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The name for an instanton gauge field configuration in SU(2)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 4\mathbb{R}^4gauge field configurations for gauge group the special unitary group SU(2)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)SU(2)-gauge field whose underlying SU(2)SU(2)-principal bundle has second Chern class=instanton number equal to ±1H 4(S 4,)\pm1 \in \mathbb{Z} \simeq H^4(S^4, \mathbb{Z}).

The physics literature typically focuses on describing this SU(2)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)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)SU(2) is diffeomorphic to S 3S^3, this allows to encode the classes of SU(2)SU(2)-principal bundles/SU(2)SU(2)-instantons on S 4S^4 in terms of homotopy classes of maps S 3S 3S^3 \to S^3, 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.

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