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Baryons in one theory are made of Nc distinct quarks; there are thus Nf-choose-Nc baryons. Baryons in the other theory are made of Nf-Nc distinct quarks. There are thus Nf-choose-(Nf-Nc) baryons. The numbers match. More precisely the map between baryons maps a set of Nc flavours to the complement
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I don't think it's clear what the statement, "The mesons and baryons are preserved by the duality," means. In particular, baryons in one theory are made of a different number of quarks than in the other theory. Does the statement just mean something like "both theories have mesons and baryons"?
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