# About defining “baryons” and “mesons”

I want to understand the proof of the claims (of the construction as well as of its uniqueness) of gauge singlet states given around equation 2.13 (page 10) of this paper.

• Also does the listing of gauge singlet states there depend on the fact that these are superconformal primaries? (Are they claiming that any gauge singlet state is a primary?)

What exactly is the connection between the construction of the gauge singlets and them being superconformal primaries?

Let me repeat the claims here again,

• If you have $N_f$ fields in the fundamental representation of $U(N_c)$ then apparently these can't be combined (tensored?) into an $U(N_c)$ invariant (gauge singlet).

• But $N_f$ in the fundamental of $SU(N_c)$ can be combined into "baryons" - gauge singlets of $SU(N_c)$ as, $\epsilon_{i_1\dots i_{N_f-N_c}j_1\dots j_{N_c}}\epsilon^{a_1\dots a_{N_c}}$ $\prod_{k=1}^{N_c} \phi^{j_k}_{a_k}$

• If with the same $SU(N_c)$ the $N_f$ fields happen to be in the adjoint of $SU(N_c)$ then there exists forms invariant under $SU(N_c)$ given as $Tr[\prod_{k=1}^n \phi_{i_k}]$ (for any $n$ of these $N_f$ fields)

• If one has a pair of fields in the fundamental and the anti-fundamental of $SU(N_c)$ then the gauge invariant operators under $SU(N_c)$ are given as the "mesons" - $\phi^i_a \bar{\phi}^a_j$ (where $a$ is the $N_c$ index and $i,j$ is the $N_f$ index)

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I wonder if some Hilbert's invariant theory has gone into these claims. If yes, how? I guess somewhere it is being used that the gauge invariant states are finitely generated since these are invariant under the action of these reductive gauge groups.

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Hi user6818 - this is a lot of different questions. Could you edit your post to reduce it to ask only one thing, or a couple of closely related ones? You can always post the others separately. –  David Z Feb 1 at 5:44
I may be out of my depth here, but I think the requirement for field unitarity is related to the invariance of the scalar observables, i.e., the field energy density. Non-unitary transformations could break this conservation, and would lead to non-physical results. –  KDN Feb 13 at 12:54