Gauge theory (mathematics)
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 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 formula_1 where formula_2 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 formula_3. Such instantons are defined by a choice of 5 parameters, the center formula_4 and scale formula_5, corresponding to the formula_6-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 formula_7. 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 formula_8 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 formula_8 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 formula_10, or a principal formula_10-bundle, consists of a quintuple formula_12 where formula_13 is a smooth fibre bundle with fibre space isomorphic to a Lie group formula_10, and formula_15 represents a free and transitive right group action of formula_10 on formula_17 which preserves the fibres, in the sense that for all formula_18, formula_19 for all formula_20. Here formula_17  is the "total space", and formula_22 the "base space". Using the right group action for each formula_23 and any choice of formula_24, the map formula_25 defines a diffeomorphism formula_26 between the fibre over formula_27 and the Lie group formula_10 as smooth manifolds. Note however there is no natural way of equipping the fibres of formula_17 with the structure of Lie groups, as there is no natural choice of element formula_24 for every formula_23. The simplest examples of principal bundles are given when formula_32 is the circle group. In this case the principal bundle has dimension formula_33 where formula_34. Another natural  example occurs when formula_35 is the frame bundle of the tangent bundle of the manifold formula_22, or more generally the frame bundle of a vector bundle over formula_22. In this case the fibre of formula_17 is given by the general linear group formula_39. Since a principal bundle is a fibre bundle, it locally has the structure of a product. That is, there exists an open covering formula_40 of formula_22 and diffeomorphisms formula_42 commuting with the projections formula_43 and formula_44, such that the "transition functions" formula_45 defined by formula_46 satisfy the cocycle condition on any triple overlap formula_48. 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 formula_49 along the intersections formula_50 using the transition functions. The cocycle condition ensures precisely that this defines an equivalence relation on the disjoint union formula_51 and therefore that the quotient space formula_52 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" formula_53 satisfying formula_54 is an  equivalent method of specifying a local trivialisation map. Namely, one can define formula_55 where formula_56 is the unique group element such that formula_57. Vector bundles. A vector bundle is a triple formula_58 where formula_59 is a fibre bundle with fibre given by a vector space formula_60 where formula_61 is a field. The number formula_62 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 formula_40 is such a cover, then under the isomorphism one obtains formula_62 distinguished local sections of formula_66 corresponding to the formula_62  coordinate basis vectors formula_68 of formula_60, denoted formula_70. These are defined by the equation To specify a trivialisation it is therefore equivalent to give a collection of formula_62 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 formula_73 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 formula_75, one obtains exactly the frame  bundle of formula_66, a principal formula_75-bundle. Associated bundles. Given a principal formula_10-bundle formula_17 and a representation formula_15 of formula_10 on a vector space formula_82, one can construct an associated vector bundle formula_83 with fibre the vector space formula_82. To define this vector bundle, one considers the right action on the product formula_85 defined by formula_86 and defines formula_87 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 formula_17 has transition functions formula_89 with respect to a local trivialisation formula_40, then one  constructs the associated vector bundle using the transition functions formula_91. The associated bundle construction can be performed for any fibre space formula_92, not just a vector space, provided formula_93 is a group homomorphism. One key example is the "capital A adjoint bundle" formula_94 with fibre formula_10, constructed using the group homomorphism formula_96 defined by conjugation formula_97. Note that despite having fibre formula_10, the Adjoint bundle is neither a principal bundle, or isomorphic as a fibre bundle to formula_17 itself. For example, if formula_10 is Abelian, then the conjugation action is trivial and formula_94 will be the trivial formula_10-fibre bundle  over formula_22 regardless of whether or not formula_17 is trivial as a fibre bundle. Another key example is the "lowercase a adjoint bundle" formula_105 constructed using the adjoint representation formula_106 where formula_107 is the Lie algebra of formula_10. 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 formula_109 commuting with the projection operator formula_43 and the right action formula_15. For a vector bundle a gauge transformation is similarly defined by a diffeomorphism formula_112 commuting with the projection operator formula_43 which  is a linear isomorphism of vector spaces on each fibre. The gauge transformations (of formula_17 or formula_66) form a group under composition, called the gauge group, typically denoted formula_116. This group can be characterised as the space of global sections formula_117 of the adjoint bundle, or formula_118 in the case of a vector bundle, where formula_119 denotes the frame bundle. One can also define a local gauge transformation as a local bundle isomorphism over a trivialising open subset formula_120. This can be uniquely specified as a map formula_121 (taking formula_122 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 formula_120, the transition function is precisely a local gauge transformation formula_125. 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 formula_126 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 formula_10 itself, there is no canonical way of specifying which sections are constant. A choice of local trivialisation leads to one possible choice, where if formula_17 is trivial over a set formula_120, then a local section could be said to be horizontal if it is constant with respect to this trivialisation, in the sense that formula_130 for all formula_131 and one formula_20. In particular a trivial principal bundle formula_133 comes equipped with a trivial connection. In general a connection is given by a choice of horizontal subspaces formula_134 of the tangent spaces at every point formula_18, such  that at every point one has formula_136 where formula_82 is the vertical bundle defined by formula_138. These horizontal subspaces must be compatible with the principal bundle structure by requiring that the horizontal distribution formula_139 is invariant under the right group action: formula_140 where formula_141 denotes right multiplication by formula_142. A section formula_143 is said to be horizontal if formula_144 where formula_143 is identified with its image inside formula_17, which is a submanifold of formula_17 with tangent bundle formula_148. Given a vector field formula_149, there is a unique horizontal lift formula_150. The curvature of the connection formula_139 is given by the  two-form with values in the adjoint bundle formula_152 defined by where formula_154 is the Lie bracket of vector fields. Since the vertical bundle consists of the tangent spaces to the fibres of formula_17 and these fibres are isomorphic to the Lie group formula_10 whose tangent bundle is canonically identified with formula_157, there is a unique Lie algebra-valued two-form formula_158 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 formula_159 fails to embed inside  formula_17 as a horizontal submanifold locally. The choice of horizontal subspaces may be equivalently expressed by a projection operator formula_161 which is equivariant in the correct sense, called the connection one-form. For a horizontal distribution formula_139, this is defined by formula_163 where formula_164 denotes the decomposition of a tangent vector with respect to the direct sum decomposition formula_165. Due to the equivariance, this projection one-form may be taken to be Lie algebra-valued, giving some formula_166. A local trivialisation for formula_17 is equivalently given by a local section formula_168 and the connection one-form and curvature can be  pulled back along this smooth map. This gives the local connection one-form formula_169 which takes values in the adjoint bundle of formula_17. Cartan's structure equation says that the curvature may be expressed in terms of the local one-form formula_171 by the expression where we use the Lie bracket on the Lie algebra bundle formula_105 which is identified with formula_174 on the local trivialisation formula_120. Under a local gauge transformation formula_176 so that formula_177, the local connection one-form transforms by the expression where formula_179 denotes the Maurer–Cartan form of the Lie group formula_10. In the case  where formula_10 is a matrix Lie group, one has the simpler expression formula_182 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 formula_183-linear differential operator such that for all formula_186 and sections formula_187. The covariant derivative of a section formula_143 in the direction of a vector field formula_189 is defined by where on the right we use  the natural pairing between formula_191 and formula_192. This is a new section of the vector bundle formula_66, thought of as the derivative of formula_143 in the direction of formula_189. The operator formula_196 is the covariant derivative operator in the direction of formula_189. The curvature of formula_198 is given by the operator formula_199 with values in the endomorphism bundle, defined by In a local trivialisation the exterior derivative formula_201 acts as a trivial connection (corresponding in the principal bundle picture to the trivial connection discussed above). Namely for a local frame formula_202 one defines where here we have used Einstein notation  for a local section formula_204. Any two connections formula_205 differ by an formula_206-valued one-form formula_207. To see this, observe that the difference of two connections is formula_208-linear: 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 formula_210. This space is commonly denoted formula_211. Applying this observation locally, every connection over a trivialising subset formula_120 differs from the trivial connection formula_201 by some local connection one-form  formula_214, with the property that formula_215 on formula_120. In terms of this local connection form, the curvature may be written as 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 formula_218 where on the right we now perform wedge of one-forms and commutator of endomorphisms. Under a gauge transformation formula_219 of the vector bundle formula_66, a connection formula_198 transforms into a connection formula_222 by the conjugation formula_223. The difference formula_224 where here formula_198 is acting on the endomorphisms of  formula_66. Under a "local" gauge transformation formula_142 one obtains the same expression 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 formula_139 has local connection forms formula_230, and formula_231 is a representation of formula_10 defining an associated vector bundle formula_83, then the induced local connection one-forms are defined by Here formula_235 is the induced Lie algebra homomorphism from formula_236, and we use the fact that this  map induces a homomorphism of vector bundles formula_237. The induced curvature can be simply defined by 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 formula_107 is sent to the commutator of endomorphisms of formula_240 under the Lie algebra homomorphism formula_235. 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 formula_211 modelled on the vector space formula_243 (or formula_244 in the case  of vector bundles). Two connections formula_245 are said to be gauge equivalent if there exists a gauge transformation formula_219 such that formula_247. 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 formula_248, which is in general neither a Hausdorff space or a smooth manifold. Many interesting properties of the base manifold formula_22 can be encoded in the geometry and topology of moduli spaces of connections on principal bundles and vector bundles over formula_22. Invariants of formula_22, such as Donaldson invariants or Seiberg–Witten invariants  can be obtained by computing numeral quantities derived from moduli spaces of connections over formula_22. The most famous application of this idea is Donaldson's theorem, which uses the moduli space of Yang–Mills connections on a principal formula_253-bundle over a simply connected four-manifold formula_22 to study its intersection form. For this work Donaldson was awarded a Fields Medal. Notational conventions. There are various notational conventions used for connections on vector bundles and principal bundles which will be summarised here. Dictionary of mathematical and physical terminology. The mathematical and physical fields of gauge theory involve the study of  the same objects, but use different terminology to describe them. Below is a summary of how these terms relate to each other. As a demonstration of this dictionary, consider an interacting term of an electron-position particle field and the electromagnetic field in the Lagrangian of quantum electrodynamics: Mathematically this might be rewritten where formula_207 is a connection on a principal formula_311 bundle formula_17, formula_313 is a section of an associated spinor bundle and formula_314 is the induced Dirac operator of the induced covariant derivative formula_269 on this associated bundle. The first term is an interacting term in the Lagrangian  between the spinor field (the field representing the electron-positron) and the gauge field (representing the electromagnetic field). The second term is the regular Yang–Mills functional which describes the basic non-interacting properties of the electromagnetic field (the connection formula_207). The term of the form formula_317 is an example of what in physics is called minimal coupling, that is, the simplest possible interaction between a matter field formula_313 and a gauge field formula_207. Yang–Mills theory. The predominant theory that occurs in mathematical gauge theory is Yang–Mills theory. This theory involves the study of connections which are critical  points of the Yang–Mills functional defined by where formula_321 is an oriented Riemannian manifold with formula_322 the Riemannian volume form and formula_323 an formula_324-norm on the adjoint bundle formula_105. This functional is the square of the formula_324-norm of the curvature of the connection formula_207, so connections which are critical points of this function are those with curvature as small as possible (or higher local minima of formula_328). These critical points are characterised as solutions of the associated Euler–Lagrange equations, the Yang–Mills equations where formula_272 is the induced exterior covariant derivative of formula_269 on formula_105 and  formula_333 is the Hodge star operator. Such solutions are called Yang–Mills connections and are of significant geometric interest. The Bianchi identity asserts that for any connection, formula_334. By analogy for differential forms a harmonic form formula_261 is characterised by the condition If one defined a harmonic connection by the condition that the then study of Yang–Mills connections is similar in nature to that of harmonic forms. Hodge theory provides a unique harmonic representative of every de Rham cohomology class formula_338. Replacing a cohomology class by a gauge orbit formula_339, the study of Yang–Mills connections can be seen  as trying to find unique representatives for each orbit in the quotient space formula_248 of connections modulo gauge transformations. Self-duality and anti-self-duality equations. In dimension four the Hodge star operator sends two-forms to two-forms, formula_341, and squares to the identity operator, formula_342. Thus the Hodge star operating on two-forms has eigenvalues formula_343, and the two-forms on an oriented Riemannian four-manifold split as a direct sum into the self-dual and anti-self-dual two-forms, given by the formula_345 and formula_346 eigenspaces of the Hodge star operator respectively. That is, formula_347 is  self-dual if formula_348, and anti-self dual if formula_349, and every differential two-form admits a splitting formula_350 into self-dual and anti-self-dual parts. If the curvature of a connection formula_207 on a principal bundle over a four-manifold is self-dual or anti-self-dual then by the Bianchi identity formula_352, so the connection is automatically a Yang–Mills equations. The equation is a first order partial differential equation for the connection formula_207, and therefore is simpler to study than the full second order Yang–Mills equation. The equation formula_355 is called the self-duality equation , and the equation formula_356 is called the anti-self-duality equation, and solutions to these equations are self-dual connections or anti-self-dual connections respectively. Dimensional reduction. One way to derive new and interesting gauge-theoretic equations is to apply the process of "dimensional reduction" to the Yang–Mills equations. This process involves taking the Yang–Mills equations over a manifold formula_22 (usually taken to be the Euclidean space formula_358), and imposing that the solutions of the equations be invariant under a group of translational or other symmetries. Through this process the Yang–Mills equations lead to the Bogomolny  equations describing monopoles on formula_359, Hitchin's equations describing Higgs bundles on Riemann surfaces, and the Nahm equations on real intervals, by imposing symmetry under translations in one, two, and three directions respectively. Gauge theory in one and two dimensions. Here the Yang–Mills equations when the base manifold formula_22 is of low dimension is discussed. In this setting the equations simplify dramatically due to the fact that in dimension one there are no two-forms, and in dimension two the Hodge star operator on two-forms acts as formula_361. Yang–Mills theory. One may study the Yang–Mills equations  directly on a manifold of dimension two. The theory of Yang–Mills equations when the base manifold is a compact Riemann surface was carried about by Michael Atiyah and Raoul Bott. In this case the moduli space of Yang–Mills connections over a complex vector bundle formula_66 admits various rich interpretations, and the theory serves as the simplest case to understand the equations in higher dimensions. The Yang–Mills equations in this case become for some topological constant formula_364 depending on formula_66. Such connections are called "projectively flat", and in the case where the vector bundle is topologically trivial (so  formula_366) they are precisely the flat connections. When the rank and degree of the vector bundle are coprime, the moduli space formula_367 of Yang–Mills connections is smooth and has a natural structure of a symplectic manifold. Atiyah and Bott observed that since the Yang–Mills connections are projectively flat, their holonomy gives projective unitary representations of the fundamental group of the surface, so that this space has an equivalent description as a moduli space of projective unitary representations of the fundamental group of the Riemann surface, a character variety. The theorem of Narasimhan and Seshadri gives an alternative description  of this space of representations as the moduli space of stable holomorphic vector bundles which are smoothly isomorphic to the formula_66. Through this isomorphism the moduli space of Yang–Mills connections gains a complex structure, which interacts with the symplectic structure of Atiyah and Bott to make it a compact Kähler manifold. Simon Donaldson gave an alternative proof of the theorem of Narasimhan and Seshadri that directly passed from Yang–Mills connections to stable holomorphic structures. Atiyah and Bott used this rephrasing of the problem to illuminate the intimate relationship between the extremal Yang–Mills connections and the stability of  the vector bundles, as an infinite-dimensional moment map for the action of the gauge group formula_116, given by the curvature map formula_370 itself. This observation phrases the Narasimhan–Seshadri theorem as a kind of infinite-dimensional version of the Kempf–Ness theorem from geometric invariant theory, relating critical points of the norm squared of the moment map (in this case Yang–Mills connections) to stable points on the corresponding algebraic quotient (in this case stable holomorphic vector bundles). This idea has been subsequently very influential in gauge theory and complex geometry since its introduction. Nahm equations. The Nahm equations , introduced by Werner Nahm, are obtained as the dimensional reduction of the anti-self-duality in four dimensions to one dimension, by imposing translational invariance in three directions. Concretely, one requires that the connection form formula_371 does not depend on the coordinates formula_372. In this setting the Nahm equations between a system of equations on an interval formula_373 for four matrices formula_374 satisfying the triple of equations It was shown by Nahm that the solutions to these equations (which can be obtained fairly easily as they are a system of ordinary differential equations) can be used to construct solutions to  the Bogomolny equations, which describe monopoles on formula_359. Nigel Hitchin showed that solutions to the Bogomolny equations could be used to construct solutions to the Nahm equations, showing solutions to the two problems were equivalent. Donaldson further showed that solutions to the Nahm equations are equivalent to rational maps of degree formula_2 from the complex projective line formula_378 to itself, where formula_2 is the charge of the corresponding magnetic monopole. The moduli space of solutions to the Nahm equations has the structure of a hyperkähler manifold. Hitchin's equations and Higgs bundles. Hitchin's equations, introduced by Nigel Hitchin, are  obtained as the dimensional reduction of the self-duality equations in four dimensions to two dimensions by imposing translation invariance in two directions. In this setting the two extra connection form components formula_380 can be combined into a single complex-valued endomorphism formula_381, and when phrased in this way the equations become conformally invariant and therefore are natural to study on a compact Riemann surface rather than formula_382. Hitchin's equations state that for a pair formula_383 on a complex vector bundle formula_384 where formula_385, that where formula_387 is the formula_388-component of formula_272. Solutions of Hitchin's equations are  called Hitchin pairs. Whereas solutions to the Yang–Mills equations on a compact Riemann surface correspond to projective "unitary" representations of the surface group, Hitchin showed that solutions to Hitchin's equations correspond to projective "complex" representations of the surface group. The moduli space of Hitchin pairs naturally has (when the rank and degree of the bundle are coprime) the structure of a Kähler manifold. Through an analogue of Atiyah and Bott's observation about the Yang–Mills equations, Hitchin showed that Hitchin pairs correspond to so-called stable Higgs bundles, where a Higgs bundle is a pair formula_390 where  formula_384 is a holomorphic vector bundle and formula_392 is a holomorphic endomorphism of formula_66 with values in the canonical bundle of the Riemann surface formula_394. This is shown through an infinite-dimensional moment map construction, and this moduli space of Higgs bundles also has a complex structure, which is different to that coming from the Hitchin pairs, leading to two complex structures on the moduli space formula_367 of Higgs bundles. These combine to give a third making this moduli space a hyperkähler manifold. Hitchin's work was subsequently vastly generalised by Carlos Simpson, and the correspondence between solutions to Hitchin 's equations and Higgs bundles over an arbitrary Kähler manifold is known as the nonabelian Hodge theorem. Gauge theory in three dimensions. Monopoles. The dimensional reduction of the Yang–Mills equations to three dimensions by imposing translational invariance in one direction gives rise to the Bogomolny equations for a pair formula_396 where formula_397 is a family of matrices. The equations are When the principal bundle formula_399 has structure group formula_311 the circle group, solutions to the Bogomolny equations model the Dirac monopole describing a magnetic monopole in classical electromagnetism. The work of Nahm and Hitchin shows that when the structure  group is the special unitary group formula_253 solutions to the monopole equations correspond to solutions to the Nahm equations, and by work of Donaldson these further correspond to rational maps from formula_378 to itself of degree formula_2 where formula_2 is the charge of the monopole. This charge is defined as the limit of the integral of the pairing formula_406 over spheres formula_407 in formula_359 of increasing radius formula_284. Chern–Simons theory. Chern–Simons theory in 3 dimensions is a topological quantum field theory with an action functional proportional to the integral of the Chern–Simons form, a three-form defined  by Classical solutions to the Euler–Lagrange equations of the Chern–Simons functional on a closed 3-manifold formula_22 correspond to flat connections on the principal formula_10-bundle formula_413. However, when formula_22 has a boundary the situation becomes more complicated. Chern–Simons theory was used by Edward Witten to express the Jones polynomial, a knot invariant, in terms of the vacuum expectation value of a Wilson loop in formula_253 Chern–Simons theory on the three-sphere formula_416. This was a stark demonstration of the power of gauge theoretic problems to provide new insight in topology, and was one of the  first instances of a topological quantum field theory. In the quantization of the classical Chern–Simons theory, one studies the induced flat or projectively flat connections on the principal bundle restricted to surfaces formula_417 inside the 3-manifold. The classical state spaces corresponding to each surface are precisely the moduli spaces of Yang–Mills equations studied by Atiyah and Bott. The geometric quantization of these spaces was achieved by Nigel Hitchin and Axelrod–Della Pietra–Witten independently, and in the case where the structure group is complex, the configuration space is the moduli space of Higgs bundles and its quantization  was achieved by Witten. Floer homology. Andreas Floer introduced a type of homology on a 3-manifolds defined in analogy with Morse homology in finite dimensions. In this homology theory, the Morse function is the Chern–Simons functional on the space of connections on an formula_253 principal bundle over the 3-manifold formula_22. The critical points are the flat connections, and the flow lines are defined to be the Yang–Mills instantons on formula_420 that restrict to the critical flat connections on the two boundary components. This leads to instanton Floer homology. The Atiyah–Floer conjecture asserts that instanton Floer  homology agrees with the Lagrangian intersection Floer homology of the moduli space of flat connections on the surface formula_421 defining a Heegaard splitting of formula_22, which is symplectic due to the observations of Atiyah and Bott. In analogy with instanton Floer homology one may define Seiberg–Witten Floer homology where instantons are replaced with solutions of the Seiberg–Witten equations. By work of Clifford Taubes this is known to be isomorphic to embedded contact homology and subsequently Heegaard Floer homology. Gauge theory in four dimensions. Gauge theory has been most intensively studied in four dimensions. Here the mathematical study of  gauge theory overlaps significantly with its physical origins, as the standard model of particle physics can be thought of as a quantum field theory on a four-dimensional spacetime. The study of gauge theory problems in four dimensions naturally leads to the study of topological quantum field theory. Such theories are physical gauge theories that are insensitive to changes in the Riemannian metric of the underlying four-manifold, and therefore can be used to define topological (or smooth structure) invariants of the manifold. Anti-self-duality equations. In four dimensions the Yang–Mills equations admit a simplification to the first  order anti-self-duality equations formula_356 for a connection formula_207 on a principal bundle formula_413 over an oriented Riemannian four-manifold formula_22. These solutions to the Yang–Mills equations represent the absolute minima of the Yang–Mills functional, and the higher critical points correspond to the solutions formula_329 that do "not" arise from anti-self-dual connections. The moduli space of solutions to the anti-self-duality equations, formula_428, allows one to derive useful invariants about the underlying four-manifold. This theory is most effective in the case where formula_22 is simply connected. For example, in this case Donaldson's  theorem asserts that if the four-manifold has negative-definite intersection form (4-manifold), and if the principal bundle has structure group the special unitary group formula_253 and second Chern class formula_431, then the moduli space formula_428 is five-dimensional and gives a cobordism between formula_22 itself and a disjoint union of formula_434 copies of formula_435 with its orientation reversed. This implies that the intersection form of such a four-manifold is diagonalisable. There are examples of simply connected topological four-manifolds with non-diagonalisable intersection form, such as the E8 manifold, so Donaldson's theorem implies the existence of  topological four-manifolds with no smooth structure. This is in stark contrast with two or three dimensions, in which topological structures and smooth structures are equivalent: any topological manifold of dimension less than or equal to 3 has a unique smooth structure on it. Similar techniques were used by Clifford Taubes and Donaldson to show that Euclidean space formula_8 admits uncountably infinitely many distinct smooth structures. This is in stark contrast to any dimension other than four, where Euclidean space has a unique smooth structure. An extension of these ideas leads to Donaldson theory, which constructs further invariants of smooth  four-manifolds out of the moduli spaces of connections over them. These invariants are obtained by evaluating cohomology classes on the moduli space against a fundamental class, which exists due to analytical work showing the orientability and compactness of the moduli space by Karen Uhlenbeck, Taubes, and Donaldson. When the four-manifold is a Kähler manifold or algebraic surface and the principal bundle has vanishing first Chern class, the anti-self-duality equations are equivalent to the Hermitian Yang–Mills equations on the complex manifold formula_22. The Kobayashi–Hitchin correspondence proven for algebraic surfaces by Donaldson, and in general by  Uhlenbeck and Yau, asserts that solutions to the HYM equations correspond to stable holomorphic vector bundles. This work gave an alternate algebraic description of the moduli space and its compactification, because the moduli space of "semistable" holomorphic vector bundles over a complex manifold is a projective variety, and therefore compact. This indicates one way of compactifying the moduli space of connections is to add in connections corresponding to semi-stable vector bundles, so-called "almost Hermitian Yang–Mills connections". Seiberg–Witten equations. During their investigation of supersymmetry in four dimensions, Edward Witten and Nathan Seiberg uncovered a system of equations  now called the Seiberg–Witten equations, for a connection formula_207 and spinor field formula_313. In this case the four-manifold must admit a Spin structure, which defines a principal Spin bundle formula_17 with determinant line bundle formula_441, and an associated spinor bundle formula_442. The connection formula_207 is on formula_441, and the spinor field formula_445. The Seiberg–Witten equations are given by Solutions to the Seiberg–Witten equations are called monopoles. The moduli space of solutions to the Seiberg–Witten equations, formula_447 where formula_448 denotes the choice of Spin structure, is used to derive the Seiberg–Witten invariants. The Seiberg–Witten  equations have an advantage over the anti-self-duality equations, in that the equations themselves may be perturbed slightly to give the moduli space of solutions better properties. To do this, an arbitrary self-dual two-form is added on to the first equation. For generic choices of metric formula_142 on the underlying four-manifold, and choice of perturbing two-form, the moduli space of solutions is a compact smooth manifold. In good circumstances (when the manifold formula_22 is of "simple type"), this moduli space is zero-dimensional: a finite collection of points. The Seiberg–Witten invariant in this case  is simply the number of points in the moduli space. The Seiberg–Witten invariants can be used to prove many of the same results as Donaldson invariants, but often with easier proofs which apply in more generality. Gauge theory in higher dimensions. Hermitian Yang–Mills equations. A particular class of Yang–Mills connections are possible to study over Kähler manifolds or Hermitian manifolds. The Hermitian Yang–Mills equations generalise the anti-self-duality equations occurring in four-dimensional Yang–Mills theory to holomorphic vector bundles over Hermitian complex manifolds in any dimension. If formula_451 is a holomorphic vector bundle over  a compact Kähler manifold formula_452, and formula_207 is a Hermitian connection on formula_66 with respect to some Hermitian metric formula_455. The Hermitian Yang–Mills equations are where formula_364 is a topological constant depending on formula_66. These may be viewed either as an equation for the Hermitian connection formula_207 or for the corresponding Hermitian metric formula_455 with associated Chern connection formula_207. In four dimensions the HYM equations are equivalent to the ASD equations. In two dimensions the HYM equations correspond to the Yang–Mills equations considered by Atiyah and Bott. The Kobayashi–Hitchin correspondence asserts that solutions of the HYM equations  are in correspondence with polystable holomorphic vector bundles. In the case of compact Riemann surfaces this is the theorem of Narasimhan and Seshadri as proven by Donaldson. For algebraic surfaces it was proven by Donaldson, and in general it was proven by Karen Uhlenbeck and Shing-Tung Yau. This theorem is generalised in the nonabelian Hodge theorem by Simpson, and is in fact a special case of it where the Higgs field of a Higgs bundle formula_390 is set to zero. Exceptional holonomy instantons. The effectiveness of solutions of the Yang–Mills equations in defining invariants of four-manifolds has  led to interest that they may help distinguish between exceptional holonomy manifolds such as G2 manifolds in dimension 7 and Spin(7) manifolds in dimension 8, as well as related structures such as Calabi–Yau 6-manifolds and nearly Kähler manifolds. String theory. New gauge-theoretic problems arise out of superstring theory models. In such models the universe is 10 dimensional consisting of four dimensions of regular spacetime and a 6-dimensional Calabi–Yau manifold. In such theories the fields which act on strings live on bundles over these higher dimensional spaces, and one is interested in gauge-theoretic problems relating  to them. For example, the limit of the natural field theories in superstring theory as the string radius approaches zero (the so-called "large volume limit") on a Calabi–Yau 6-fold is given by Hermitian Yang–Mills equations on this manifold. Moving away from the large volume limit one obtains the deformed Hermitian Yang–Mills equation, which describes the equations of motion for a D-brane in the B-model of superstring theory. Mirror symmetry predicts that solutions to these equations should correspond to special Lagrangian submanifolds of the mirror dual Calabi–Yau.