BPST instanton

In theoretical physics, the BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. It is a classical solution to the equations of motion of SU(2) Yang–Mills theory in Euclidean space-time (i.e. after Wick rotation), meaning it describes a transition between two different topological vacua of the theory. It was originally hoped to open the path to solving the problem of confinement, especially since Polyakov had proven in 1975 that instantons are the cause of confinement in three-dimensional compact-QED. This hope was not realized, however.

Description

The instanton

The BPST instanton is an essentially non-perturbative classical solution of the Yang–Mills field equations. It is found when minimizing the Yang–Mills SU(2) Lagrangian density:

L

=

−

1

4

F

μ

ν

a

F

μ

ν

a

{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }^{a}F_{\mu \nu }^{a}}

with Fμνa = ∂μAνa – ∂νAμa + gεabcAμbAνc the field strength. The instanton is a solution with finite action, so that Fμν must go to zero at space-time infinity, meaning that Aμ goes to a pure gauge configuration. Space-time infinity of our four-dimensional world is S3. The gauge group SU(2) has exactly the same structure, so the solutions with Aμ pure gauge at infinity are mappings from S3 onto itself.