nLab
BPTS-instanton

Idea

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

The BPTS construction considers – on a 4-dimensional Minkowski spacetime Wick rotated to the Euclidean ${ℝ}^4$ – gauge field configurations for gauge group the special unitary group $\mathrm{SU}(2)$ that have vanishing field strength outside some finite radius. These are then equivalently configurations on the 4-sphere. The BPTS instanton is the $\mathrm{SU}(2)$-gauge field whose underlying $\mathrm{SU}(2)$-principal bundle has second Chern class=instanton number equal to $\pm 1\in \mathbb{Z}\simeq H^4(S^4,\mathbb{Z})$.

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

Related concepts

• Yang-Mills instanton

  • instanton in QCD

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.