group theory - Coadjoint orbits in physics

I am looking for some application of coadjoint orbits in physics. If you know some of them please let me know.

Answer

The Wilson loop observables inside 3d Chern-Simons gauge field theory are secretly themselves the quantization of a 1d field theory in terms of coadjoint orbits.

This possibly still surprising-sounding statement was hinted at already on p. 22 of the seminal

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  Edward Witten, Quantum Field Theory and the Jones Polynomial Commun. Math. Phys. 121 (3) (1989) 351–399. MR0990772 (project EUCLID)

  
  

  A detailed discussion of how this works is in section 4 of

  
  


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  Chris Beasley, Localization for Wilson Loops in Chern-Simons Theory, in J. Andersen, H. Boden, A. Hahn, and B. Himpel (eds.) Chern-Simons Gauge Theory: 20 Years After, , AMS/IP Studies in Adv. Math., Vol. 50, AMS, Providence, RI, 2011. (arXiv:0911.2687)

  
  
  

following

• S. Elitzur, Greg Moore, A. Schwimmer, and Nathan Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108–134.

The idea is indicated on the nLab here.

As also discussed there, the statement that there is a coadjoint orbit 1d quantum field theory sort of "inside" 3d Chern-Simons theory has a nice interpretation from a point of view of extended quantum field theory. This we have discussed in section 3.4.5 of

So given the ubiquity of Chern-Simons theory in QFT, and the fact that much of what is interesting about it is encoded in its Wilson loop observables, this means that quantization of coadjoint orbits plays a similarly important role. For instance given that all of rational 2d conformal field theory is dually encoded, via the FRS theorem, by 3d Chern-Simons theory in such a way that CFT field insertions are mapped to the CS Wilson loops, this means that quantized coadjoint orbits are at work behind the scenes in much of 2d CFT.