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| Yang–Mills functional | |
|---|---|
| Name | Yang–Mills functional |
| Field | Mathematical physics, Differential geometry |
| Introduced | 1954 |
| Key people | Chen Ning Yang, Robert Mills, Michael Atiyah, Isadore Singer, Simon Donaldson, Karen Uhlenbeck, Edward Witten, Nathan Seiberg, Alexander Grothendieck |
Yang–Mills functional The Yang–Mills functional is an action functional in mathematical physics and differential geometry defined on connections of principal fiber bundles, central to gauge theory, topology, and quantum field theory. It provides a variational framework whose critical points produce the Yang–Mills equations, which underpin results in Donaldson theory, Seiberg–Witten theory, and conjectures in Millennium Prize Problems such as existence and mass gap. The functional links developments by Chen Ning Yang, Robert Mills, Michael Atiyah, Isadore Singer, and analytic advances by Karen Uhlenbeck and Richard Hamilton.
The Yang–Mills functional is defined for a connection on a principal Galois-related principal bundle with structure group Lie groups like SU(2), SU(3), U(1), SO(3), or E8 over a Riemannian manifold such as Euclidean space, Minkowski space, S^4, Torus, or Calabi–Yau manifold. Using local frames associated to Élie Cartan and curvature forms introduced by Élie Cartan and Élie Cartan's school, the functional integrates the norm squared of the curvature tensor analogous to energy functionals in Yang–Mills physics originally proposed by Chen Ning Yang and Robert Mills in 1954. Its formulation employs tools from Hodge theory, de Rham cohomology, Chern–Weil theory, and Atiyah–Bott's symplectic approach to moduli.
Critical points of the functional satisfy Euler–Lagrange equations known as the Yang–Mills equations, studied by James Simons, Michael Atiyah, Isadore Singer, and analysts like Karen Uhlenbeck, S. K. Donaldson, and Nicolaas Kuiper. Self-dual and anti-self-dual connections, called instantons in works by Alexander Belavin, Alexander Polyakov, and Gerard 't Hooft, solve first-order reductions of these equations on four-manifolds such as S^4 and CP^2. Existence and uniqueness results trace to techniques by Mikhail Gromov, Yakov Sinai, Richard Hamilton, and compactness arguments by Karen Uhlenbeck and Clifford Taubes.
Moduli spaces of Yang–Mills connections have been constructed and analyzed by Michael Atiyah, Simon Donaldson, Isadore Singer, Nicholas Katz, and Edward Witten, linking to Floer homology, Donaldson invariants, and Seiberg–Witten invariants. The study uses methods from Geometric Invariant Theory of David Mumford, spectral flow results by Daniel Quillen, and cobordism techniques related to John Milnor and Stephen Smale. Compactification and Uhlenbeck compactness involve insights by Karen Uhlenbeck, Clifford Taubes, Daniel Freed, and Karen Uhlenbeck's collaborators, with ties to Atiyah–Bott localization in the context of gauge group actions by Witten and Edward Frenkel.
Elliptic regularity for Yang–Mills connections invokes results from Sergei Sobolev, Laurent Schwartz, Agmon and estimates reminiscent of those in Elliptic operator theory by Atiyah, Singer, and Grigoriy Perelman's heat kernel methods. Removability of singularities and energy quantization were established by Karen Uhlenbeck, Tao, and T. C. Collins with input from Camillo De Lellis and László Székelyhidi. Parabolic Yang–Mills flow analysis parallels work on Ricci flow by Richard Hamilton and singularity analysis by Grigori Perelman and uses monotonicity formulas akin to those used by Charles Fefferman and Elias Stein.
Applications of the Yang–Mills functional permeate four-manifold topology through Donaldson theory, influencing classification results by Simon Donaldson, Michael Freedman, and implications for exotic smooth structures studied by Ciprian Manolescu. Connections to Mirror symmetry and String theory were developed by Edward Witten, Nathan Seiberg, Cumrun Vafa, and Shing-Tung Yau, while implications for index theory and Lefschetz fixed-point results reflect contributions by Atiyah and Isadore Singer. Relations to Chern–Simons theory explored by Edward Witten and knot invariants by Vladimir Drinfeld and Maxim Kontsevich connect to low-dimensional topology through William Thurston's geometric program.
Explicit instanton solutions on S^4 and R^4 were constructed by Alexander Belavin, I. M. Singer, and Nicolaas Kuiper; monopole solutions relate to work by Paul Dirac, Gerard 't Hooft, and Alexander Polyakov. Abelian Yang–Mills reductions recover classical electrodynamics from James Clerk Maxwell's equations in the U(1) case. Symmetric solutions exploiting Lie group symmetries were studied by Élie Cartan's theory, and calorons and periodic instantons were developed by Nick Manton and Pierre van Baal.
Generalizations include Yang–Mills–Higgs functionals studied by Peter Higgs, Gerard 't Hooft, and Steven Weinberg, dimensional reductions resulting in vortex equations analyzed by E. Witten and N. Seiberg, and higher-dimensional analogues explored by Shing-Tung Yau and Simon Donaldson. Analytical techniques harness Morse theory traditions from Marston Morse, infinite-dimensional Morse theory by Raoul Bott, and variational methods by John Nash and Richard Palais. Quantum gauge-theoretic approaches connect to Renormalization Group work by Kenneth Wilson and path integral formalism advanced by Richard Feynman.