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Baryons in quantum chromodynamics can be described as a bound state of three quarks. But they can also be described as a topological soliton of the pion field. How can both descriptions be equivalent?

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The skyrmion description only appears in an effective, non-renormalizable theory where pions are treated as elementary particles. Using a more high-energy viewpoint, however, pions themselves are made out of quarks and gluons, and the short-distance QCD rescription using quarks and gluons is renormalizable and much more accurate.

However, the pion effective field theory is still marginally applicable at the scale of the proton, so it must inevitably have a description for that. It is not surprising that it must be topologically nontrivial because baryons carry new kinds of charges, the baryon charge, that would vanish in topologically trivial configurations of pion fields.

There is no contradiction - the bound state of quarks and gluons that make up the proton is not "any bound state". It has some specific properties and quantum numbers, and those inevitably mean that the best approximation in an effective pion theory has to be a skyrmion. So this is not a real duality.

There are very similar descriptions of baryons (among many other objects) that are full-fledged dualities. In AdS/CFT, the baryons are branes wrapped on an internal sphere:

In AdS/CFT, the quark-gluon composition of the baryons really becomes invisible in the bulk, so the branes as baryons can't be reduced to quarks and gluons in any direct way. The branes are wrapped, much like the skyrmions are wrapped around some place of the configuration space, but I don't think that there is a direct isomorphism between these two wrappings.

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