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} = ±
^{±}*F*_{ab}.

* __Alternative characterization__:
Self-dual Yang-Mills equations are equivalent to the consistency conditions for the system

(*D*_{y} −
*ξ** D*_{v}) *ψ* = 0 ,
(*D*_{u} −
*ξ D*_{z}) *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 S^{4} defined by the Hopf
fibration; The standard metric on S^{7} defines
a connection *A* on it; If *s* is the south pole of S^{4},
define the inverse stereographic projection *ψ*:
\(\mathbb R\)^{4} → S^{4} \ {*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; solutions
of 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.

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