Types of quantum field thories
The BPTS construction considers – on a 4-dimensional Minkowski spacetime Wick rotated to the Euclidean – gauge field configurations for gauge group the special unitary group that have vanishing field strength outside some finite radius. These are then equivalently configurations on the 4-sphere. The BPTS instanton is the -gauge field whose underlying -principal bundle has second Chern class=instanton number equal to .
The physics literature typically focuses on describing this -bundle in terms of the Cech cocycle which after covering the 4-sphere with two 4-balls (two “hemispheres”) is given by an -vaued transition function on the intersection of these two balls, which has the homotopy type of the 3-spehere. Since also the manifold underlying the special unitary Lie group is diffeomorphic to , this allows to encode the classes of -principal bundles/-instantons on in terms of homotopy classes of maps , and this is what much of the literature focuses on.
The original articles are
A. A. Belavin, A.M. Polyakov, A.S. Schwartz, Yu.S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1), 85-87 (1975) doi
A. A. Belavin, V.A. Fateev, A.S. Schwarz, Yu.S. Tyupkin, Quantum fluctuations of multi-instanton solutions, Phys. Lett. B 83 (3-4), 317-320 (1979) doi
For surveys and introductions see the references at Yang-Mills instanton.