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

(*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; 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 4

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send feedback and suggestions to bombelli at olemiss.edu – modified 31 mar 2019