nLab
BPTS-instanton

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

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.

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)