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.Yang, C.N. and Mills, R.L., 1954. Conservation of isotopic spin and isotopic gauge invariance. Physical review, 96(1), p. 191.
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.Atiyah, M.F., Hitchin, N.J. and Singer, I.M., 1977. Deformations of instantons. Proceedings of the National Academy of Sciences, 74(7), pp. 2662–2663.Atiyah, M.F., Hitchin, N.J. and Singer, I.M., 1978. Self-duality in four-dimensional Riemannian geometry. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 362(1711), pp. 425–461. In this work the moduli space of self-dual connections (instantons) on Euclidean space was studied, and shown to be of dimension where is a positive integer parameter. This linked up with the discovery by physicists of BPST instantons, vacuum solutions to the Yang–Mills equations in four dimensions with . Such instantons are defined by a choice of 5 parameters, the center and scale , corresponding to the -dimensional moduli space. A BPST instanton is depicted to the right.
Around the same time Atiyah and Richard Ward discovered links between solutions to the self-duality equations and algebraic bundles over the complex projective space .Atiyah, M.F. and Ward, R.S., 1977. Instantons and algebraic geometry. Communications in Mathematical Physics, 55(2), pp. 117–124. Another significant early discovery was the development of the ADHM construction by Atiyah, Vladimir Drinfeld, Hitchin, and Yuri Manin.Atiyah, M.F., Hitchin, N.J., Drinfeld, V.G. and Manin, Y.I., 1978. Construction of instantons. Physics Letters A, 65(3), pp. 185–187. This construction allowed for the solution to the anti-self-duality equations on Euclidean space from purely linear algebraic data.
Significant breakthroughs encouraging the development of mathematical gauge theory occurred in the early 1980s. At this time the important work of Atiyah and Raoul Bott about the Yang–Mills equations over Riemann surfaces showed that gauge theoretic problems could give rise to interesting geometric structures, spurring the development of infinite-dimensional moment maps, equivariant Morse theory, and relations between gauge theory and algebraic geometry.Atiyah, M.F. and Bott, R., 1983. The yang-mills equations over riemann surfaces. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 308(1505), pp. 523–615. Important analytical tools in geometric analysis were developed at this time by Karen Uhlenbeck, who studied the analytical properties of connections and curvature proving important compactness results.Uhlenbeck, K.K., 1982. Connections with Lp bounds on curvature. Communications in Mathematical Physics, 83(1), pp. 31–42. 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.Donaldson, S.K., 1983. An application of gauge theory to four-dimensional topology. Journal of Differential Geometry, 18(2), pp. 279–315.Donaldson, S.K., 1990. Polynomial invariants for smooth four-manifolds. Topology, 29(3), pp. 257–315. With these invariants, novel results such as the existence of topological manifolds admitting no smooth structures, or the existence of many distinct smooth structures on the Euclidean space could be proved. For this work Donaldson was awarded the Fields Medal 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.Witten, E., 1989. Quantum field theory and the Jones polynomial. Communications in Mathematical Physics, 121(3), pp. 351–399. This work and the discovery of Donaldson invariants, as well as novel work of Andreas Floer on Floer homology, 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.Witten, Edward (1994), "Monopoles and four-manifolds.", Mathematical Research Letters, 1 (6): 769–796, arXiv:hep-th/9411102, Bibcode:1994MRLet...1..769W, doi:10.4310/MRL.1994.v1.n6.a13, MR 1306021, archived from the original on 2013-06-29Vafa, C. and Witten, E., 1994. A strong coupling test of S-duality. arXiv preprint hep-th/9408074. 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.Simon K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proceedings of the London Mathematical Society (3) 50 (1985), 1-26.Karen Uhlenbeck and Shing-Tung Yau, On the existence of Hermitian–Yang-Mills connections in stable vector bundles.Frontiers of the mathematical sciences: 1985 (New York, 1985). Communications on Pure and Applied Work of Nigel Hitchin and Carlos Simpson on Higgs 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.Hitchin, N.J., 1987. The self-duality equations on a Riemann surface. Proceedings of the London Mathematical Society, 3(1), pp. 59–126.Simpson, Carlos T. Higgs bundles and local systems. Publications Mathématiques de l'IHÉS, Volume 75 (1992), pp. 5–95. http://www.numdam.org/item/PMIHES_1992__75__5_0/ 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.Donaldson, S.K., Donaldson, S.K. and Kronheimer, P.B., 1990. The Geometry of Four-Manifolds. Oxford University Press.
Principal bundles
The central objects of study in gauge theory are principal bundles and vector bundles. The choice of which to study is essentially arbitrary, as one may pass between them, but principal bundles are the natural objects from the physical perspective to describe gauge 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 , or a principal -bundle, consists of a quintuple where is a smooth fibre bundle with fibre space isomorphic to a Lie group , and represents a free and transitive right group action of on which preserves the fibres, in the sense that for all , for all . Here is the total space, and the base space. Using the right group action for each and any choice of , the map defines a diffeomorphism between the fibre over and the Lie group as smooth manifolds. Note however there is no natural way of equipping the fibres of with the structure of Lie groups, as there is no natural choice of element for every .
The simplest examples of principal bundles are given when is the circle group. In this case the principal bundle has dimension where . Another natural example occurs when is the frame bundle of the tangent bundle of the manifold , or more generally the frame bundle of a vector bundle over . In this case the fibre of is given by the general linear group .
Since a principal bundle is a fibre bundle, it locally has the structure of a product. That is, there exists an open covering of and diffeomorphisms commuting with the projections and , such that the transition functions defined by satisfy the cocycle condition
on any triple overlap . In order to define a principal bundle it is enough to specify such a choice of transition functions, The bundle is then defined by gluing trivial bundles along the intersections using the transition functions. The cocycle condition ensures precisely that this defines an equivalence relation on the disjoint union and therefore that the quotient space 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 satisfying is an equivalent method of specifying a local trivialisation map. Namely, one can define where is the unique group element such that .
Vector bundles
A vector bundle is a triple where is a fibre bundle with fibre given by a vector space where is a field. The number is the rank of the vector bundle. Again one has a local description of a vector bundle in terms of a trivialising open cover. If is such a cover, then under the isomorphism
one obtains distinguished local sections of corresponding to the coordinate basis vectors of , denoted . These are defined by the equation