Self-Dual and Anti-Self-Dual Gauge Fields  

In General > s.a. integrable systems; yang-mills gauge theory.
$ Def: Solutions of the Yang-Mills equations such that \(F_{ab} = \pm\, ^*F_{ab}\); > s.a. duality in field theories.
* Alternative characterization: Self-dual Yang-Mills equations are equivalent to the consistency conditions for the system

(Dyξ Dv) ψ = 0 ,    (Duξ Dz) ψ = 0 .

* Motivation: They minimize the action.
* Symmetry reductions: They give integrable systems [@ Ward PTRS(85)]; In the stationary axisymmetric case, one gets the Ernst equation [@ Witten PRD(79)]; {& M Díaz, seminar}.
@ General references: Taubes JDG(82), JDG(84); Donaldson PLMS(85); Lerner CMP(90); Kalitzin & Sokatchev PLB(91); Taubes 96 [anti-self-dual]; Selivanov ht/97-proc [perturbiner], PLB(98)ht/97 [coupled to gravity, perturbiner]; Popov RVMP(99)ht/98; Inami et al NPB(06)ht [Higgs phase, non-integrability]; Adam et al JHEP(08)-a0804 [conserved quantities].
@ Symmetries: Sorokin ht/97-talk; Ivanova JMP(98), JNMP(98) [algebra]; Mansfield & Wardlow JHEP(09)-a0903 [infinite-dimensional algebra].
@ Action, Lagrangian: Berkovits & Hull JHEP(98) [covariant]; Nieto & Socorro PRD(99)ht/98 [and gravity, MacDowell-Mansouri formalism]; Chen & Ho NPB(10)-a1001, Huang NPB(12) [various dimensionalities]; Bandos JHEP(14)-a1406 [chiral bosons in topologically non-trivial spacetimes].
@ Reductions: Sasa JPA(99); Ablowitz et al JMP(03) [and integrable systems].
@ Deformations: García-Compeán et al APPB(98)ht/97.
@ Electromagnetism: Hillion RPMP(09); Chubykalo et al AJP(10)aug [properties and applications].
@ Maxwell theory in curved spacetime: Dotti & Kozameh JMP(96); Torres Del Castillo GRG(99) [Debye potentials].
@ Related topics: Movshev a0812; Alexandru & Horváth PLB(12)-a1110 [dynamical tendency for self-duality]; > s.a. lattice gauge theory.

Solutions of Yang-Mills Equations > s.a. Bogomolny Equation.
* Construction: One can get an anti-self-dual connection on an SU(2) bundle as follows; Start with the SU(2)-bundle over S4 defined by the Hopf fibration; The standard metric on S7 defines a connection A on it; If s is the south pole of S4, define the inverse stereographic projection ψ: \(\mathbb R\)4 → S4 \ {s}; Then φ* A defines the anti-self-dual SU(2) connection.
@ General references: Korepin & Oota JPA(96) [scattering of plane waves]; Castro & Plebański JMP(99)ht/97 [SU(∞) Moyal anti-self-dual Yang-Mills]; Kamata & Nakamula PLB(99)ht; Khater & Sayed IJTP(02), et al IJTP(04) [SU(2) and SU(3)]; Khater et al IJTP(06) [and new representation]; > s.a. monopoles; solutionsof gauge theories [space of solutions, metric]; Prasad-Sommerfield Solution.
@ Instantons: Radu et al PRD(08)-a0707 [instantons in even dimensions]; > s.a. gravitational instantons.
@ Supersymmetric: Loginov JPA(04) [and supergravity, super-Yang-Mills].
@ Other types: Sushch a0909 [discrete equations on a double complex]; Mosna & Tavares PRD(09)-a0909 [in Euclidean Schwarzschild space]; Grant CMP(10) [reducible connections, and non-local symmetries].

Other Self-Dual Fields > s.a. non-commutative gravity; supergravity.
@ Half-flat gravity: García-Compeán et al GRG(05) [non-commutative, topological]; > s.a. self-dual solutions in general relativity.
@ Self-dual n-forms in general: Sen a1903 [in 4n+2 dimensions, action and Hamiltonian].

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