Abstract
We initiate a systematic, non-perturbative study of the large-N expansion in the two-dimensional SU(N)×SU(N) principal Chiral model (PCM). Starting with the known infinite-N solution for the ground state at fixed chemical potential, 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 noncritical string theory and suggests that the double-scaled PCM is dual to a noncritical string with a (2+1)-dimensional target space where an additional dimension emerges dynamically from the SU(N) Dynkin diagram.
Highlights
We initiate a systematic, non-perturbative study of the large-N expansion in the two-dimensional SUðNÞ × SUðNÞ principal Chiral model (PCM)
The resulting double-scaling limit bears striking similarities to the c 1⁄4 1 noncritical string theory and suggests that the double-scaled PCM is dual to a noncritical string with a (2 þ 1)-dimensional target space where an additional dimension emerges dynamically from the SUðNÞ Dynkin diagram
The linear integral equation of the Bethe ansatz [6] describing the finite-density state of PCM appears to be exactly solvable in the planar N → ∞ limit at any density, at least for a particular configuration of the chemical potentials arranged along the first Perron-Frobenius mode on the AN−1 Dynkin diagram [15,16]
Summary
Laboratoire de physique de l’École normale superieure, ENS, Universite PSL, CNRS, Sorbonne Universite, Universite Paris-Diderot, Sorbonne Paris Cite, 24 rue Lhomond, 75005 Paris, France and Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland. Like QCD, PCM is asymptotically free, generates a mass gap by dimensional transmutation, and features a nontrivial topological expansion in ’t Hooft’s large-N limit. The latter point suggests that in the strong-coupling regime, when planar diagrams become dense, the theory may have a dual string description. While consistent with the expected asymptotic freedom at large densities, the solution reveals a remarkable nonperturbative behavior at threshold, the smallest possible value of the chemical potential that leaves only a few excitations above the vacuum. As noticed already in [16], the log-behavior is reminiscent of the c 1⁄4 1 bosonic string theory in its dual matrix quantum-mechanical (MQM) formulation [17]
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