Instantons |

**In General** > s.a. gravitational
instantons [including supergravity].

@ __References__: Jentschura & Zinn-Justin JPA(01)mp [higher-order
corrections];
Paradis et al PRA(05)qp/04 [and
tunneling].

**In Gauge Theories** > s.a. QCD
phenomenology; quantum particles.

* __Idea__: Instantons are
(anti)self-dual solutions of the Yang-Mills field equations in Euclideanized
spacetime, satisfying certain boundary conditions that physically guarantee that they will have a
finite action, and they can formally be extended from \(\mathbb R\)^{4}
to S^{4}.

* __History__: Initially
called also pseudo-particles; They have become very useful in the theory of 4-manifolds.

* __Boundary conditions__:
We impose *A*_{a} \(\mapsto\)
*g*^{–1} ∂_{a}
*g* (pure gauge) and *F*_{ab} → 0 at infinity.

* __Field equations__: The
(anti)self-duality conditions guarantee that the field equations *D***F* =
0 are satisfied, since in this case they are equivalent to the Bianchi identities
*DF* = 0; Recall also that these equations depend only on the conformal structure
of the metric; But it can also be shown that (anti)self-dual solutions are global minima
of the action for a given topological sector, characterized by the integer
*k* = –C_{2}(*P*), *P* being
the bundle over S^{4} to which the gauge field belongs.

* __Reason for Euclideanization__:
The reason why Euclideanized spacetime is used (besides convenience for calculations)
is that in Lorentzian spacetime the (anti)self-duality condition (which would read
*F* = ± i **F*, instead of *F* = ± **F*) would
mean i\(\cal G\) = \(\cal G\), where \(\cal G\) is the Lie algebra of the gauge group,
and this requires that the Lie group be non-compact, contrary to what is usually assumed.

* __Applications__: They
are used in calculating tunneling probability amplitudes by the semiclassical (WKB) formula Γ ~
exp(–*I*/\(\hbar\)), with *I* the Euclidean action.

@ __In Yang-Mills theory__: Schwarz PLB(77),
Jackiw & Rebbi PLB(77)
[SU(2), degrees of freedom]; Ford et al NPB(01)ht/00 [ADHM
construction for SU(2) on torus]; Tyurin RMS(02)
[rev, mathematical]; Bonora et al JPA(03)ht/02 [2D,
stringy]; Forkel ht/04 [closed
FLRW models]; Colladay & McDonald
JMP(04)
[and Lorentz violation]; Vandoren & van Nieuwenhuizen a0802-ln
[in super-Yang-Mills theory]; Oh et al JHEP(11) [from gravitational instantons]; Cano et al JHEP(17)-a1704 [gravitating]; > s.a. self-dual
solutions.

@ __In QCD__: Shuryak NPB(82)
+ next two; Brown et al PRD(99)hp/98 [and
QCD vacuum]; Schäfer & Shuryak RMP(98);
't Hooft ht/99; > s.a. U(1)
problem.

@ __Relation with monopoles__: Garland & Murray CMP(89); Di Giacomo & Hasegawa PRD(15)-a1501 [and chiral symmetry breaking].

@ __Related topics__: Demetrian PS(06)
[Coleman-de Luccia, second-order].

**References** > s.a. 4D manifolds;
quaternions; riemannian geometry.

@ __Books and reviews__: Rajaraman 82; in Coleman 85;
Freed & Uhlenbeck 90;
Dorey et al PRP(02)ht [multi-instanton calculus];
Ritter mp/03;
Jardim m.DG/05-en
[topological aspects]; Weinberg 12.

@ __Original papers__: Belavin et al PLB(75);
Brown et al PRD(77).

@ __Mathematical__: Atiyah & Ward CMP(77);
Atiyah et al PRS(78).

@ __Instanton moduli space__: Blau et al ht/01 [information
metric and AdS-cft].

@ __As a tool in quantum field theory__:
Mueller-Kirsten et al JHEP(01) [and WKB].

@ __In curved spacetime__: Gibbons [Taub-NUT]; Kim & Kim NCB(99)
[in de Sitter]; Etesi & Hausel JGP(01)ht/00 [abelian,
euclidean Schwarzschild].

@ __Fuzzy / non-commutative__: Balachandran & Vaidya IJMPA(01)ht/99;
Schwarz CMP(01)ht; > s.a. non-commutative
fields; spacetime foam.

@ __Other types__: Landi & Marmo PLB(88) [algebraic instantons]; Baraglia & Hekmati AiM(16)-a1401 [contact instantons, moduli space].

@ __Related topics__: Lo et al PRD(79) [multi-instantons];
Shuryak NPB(88)
+ next three [ensemble]; Moch et al NPB(97)
[and deep inelastic scattering];
García Pérez et al PLB(00)
[size distribution]; Jardim JGP(04)
[Nahm transform]; Hanany & Kalveks JHEP(15)-a1509 [Hilbert series].

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