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 Fab = +/–Fab.
* Alternative characterization: Self-dual Yang-Mills equations are equivalent to the consistency conditions for the system

(Dy Dv) = 0 ,   (Du Dz) = 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 [@ L Witten 79]; {& M Díaz, seminar}.
@ General references: Taubes JDG(82), JDG(84); Donaldson PLMS(85); Lerner CMP(90); Kalitzin & Sokatchev PLB(91); Selivanov ht/97-in [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-in; Ivanova JMP(98), JNMP(98) [algebra]; Mansfield & Wardlow a0903 [infinite-dimensional algebra].
@ Action: Berkovits & Hull JHEP(98) [covariant]; Nieto & Socorro PRD(99)ht/98 [and gravity, MacDowell-Mansouri formalism].
@ 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).
@ Maxwell theory in curved spacetime: Dotti & Kozameh JMP(96); Torres Del Castillo GRG(99) [Debye potentials].
@ Related topics: Movshev a0812.

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 : R4 → S4 \ {s}; Then * A defines the anti-self-dual SU(2) connection.
@ General references: Korepin & Oota JPA(96) [scattering of plane waves]; Castro & Plebanski 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; solutions of gauge theories [space of solutions, metric]; > s.a. 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 a0909 [in Euclidean Schwarzschild space].

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.


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