidean space R 4 {\displaystyle \mathbb {R} ^{4}} 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 maps, equivariant Morse theory, and relations between gauge theory and algebraic geometry. Important analytical tools in geometric analysis were developed at this time by Karen Uhlenbeck, 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 and Edward Witten. Donaldson used a combination of algebraic geometry and geometric analysis techniques to construct new invariants of four manifolds, now known as Donaldson invariants. 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 R 4 {\displaystyle \mathbb {R} ^{4}} could be proved. For this work Donaldson was awarded the Fields Medal in 1986. Witten similarly observed the power of gauge theory to describe topological invariants, by relating quantities arising from Chern–Simons theory in three dimensions to the Jones polynomial, an invariant of knots. This work and the discovery of Donaldson invariants, as well as novel work of Andreas Floer on Floer homology, inspired the study of topological quantum field theory. 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. Strong links to algebraic geometry were realised by the work of Donaldson, Uhlenbeck, and Shing-Tung Yau on the Kobayashi–Hitchin correspondence relating Yang–Mills connections to stable vector bundles. Work of Nigel Hitchin and Carlos Simpson on Higgs bundles 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 through the Hitchin system. Links to string theory and Mirror symmetry were realised, where gauge theory is essential to phrasing the homological mirror symmetry conjecture and the AdS/CFT correspondence.
Fundamental objects of interest 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
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 {\displaystyle G} , or a principal G {\displaystyle G} -bundle, consists of a quintuple ( P , X , π , G , ρ ) {\displaystyle (P,X,\pi ,G,\rho )} where π : P → X {\displaystyle \pi :P\to X} is a smooth fibre bundle with fibre space isomorphic to a Lie group G {\displaystyle G} , and ρ {\displaystyle \rho } represents a free and transitive right group action of G {\displaystyle G} on P {\displaystyle P} which preserves the fibres, in the sense that for all p ∈ P {\displaystyle p\in P} , π ( p g ) = π ( p ) {\displaystyle \pi (pg)=\pi (p)} for all g ∈ G {\displaystyle g\in G} . Here P {\displaystyle P} is the total space, and X {\displaystyle X} the base space. Using the right group action for each x ∈ X {\displaystyle x\in X} and any choice of p ∈ P x {\textstyle p\in P_{x}} , the map g ↦ p g {\displaystyle g\mapsto pg} defines a diffeomorphism P x ≅ G {\displaystyle P_{x}\cong G} between the fibre over x {\displaystyle x} and the Lie group G {\displaystyle G} as smooth manifolds. Note however there is no natural way of equipping the fibres of P {\displaystyle P} with the structure of Lie groups, as there is no natural choice of element p ∈ P x {\displaystyle p\in P_{x}} for every x ∈ X {\displaystyle x\in X} . The simplest examples of principal bundles are given when G = U ( 1 ) {\displaystyle G=\operatorname {U} (1)} is the circle group. In this case the principal bundle has dimension dim P = n + 1 {\displaystyle \dim P=n+1} where dim X = n {\displaystyle \dim X=n} . Another natural example occurs when P = F ( T X ) {\displaystyle P={\mathcal {F}}(TX)} is the frame bundle of the tangent bundle of the manifold X {\displaystyle X} , or more generally the frame bundle of a vector bundle over X {\displaystyle X} . In this case the fibre of P {\displaystyle P} is given by the general linear group GL ( n , R ) {\displaystyle \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 α } {\displaystyle \{U_{\alpha }\}} of X {\displaystyle X} and diffeomorphisms φ α : P U α → U α × G {\displaystyle \varphi _{\alpha }:P_{U_{\alpha }}\to U_{\alpha }\times G} commuting with the projections π {\displaystyle \pi } and pr 1 {\displaystyle \operatorname {pr} _{1}} , such that the transition functions g α β : U α ∩ U β → G {\displaystyle g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to G} defined by φ α ∘ φ β − 1 ( x , g ) = ( x , g α β ( x ) g ) {\displaystyle \varphi _{\alpha }\circ \varphi _{\beta }^{-1}(x,g)=(x,g_{\alpha \beta }(x)g)} satisfy the cocycle condition
g α β ( x ) g β γ ( x ) = g α γ ( x ) {\displaystyle g_{\alpha \beta }(x)g_{\beta \gamma }(x)=g_{\alpha \gamma }(x)}
on any triple overlap U α ∩ U β ∩ U γ {\displaystyle 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 α × G {\displaystyle U_{\alpha }\times G} along the intersections U α ∩ U β {\displaystyle U_{\alpha }\cap U_{\beta }} using the transition functions. The cocycle condition ensures precisely that this defines an equivalence relation on the disjoint union ⨆ α U α × G {\displaystyle \bigsqcup _{\alpha }U_{\alpha }\times G} and therefore that the quotient space P = ⨆ α U α × G / ∼ {\displaystyle 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 α : U α → P U α {\displaystyle s_{\alpha }:U_{\alpha }\to P_{U_{\alpha }}} satisfying π ∘ s α = Id {\displaystyle \pi \circ s_{\alpha }=\operatorname {Id} } is an equivalent method of specifying a local trivialisation map. Namely, one can define φ α ( p ) = ( π ( p ) , s ~ α ( p ) ) {\displaystyle \varphi _{\alpha }(p)=(\pi (p),{\tilde {s}}_{\alpha }(p))} where s ~ α ( p ) ∈ G {\displaystyle {\tilde {s}}_{\alpha }(p)\in G} is the unique group element such that p s ~ α ( p ) − 1 = s α ( π ( p ) ) {\displaystyle p{\tilde {s}}_{\alpha }(p)^{-1}=s_{\alpha }(\pi (p))} .
Vector bundles
A vector bundle is a triple ( E , X , π ) {\displaystyle (E,X,\pi )} where π : E → X {\displaystyle \pi :E\to X} is a fibre bundle with fibre given by a vector space K r {\displaystyle \mathbb {K} ^{r}} where K = R , C {\displaystyle \mathbb {K} =\mathbb {R} ,\mathbb {C} } is a field. The number r {\displaystyle 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 α } {\displaystyle \{U_{\alpha }\}} is such a cover, then under the isomorphism
φ α : E U α → U α × K r {\displaystyle \varphi _{\alpha }:E_{U_{\alpha }}\to U_{\alpha }\times \mathbb {K} ^{r}}
one obtains r {\displaystyle r} distinguished local sections of E {\displaystyle E} corresponding to the r {\displaystyle r} coordinate basis vectors e 1 , … , e r {\displaystyle e_{1},\dots ,e_{r}} of K r {\displaystyle \mathbb {K} ^{r}} , denoted e 1 , … , e r {\displaystyle {\boldsymbol {e}}_{1},\dots ,{\boldsymbol {e}}_{r}} . These are defined by the equation
φ α ( e i ( x ) ) = ( x , e i ) . {\displaystyle \varphi _{\alpha }({\boldsymbol {e}}_{i}(x))=(x,e_{i}).}
To specify a trivialisation it is therefore equivalent to give a collection of r {\displaystyle 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 α β : U α ∩ U β → GL ( r , K ) {\displaystyle g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to \operatorname {GL} (r,\mathbb {K} )} for a vector bundle, defined by
φ α ∘ φ β − 1 ( x , v ) = ( x , g α β ( x ) v ) . {\displaystyle \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 GL ( r , K ) {\displaystyle \operatorname {GL} (r,\mathbb {K} )} , one obtains exactly the frame bundle of E {\displaystyle E} , a principal GL ( r , K ) {\displaystyle \operatorname {GL} (r,\mathbb {K} )} -bundle.
Associated bundles
Given a principal G {\displaystyle G} -bundle P {\displaystyle P} and a representation ρ {\displaystyle \rho } of G {\displaystyle G} on a vector space V {\displaystyle V} , one can construct an associated vector bundle E = P × ρ V {\displaystyle E=P\times _{\rho }V} with fibre the vector space V {\displaystyle V} . To define this vector bundle, one considers the right action on the product P × V {\displaystyle P\times V} defined by ( p , v ) g = ( p g , ρ ( g − 1 ) v ) {\displaystyle (p,v)g=(pg,\rho (g^{-1})v)} and defines P × ρ V = ( P × V ) / G {\displaystyle 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 {\displaystyle P} has transition functions g α β {\displaystyle g_{\alpha \beta }} with respect to a local trivialisation { U α } {\displaystyle \{U_{\alpha }\}} , then one constructs the associated vector bundle using the transition functions ρ ∘ g α β : U α ∩ U β → GL ( V ) {\displaystyle \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 {\displaystyle F} , not just a vector space, provided ρ : G → Aut ( F ) {\displaystyle \rho :G\to \operatorname {Aut} (F)} is a group homomorphism. One key example is the capital A adjoint bundle Ad ( P ) {\displaystyle \operatorname {Ad} (P)} with fibre G {\displaystyle G} , constructed using the group homomorphism ρ : G → Aut ( G ) {\displaystyle \rho :G\to \operatorname {Aut} (G)} defined by conjugation g ↦ ( h ↦ g h g − 1 ) {\displaystyle g\mapsto (h\mapsto ghg^{-1})} . Note that despite having fibre G {\displaystyle G} , the Adjoint bundle is neither a principal bundle, nor isomorphic as a fibre bundle to P {\displaystyle P} itself. For example, if G {\displaystyle G} is Abelian, then the conjugation action is trivial and Ad ( P ) {\displaystyle \operatorname {Ad} (P)} will be the trivial G {\displaystyle G} -fibre bundle over X {\displaystyle X} regardless of whether or not P {\displaystyle P} is trivial as a fibre bundle. Another key example is the lowercase a adjoint bundle ad ( P ) {\displaystyle \operatorname {ad} (P)} constructed using the adjoint representation ρ : G → Aut ( g ) {\displaystyle \rho :G\to \operatorname {Aut} ({\mathfrak {g}})} where g {\displaystyle {\mathfrak {g}}} is the Lie algebra of G {\displaystyle 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 φ : P → P {\displaystyle \varphi :P\to P} commuting with the projection operator π {\displaystyle \pi } and the right action ρ {\displaystyle \rho } . For a vector bundle a gauge transformation is similarly defined by a diffeomorphism φ : E → E {\displaystyle \varphi :E\to E} commuting with the projection operator π {\displaystyle \pi } which is a linear isomorphism of vector spaces on each fibre. The gauge transformations (of P {\displaystyle P} or E {\displaystyle E} ) form a group under composition, called the gauge group, typically denoted G {\displaystyle {\mathcal {G}}} . This group can be characterised as the space of global sections G = Γ ( Ad ( P ) ) {\displaystyle {\mathcal {G}}=\Gamma (\operatorname {Ad} (P))} of the adjoint bundle, or G = Γ ( Ad ( F ( E ) ) ) {\displaystyle {\mathcal {G}}=\Gamma (\operatorname {Ad} ({\mathcal {F}}(E)))} in the case of a vector bundle, where F ( E ) {\displaystyle {\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 α {\displaystyle U_{\alpha }} . This can be uniquely specified as a map g α : U α → G {\displaystyle g_{\alpha }:U_{\alpha }\to G} (taking G = GL ( r , K ) {\displaystyle G=\operatorname {GL} (r,\mathbb {K} )} in the case of vector bundles), where the induced bundle isomorphism is defined by
φ α ( p ) = p g α ( π ( p ) ) {\displaystyle \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 α {\displaystyle U_{\alpha }} , the transition function is precisely a local gauge transformation g α α : U α → G {\displaystyle 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 → P {\displaystyle 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 {\displaystyle 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 {\displaystyle P} is trivial over a set U α {\displaystyle 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 φ α ( s ( x ) ) = ( x , g ) {\displaystyle \varphi _{\alpha }(s(x))=(x,g)} for all x ∈ U α {\displaystyle x\in U_{\alpha }} and one g ∈ G {\displaystyle g\in G} . In particular a trivial principal bundle P = X × G {\displaystyle P=X\times G} comes equipped with a trivial connection. In general a connection is given by a choice of horizontal subspaces H p ⊂ T p P {\displaystyle H_{p}\subset T_{p}P} of the tangent spaces at every point p ∈ P {\displaystyle p\in P} , such that at every point one has T p P = H p ⊕ V p {\displaystyle T_{p}P=H_{p}\oplus V_{p}} where V {\displaystyle V} is the vertical bundle defined by V = ker d π {\displaystyle V=\ker d\pi } . These horizontal subspaces must be compatible with the principal bundle structure by requiring that the horizontal distribution H {\displaystyle H} is invariant under the right group action: H p g = d ( R g ) ( H p ) {\displaystyle H_{pg}=d(R_{g})(H_{p})} where R g : P → P {\displaystyle R_{g}:P\to P} denotes right multiplication by g {\displaystyle g} . A section s {\displaystyle s} is said to be horizontal if T p s ⊂ H p {\displaystyle T_{p}s\subset H_{p}} where s {\displaystyle s} is identified with its image inside P {\displaystyle P} , which is a submanifold of P {\displaystyle P} with tangent bundle T s {\displaystyle Ts} . Given a vector field v ∈ Γ ( T X ) {\displaystyle v\in \Gamma (TX)} , there is a unique horizontal lift v # ∈ Γ ( H ) {\displaystyle v^{\#}\in \Gamma (H)} . The curvature of the connection H {\displaystyle H} is given by the two-form with values in the adjoint bundle F ∈ Ω 2 ( X , ad ( P ) ) {\displaystyle F\in \Omega ^{2}(X,\operatorname {ad} (P))} defined by
F ( v 1 , v 2 ) = [ v 1 # , v 2 # ] − [ v 1 , v 2 ] # {\displaystyle F(v_{1},v_{2})=[v_{1}^{\#},v_{2}^{\#}]-[v_{1},v_{2}]^{\#}}
where [ ⋅ , ⋅ ] {\displaystyle [\cdot ,\cdot ]} is the Lie bracket of vector fields. Since the vertical bundle consists of the tangent spaces to the fibres of P {\displaystyle P} and these fibres are isomorphic to the Lie group G {\displaystyle G} whose tangent bundle is canonically identified with T G = G × g {\displaystyle TG=G\times {\mathfrak {g}}} , there is a unique Lie algebra-valued two-form F ∈ Ω 2 ( P , g ) {\displaystyle 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 {\displaystyle H} fails to embed inside P {\display