Instanton

Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory.

Yang–Mills instantons have been explicitly constructed in many cases by means of twistor theory, which relates them to algebraic vector bundles on algebraic surfaces, and via the ADHM construction, or hyperkähler reduction (see hyperkähler manifold), a geometric invariant theory procedure.

[6] An instanton can be used to calculate the transition probability for a quantum mechanical particle tunneling through a potential barrier.

The time independent Schrödinger equation for the particle reads If the potential were constant, the solution would be a plane wave, up to a proportionality factor, with This means that if the energy of the particle is smaller than the potential energy, one obtains an exponentially decreasing function.

In path integral formulation, the transition amplitude can be expressed as Following the process of Wick rotation (analytic continuation) to Euclidean spacetime (

) with the Minkowskian path integral corresponds to calculating the transition probability to tunnel through a classically allowed region (with potential −V(X)) in the Euclidean path integral (pictorially speaking – in the Euclidean picture – this transition corresponds to a particle rolling from one hill of a double-well potential standing on its head to the other hill).

In this example, the two "vacua" (i.e. ground states) of the double-well potential, turn into hills in the Euclideanized version of the problem.

Thus, the instanton field solution of the (Euclidean, i. e., with imaginary time) (1 + 1)-dimensional field theory – first quantized quantum mechanical description – allows to be interpreted as a tunneling effect between the two vacua (ground states – higher states require periodic instantons) of the physical (1-dimensional space + real time) Minkowskian system.

Note that a naïve perturbation theory around one of those two vacua alone (of the Minkowskian description) would never show this non-perturbative tunneling effect, dramatically changing the picture of the vacuum structure of this quantum mechanical system.

This may have important consequences, for example, in the theory of "axions" where the non-trivial QCD vacuum effects (like the instantons) spoil the Peccei–Quinn symmetry explicitly and transform massless Nambu–Goldstone bosons into massive pseudo-Nambu–Goldstone ones.

[10] The eigenvalues of these equations are known and permit in the case of instability the calculation of decay rates by evaluation of the path integral.

The progress of a chemical reaction can be described as the movement of a pseudoparticle on a high dimensional potential energy surface (PES).

is the canonical partition function, which is calculated by taking the trace of the Boltzmann operator in the position representation.

A well understood and illustrative example of an instanton and its interpretation can be found in the context of a QFT with a non-abelian gauge group,[note 2] a Yang–Mills theory.

An instanton is a field configuration fulfilling the classical equations of motion in Euclidean spacetime, which is interpreted as a tunneling effect between these different topological vacua.

He showed that zero modes of the Dirac equation in the instanton background lead to a non-perturbative multi-fermion interaction in the low energy effective action.

The classical Yang–Mills action on a principal bundle with structure group G, base M, connection A, and curvature (Yang–Mills field tensor) F is where

They are The first of these is an identity, because dF = d2A = 0, but the second is a second-order partial differential equation for the connection A, and if the Minkowski current vector does not vanish, the zero on the rhs.

Such solutions usually exist, although their precise character depends on the dimension and topology of the base space M, the principal bundle P, and the gauge group G. In nonabelian Yang–Mills theories,

The name instanton derives from the fact that these fields are localized in space and (Euclidean) time – in other words, at a specific instant.

Recent research on instantons links them to topics such as D-branes and Black holes and, of course, the vacuum structure of QCD.

In his 1977 paper Quark Confinement and Topology of Gauge Groups, Alexander Polyakov demonstrated that instanton effects in 3-dimensional QED coupled to a scalar field lead to a mass for the photon.

They are responsible for many nonperturbative effects in string theory, playing a central role in mirror symmetry.

Supersymmetric gauge theories often obey nonrenormalization theorems, which restrict the kinds of quantum corrections which are allowed.

Field theoretic techniques for instanton calculations in supersymmetric theories were extensively studied in the 1980s by multiple authors.

In 1984, Ian Affleck, Michael Dine and Nathan Seiberg calculated the instanton corrections to the superpotential in their paper Dynamical Supersymmetry Breaking in Supersymmetric QCD.

By then considering perturbations by various mass terms they were able to calculate the superpotential in the presence of arbitrary numbers of colors and flavors, valid even when the theory is no longer weakly coupled.

They extended their calculation to SU(2) gauge theories with fundamental matter in Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD.

In N = 4 supersymmetric gauge theories the instantons do not lead to quantum corrections for the metric on the moduli space of vacua.

[15][16] The ansatz gives explicit expressions for the gauge field and can be used to construct solutions with arbitrarily large instanton number.