Double-Scaling Limit in Principal Chiral Model: a New Non-Critical String?

We initiate a systematic, non-perturbative study of the large-$N$ expansion in the two-dimensional $SU(N)\times SU(N)$ Principal Chiral Model (PCM). Starting with the known infinite-$N$ solution for the ground state at fixed chemical potential \cite{Fateev:1994dp,Fateev:1994ai}, we devise an iterative procedure to solve the Bethe ansatz equations order by order in $1/N$. The first few orders, which we explicitly compute, reveal a systematic enhancement pattern at strong coupling calling for the near-threshold resummation of the large-$N$ expansion. The resulting double-scaling limit bears striking similarities to the $c=1$ non-critical string theory and suggests that the double-scaled PCM is dual to a non-critical string with a $2+1$-dimensional target space where an additional dimension emerges dynamically from the $SU(N)$ Dynkin diagram