Determine the individual baryon multiplicities in candidate large-N infrared phases

Determine the individual numbers of massless symmetric-type baryons \(\mathcal{B}_S\) and antisymmetric-type baryons \(\mathcal{B}_A\), denoted by \(\mathcal{N}_S\) and \(\mathcal{N}_A\), for the SU\((N)\) chiral gauge theories whose candidate infrared phase is obtained by the large-\(N\) symmetry-breaking pattern and whose total baryon multiplicity is fixed by matching the \(\mathrm{U}_\xi(1)^3\) anomaly.

Background

For the EKK SU$(N)$ chiral gauge theories with positive $N_2$, the proposed large-$N$ infrared pattern leaves an unbroken $\mathrm{U}_\xi(1)$ symmetry and produces two classes of candidate composite fermions: $\mathcal{B}_S$, built from the symmetric-representation fermions $\psi$, and $\mathcal{B}_A$, built from the antisymmetric-representation fermions $\lambda$. Matching the cubic and gravitational $\mathrm{U}_\xi(1)$ anomalies fixes only the total number $$\mathcal{N}_{\mathcal B}=\mathcal{N}_S+\mathcal{N}_A$$ of massless baryons.

The separate values of $\mathcal{N}_S$ and $\mathcal{N}_A$ depend on the pattern by which the flavor groups $\mathrm{SU}(N_1)$, $\mathrm{SU}(N_2)$, and $\mathrm{SU}(N_3)$ break. The paper introduces parameters $n_1,n_2,n_3$ to describe possible unbroken flavor subgroups and lists candidate solutions, but explicitly identifies the individual multiplicities as unknown before specifying a particular flavor-symmetry embedding and checking the remaining anomalies.