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 S4.
* History: Initially called also pseudo-particles; They have become very useful in the theory of 4-manifolds.
* Boundary conditions: We impose Aa \(\mapsto\) g−1a g (pure gauge) and Fab → 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 = −C2(P), P being the bundle over S4 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, which requires that the Lie group be non-compact, contrary to what is usually assumed.
* Applications: They are used in calculating tunneling probability amplitudes, including false vacuum decay ones, 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.
@ Multi-instanton solutions: Lo et al PRD(79) [axially symmetric]; Nian & Qian a1912 [in curved spaces].
@ In QCD: Shuryak NPB(82) + next two; Brown et al PRD(99)hp/98 [and the QCD vacuum]; Schäfer & Shuryak RMP(98); 't Hooft ht/99; Pisarski & Rennecke PRD(20)-a1910 [role]; > 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].
@ In quantum field theory: Mueller-Kirsten et al JHEP(01) [and WKB]; Vaillant et al JChemP(19)-a1908 [instanton approximation for reaction rate, semiclassical].
@ 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: 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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