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What physical system can one use as an example while considering irreps of SU(N)? What is the correspondence between the system and the irreps?

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An irreducible representation of a compact Lie group such as $SU(N)$ may always be understood as being the quantization of a certain 1-dimensional topological gauge field theory. In that the representation space is the Hilbert space of states of the 1d TFT and the group action on that is the action of the quantum observables on the states.

In mathematics this statement is known as The orbit method. In physics this statement was famously highlighted in Edward's Witten article on Chern-Simons theory and the Jones polynomial (p. 22, 23).

Details on this story and pointers to references are on the nLab here: The orbit method. See in particular the section Wilson loops and 1d Chern-Simons σ-models with target the coadjoint orbit.

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Hi, Can you outline a path towards this subject? Where I can start learning the mathematical Prerequisites. I understand some elementary mathematics, differential forms, tangent vectors and lie algebras so on... Most of the Terms in NLab are completely unfamiliar to me. –  Prathyush Sep 14 '13 at 15:28
    
Let's see, so the orbit method as such is a classical topic in geometric representation theory, on which you could pick up any of the introductions listed here ncatlab.org/nlab/show/orbit+method#References . An introductory exposition of the point of view of higher differential geometry which you will see on the nLab in general and the orbit method in Chern-Simons theory in particular we wrote here: arxiv.org/abs/1301.2580 . A full set of lecture notes along these lines is developing here: ncatlab.org/nlab/show/geometry+of+physics . If you push me, I can add specifics. –  Urs Schreiber Sep 15 '13 at 11:50

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