# Contents

## Idea

The name for an instanton gauge field configuration in $SU(2)$-Yang-Mills theory (describing the weak nuclear force).

The BPTS construction considers – on a 4-dimensional Minkowski spacetime Wick rotated to the Euclidean $\mathbb{R}^4$gauge field configurations for gauge group the special unitary group $SU(2)$ that have vanishing field strength outside some finite radius. These are then equivalently configurations on the 4-sphere. The BPTS instanton is the $SU(2)$-gauge field whose underlying $SU(2)$-principal bundle has second Chern class=instanton number equal to $\pm1 \in \mathbb{Z} \simeq H^4(S^4, \mathbb{Z})$.

The physics literature typically focuses on describing this $SU(2)$-bundle in terms of the Cech cocycle which after covering the 4-sphere with two 4-balls (two “hemispheres”) is given by an $SU(2)$-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 $SU(2)$ is diffeomorphic to $S^3$, this allows to encode the classes of $SU(2)$-principal bundles/$SU(2)$-instantons on $S^4$ in terms of homotopy classes of maps $S^3 \to S^3$, and this is what much of the literature focuses on.

## References

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.

Created on October 10, 2012 23:26:01 by Urs Schreiber (194.78.185.20)