Self-Dual
and Anti-Self-Dual Gauge Fields| 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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