instantons

Instantons are nonperturbative fluctuations in quantum field theory, represented by finite-action solutions of the Euclidean equations of motion in nonabelian gauge theories. They describe tunneling between distinct vacuum states and contribute to the path integral beyond perturbation theory.

In four-dimensional Yang-Mills theory, instantons are localized, (anti)self-dual field configurations, with F-tilde_{μν} = ± F_{μν}. Their Euclidean action is S = 8π^2 |Q| / g^2, where the topological charge Q = (1/32π^2) ∫ d^4x F^a_{μν} F-tilde^{a μν} is an integer (Pontryagin index).

The canonical example is the BPST instanton in SU(2), a Q=1 solution whose moduli include position x0, size ρ, and gauge orientation. Each instanton carries eight bosonic zero modes in SU(2) and, for massless fermions, corresponding fermionic zero modes.

Physical consequences include the realization of the QCD vacuum structure through the theta parameter and nonperturbative interactions. Instantons generate an effective 't Hooft vertex, which can break axial U(1) symmetry and influence chiral dynamics.

At finite temperature, instantons generalize to calorons (periodic in Euclidean time); their density changes and is modeled in dilute-gas or instanton-liquid frameworks for phenomenology.

In mathematics, instantons are anti-self-dual connections on four-manifolds and are central to Donaldson theory, linking gauge theory with topology.